{"id":976,"date":"2018-12-11T13:22:54","date_gmt":"2018-12-11T18:22:54","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/decimals-2\/"},"modified":"2019-01-16T14:18:17","modified_gmt":"2019-01-16T19:18:17","slug":"decimals-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/decimals-2\/","title":{"raw":"Decimals","rendered":"Decimals"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li>Round decimals<\/li>\r\n \t<li>Add and subtract decimals<\/li>\r\n \t<li>Multiply and divide decimals<\/li>\r\n \t<li>Convert decimals, fractions, and percents<\/li>\r\n \t<li>Simplify expressions with square roots<\/li>\r\n \t<li>Identify integers, rational numbers, irrational numbers, and real numbers<\/li>\r\n \t<li>Locate fractions and decimals on the number line<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833056324\" class=\"be-prepared\">\r\n<p id=\"fs-id1167833047251\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p>\r\n\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Round Decimals<\/h3>\r\n<p id=\"fs-id1167836545783\">Decimals are another way of writing fractions whose denominators are powers of ten.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836608668\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\hfill 0.1&amp; =\\hfill &amp; \\frac{1}{10}\\hfill &amp; &amp; &amp; \\text{is \u201cone tenth\u201d}\\hfill \\\\ \\hfill 0.01&amp; =\\hfill &amp; \\frac{1}{100}\\hfill &amp; &amp; &amp; \\text{is \u201cone hundredth\u201d}\\hfill \\\\ \\hfill 0.001&amp; =\\hfill &amp; \\frac{1}{1000}\\hfill &amp; &amp; &amp; \\text{is \u201cone thousandth\u201d}\\hfill \\\\ \\hfill 0.0001&amp; =\\hfill &amp; \\frac{1}{10,000}\\hfill &amp; &amp; &amp; \\text{is \u201cone ten-thousandth\u201d}\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1167836334129\">Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. <a href=\"#CNX_IntAlg_Figure_01_04_023\" class=\"autogenerated-content\">(Figure)<\/a> shows the names of the place values to the left and right of the decimal point.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_023\"><span data-type=\"media\" id=\"fs-id1167829692275\" data-alt=\"This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_023-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.\" \/><\/span><\/div>\r\n<p id=\"fs-id1167836578683\">When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836543296\" class=\"howto\">\r\n<div data-type=\"title\">Round decimals.<\/div>\r\n<ol id=\"fs-id1167836323236\" class=\"stepwise\" type=\"1\">\r\n \t<li>Locate the given place value and mark it with an arrow.<\/li>\r\n \t<li>Underline the digit to the right of the place value.<\/li>\r\n \t<li>Is the underlined digit greater than or equal to 5?\r\n<ul data-bullet-style=\"bullet\">\r\n \t<li>Yes: add 1 to the digit in the given place value.<\/li>\r\n \t<li>No: do <u data-effect=\"underline\">not<\/u> change the digit in the given place value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Rewrite the number, deleting all digits to the right of the rounding digit.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836309901\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836525533\">\r\n<div data-type=\"problem\" id=\"fs-id1167836622858\">\r\n<p id=\"fs-id1167836699130\">Round 18.379 to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829697149\">\r\n<p id=\"fs-id1167836506417\">Round \\(18.379.\\)<\/p>\r\n<span class=\"token\">\u24d0<\/span> to the nearest hundredth\r\n<table class=\"unnumbered unstyled\" summary=\"Locate the hundredths place with an arrow. In the number 18.379, the arrow points to 7. Underline the digit to the right of the given place value. Underline 9. Because 9 is greater than or equal to 5, add 1 to the 7. Rewrite the number, deleting all digits to the right of the rounding digit. Now we have 18.38. Notice that the deleted digits were NOT replaced with zeros. So, 18.379 rounded to the nearest hundredth is 18.38.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Locate the hundredths place with an arrow.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550242\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the given\r\n<div data-type=\"newline\"><\/div>\r\nplace value.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289202\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Because 9 is greater than or equal to 5, add 1 to\r\n<div data-type=\"newline\"><\/div>\r\nthe 7.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836392196\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to the\r\n<div data-type=\"newline\"><\/div>\r\nright of the rounding digit.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515429\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Notice that the deleted digits were NOT\r\n<div data-type=\"newline\"><\/div>\r\nreplaced with zeros.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836706605\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836416714\"><span class=\"token\">\u24d1<\/span> to the nearest tenth<\/p>\r\n\r\n<table id=\"fs-id1167829691732\" class=\"unnumbered unstyled can-break\" summary=\"Locate the tenths place with an arrow. In the number 18.379, the arrow points to 3. Underline the digit to the right of the given place value. Underline 7. Because 7 is greater than or equal to 5, add 1 to the 3. Rewrite the number, deleting all digits to the right of the rounding digit. Now we have 18.4. Notice that the deleted digits were NOT replaced with zeros. So, 18.379 rounded to the nearest tenth is 18.4.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Locate the tenths place with an arrow.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836334185\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the\r\n<div data-type=\"newline\"><\/div>\r\ngiven place value.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833022051\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Because 7 is greater than or equal to 5,\r\n<div data-type=\"newline\"><\/div>\r\nadd 1 to the 3.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829598222\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to\r\n<div data-type=\"newline\"><\/div>\r\nthe right of the rounding digit.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Notice that the deleted digits were NOT\r\n<div data-type=\"newline\"><\/div>\r\nreplaced with zeros.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319802\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836620890\"><span class=\"token\">\u24d2<\/span> to the nearest whole number<\/p>\r\n\r\n<table id=\"fs-id1167836294764\" class=\"unnumbered unstyled\" summary=\"Locate the ones place with an arrow. In the number 18.379, the arrow points to 8. Underline the digit to the right of the given place value. Underline 3. Since 3 is not greater than or equal to 5, do not add 1 to 8. Rewrite the number, deleting all digits to the right of the rounding digit. So, 18.379 rounded to the nearest whole number is 18.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Locate the ones place with an arrow.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836376014\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the\r\n<div data-type=\"newline\"><\/div>\r\ngiven place value.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Since 3 is not greater than or equal to 5,\r\n<div data-type=\"newline\"><\/div>\r\ndo not add 1 to the 8.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836329514\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to\r\n<div data-type=\"newline\"><\/div>\r\nthe right of the rounding digit.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836296579\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836407551\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836414960\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836515968\">\r\n<div data-type=\"problem\" id=\"fs-id1167836530505\">\r\n<p id=\"fs-id1167836512053\">Round \\(6.582\\) to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836624764\">\r\n<p id=\"fs-id1167836526857\"><span class=\"token\">\u24d0<\/span>\\(6.58\\)<span class=\"token\">\u24d1<\/span>\\(6.6\\)<span class=\"token\">\u24d2<\/span> 7<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836415391\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829694242\">\r\n<div data-type=\"problem\" id=\"fs-id1167836554945\">\r\n<p id=\"fs-id1167836367818\">Round \\(15.2175\\) to the nearest <span class=\"token\">\u24d0<\/span> thousandth <span class=\"token\">\u24d1<\/span> hundredth <span class=\"token\">\u24d2<\/span> tenth.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836626479\">\r\n<p id=\"fs-id1167833023132\"><span class=\"token\">\u24d0<\/span>\\(15.218\\)<span class=\"token\">\u24d1<\/span>\\(15.22\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(15.2\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836392732\">\r\n<h3 data-type=\"title\">Add and Subtract Decimals<\/h3>\r\n<p id=\"fs-id1167836597180\">To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836334612\" class=\"howto\">\r\n<div data-type=\"title\">Add or subtract decimals.<\/div>\r\n<ol id=\"fs-id1167829590511\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the sum or difference.<\/li>\r\n \t<li>Write the numbers so the decimal points line up vertically.<\/li>\r\n \t<li>Use zeros as placeholders, as needed.<\/li>\r\n \t<li>Add or subtract the numbers as if they were whole numbers. Then place the\r\n<div data-type=\"newline\"><\/div>\r\ndecimal point in the answer under the decimal points in the given numbers.<\/li>\r\n \t<li>Write the sum or difference with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167833059005\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836537217\">\r\n<div data-type=\"problem\" id=\"fs-id1167836366635\">\r\n<p id=\"fs-id1167836605234\">Add or subtract: <span class=\"token\">\u24d0<\/span> \\(-23.5-41.38\\) <span class=\"token\">\u24d1<\/span> \\(14.65-20.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836556022\">\r\n<p id=\"fs-id1167836606910\"><span class=\"token\">\u24d0<\/span><\/p>\r\n<p>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; -23.5-41.38\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{The difference will be negative. To subtract, we add the}\\hfill \\\\ \\text{numerals. Write the numbers so the decimal points line}\\hfill \\\\ \\text{up vertically.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}23.5\\\\ \\underset{______}{+41.38}\\end{array}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Put 0 as a placeholder after the 5 in}\\phantom{\\rule{0.2em}{0ex}}23.5.\\hfill \\\\ \\text{Remember},\\frac{5}{10}=\\frac{50}{100}\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}0.5=0.50.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}23.50\\\\ \\underset{______}{+41.38}\\end{array}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Add the numbers as if they were whole numbers.}\\hfill \\\\ \\text{Then place the decimal point in the sum.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}23.50\\\\ \\underset{______}{+41.38}\\\\ 64.88\\end{array}\\hfill \\\\ \\\\ \\\\ \\text{Write the result with the correct sign.}\\hfill &amp; &amp; &amp; -23.5-41.38=-64.88\\hfill \\end{array}\\)\r\n<p id=\"fs-id1167836312509\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; &amp; &amp; 14.65-20\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{The difference will be negative. To subtract, we}\\hfill \\\\ \\text{subtract 14.65 from 20.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Write the numbers so the decimal points line up}\\hfill \\\\ \\text{vertically.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.7em}{0ex}}20\\hfill \\\\ \\underset{______}{-14.65}\\end{array}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Remember, 20 is a whole number, so place the}\\hfill \\\\ \\text{decimal point after the 0.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{Put in zeros to the right as placeholders.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.5em}{0ex}}20.00\\\\ \\underset{______}{-14.65}\\end{array}\\hfill \\\\ \\\\ \\\\ \\\\ \\text{Subtract and place the decimal point in the answer.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\underset{______________}{\\begin{array}{ccccc}&amp; \\hfill 9&amp; &amp; \\hfill 9&amp; \\\\ \\hfill 1&amp; \\hfill \\overline{)10}&amp; &amp; \\hfill \\overline{)10}&amp; \\hfill 10\\\\ \\hfill 2&amp; \\hfill 0&amp; .\\hfill &amp; \\hfill 0&amp; \\hfill 0\\\\ \\hfill -1&amp; \\hfill 4&amp; .\\hfill &amp; \\hfill 6&amp; \\hfill 5\\end{array}}\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\begin{array}{ccccc}&amp; \\hfill 5&amp; .\\hfill &amp; \\hfill \\phantom{\\rule{0.7em}{0ex}}3&amp; \\hfill \\phantom{\\rule{0.5em}{0ex}}5\\end{array}\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\\\ \\text{Write the result with the correct sign.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; 14.65-20=-5.35\\hfill \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833057966\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836547290\">\r\n<div data-type=\"problem\" id=\"fs-id1167836620098\">\r\n<p id=\"fs-id1167836484787\">Add or subtract: <span class=\"token\">\u24d0<\/span> \\(-4.8-11.69\\) <span class=\"token\">\u24d1<\/span>\\(9.58-10.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836622008\">\r\n<p id=\"fs-id1167836283439\"><span class=\"token\">\u24d0<\/span>\\(-16.49\\)<span class=\"token\">\u24d1<\/span>\\(-0.42\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836627381\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836558181\">\r\n<div data-type=\"problem\" id=\"fs-id1167836391088\">\r\n<p id=\"fs-id1167833018605\">Add or subtract: <span class=\"token\">\u24d0<\/span> \\(-5.123-18.47\\) <span class=\"token\">\u24d1<\/span>\\(37.42-50.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833051205\">\r\n<p id=\"fs-id1167836597977\"><span class=\"token\">\u24d0<\/span>\\(-23.593\\)<span class=\"token\">\u24d1<\/span>\\(-12.58\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829691951\">\r\n<h3 data-type=\"title\">Multiply and Divide Decimals<\/h3>\r\n<p id=\"fs-id1167836580114\">When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167833019249\" class=\"howto\">\r\n<div data-type=\"title\">Multiply decimals.<\/div>\r\n<ol id=\"fs-id1167836688746\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the product.<\/li>\r\n \t<li>Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\r\n \t<li>Place the decimal point. The number of decimal places in the product is the sum of\r\n<div data-type=\"newline\"><\/div>\r\nthe number of decimal places in the factors.<\/li>\r\n \t<li>Write the product with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836524294\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836367150\">\r\n<div data-type=\"problem\" id=\"fs-id1167836484476\">\r\n<p id=\"fs-id1167836598179\">Multiply: \\(\\left(-3.9\\right)\\left(4.075\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829718181\">\r\n<table id=\"fs-id1167836333804\" class=\"unnumbered unstyled\" summary=\"Multiply minus 3.9 with 4.075. The signs are different. The product will be negative. Write in vertical format, lining up the numbers on the right. Multiply to get 158925. Add the number of decimal places in the factors, 1 plus 3 equals 4. Place the decimal point 4 places from the right. The number becomes 15.8925. The signs of the factors are different, so the product is negative. The answer is minus 15.8925.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(-3.9\\right)\\left(4.075\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The signs are different. The product\r\n<div data-type=\"newline\"><\/div>\r\nwill be negative.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The product will be negative.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Write in vertical format, lining up the\r\n<div data-type=\"newline\"><\/div>\r\nnumbers on the right.\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507262\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836520918\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Add the number of decimal places in\r\n<div data-type=\"newline\"><\/div>\r\nthe factors (1 + 3).\r\n<div data-type=\"newline\"><\/div>\r\nPlace the decimal point 4 places from the right.<\/td>\r\n<td rowspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836533297\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507277\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The signs are the different, so the product is negative.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(-3.9\\right)\\left(4.075\\right)=-15.8925\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836406682\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836526133\">\r\n<div data-type=\"problem\" id=\"fs-id1167836521632\">\r\n<p id=\"fs-id1167829645141\">Multiply: \\(-4.5\\left(6.107\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836511260\">\r\n<p id=\"fs-id1167836507336\">\\(-27.4815\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836556209\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829716991\">\r\n<div data-type=\"problem\" id=\"fs-id1167836320388\">\r\n<p id=\"fs-id1167836704688\">Multiply: \\(-10.79\\left(8.12\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836340427\">\r\n<p id=\"fs-id1167836693316\">\\(-87.6148\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836359860\">Often, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836628738\" class=\"howto\">\r\n<div data-type=\"title\">Multiply a decimal by a power of ten.<\/div>\r\n<ol id=\"fs-id1167832940646\" class=\"stepwise\" type=\"1\">\r\n \t<li>Move the decimal point to the right the same number of places as the\r\n<div data-type=\"newline\"><\/div>\r\nnumber of zeros in the power of 10.<\/li>\r\n \t<li>Add zeros at the end of the number as needed.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836408406\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833058013\">\r\n<div data-type=\"problem\" id=\"fs-id1167836321760\">\r\n<p id=\"fs-id1167833019684\">Multiply: 5.63 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836532215\">\r\n<p id=\"fs-id1167836688719\">By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.<\/p>\r\n<p id=\"fs-id1167836546356\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<table id=\"fs-id1167829721047\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 10. There is 1 zero in 10. So move the decimal point 1 place to the right, to get 56.3.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829693057\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">There is 1 zero in 10, so move the decimal point 1 place to the right.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513010\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025149\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836399856\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<table id=\"fs-id1167836525944\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 100. There are 2 zeroes in 100. So move the decimal point 2 places to the right, to get 563.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509774\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">There are 2 zeroes in 100, so move the decimal point 2 places to the right.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693882\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524191\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836309296\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<table id=\"fs-id1167836408573\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 1000. There are 3 zeroes in 100. So move the decimal point 2 places to the right. A zero must be added to the end to get 5630.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531255\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">There are 3 zeroes in 1,000, so move the decimal point 3 place to the right.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836522133\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">A zero must be added to the end.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833060124\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829597736\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836547160\">\r\n<div data-type=\"problem\" id=\"fs-id1167836547163\">\r\n<p id=\"fs-id1167836313834\">Multiply 2.58 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836510292\">\r\n<p id=\"fs-id1167833056159\"><span class=\"token\">\u24d0<\/span> 25.8 <span class=\"token\">\u24d1<\/span> 258 <span class=\"token\">\u24d2<\/span> 2,580<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836545249\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836522719\">\r\n<div data-type=\"problem\" id=\"fs-id1167836626486\">\r\n<p id=\"fs-id1167836626488\">Multiply 14.2 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836518934\">\r\n<p id=\"fs-id1167836518936\"><span class=\"token\">\u24d0<\/span> 142 <span class=\"token\">\u24d1<\/span> 1,420 <span class=\"token\">\u24d2<\/span> 14,200<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836546040\">Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.<\/p>\r\n<p id=\"fs-id1167836518923\">We review the notation and vocabulary for division:<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836292139\" data-alt=\"In the expression a divided by b equals c, a is the dividend, b is the divisor and c is the quotient. This can be written as b right parentheses a overbar, with c on top of the bar. In this case too, a is the dividend, b is the divisor and c is the quotient.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_024_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the expression a divided by b equals c, a is the dividend, b is the divisor and c is the quotient. This can be written as b right parentheses a overbar, with c on top of the bar. In this case too, a is the dividend, b is the divisor and c is the quotient.\" \/><\/span>\r\n<p id=\"fs-id1167836288209\">We\u2019ll write the steps to take when dividing decimals for easy reference.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836521247\" class=\"howto\">\r\n<div data-type=\"title\">Divide decimals.<\/div>\r\n<ol id=\"fs-id1167836389529\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the quotient.<\/li>\r\n \t<li>Make the divisor a whole number by \u201cmoving\u201d the decimal point all the way to the right. \u201cMove\u201d the decimal point in the dividend the same number of places\u2014adding zeros as needed.<\/li>\r\n \t<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\r\n \t<li>Write the quotient with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836544303\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836504089\">\r\n<div data-type=\"problem\" id=\"fs-id1167836492443\">\r\n<p id=\"fs-id1167836492445\">Divide: \\(-25.65\u00f7\\left(-0.06\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836417299\">\r\n<p id=\"fs-id1167836521102\">Remember, you can \u201cmove\u201d the decimals in the divisor and dividend because of the Equivalent Fractions Property.<\/p>\r\n\r\n<table id=\"fs-id1167829717572\" class=\"unnumbered unstyled\" summary=\"Minus 25.65 divided by minus 0.06. The signs are the same, so the quotient is positive. Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal point in the dividend the same number of places. Hence the divisor is 6 and the dividend is 2565. Divide, placing the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. The quotient is 427.5.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836551421\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">The signs are the same.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The quotient is positive.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Make the divisor a whole number by \u201cmoving\u201d the\r\n<div data-type=\"newline\"><\/div>\r\ndecimal point all the way to the right.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">\u201cMove\u201d the decimal point in the dividend the same\r\n<div data-type=\"newline\"><\/div>\r\nnumber of places.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Divide.\r\n<div data-type=\"newline\"><\/div>\r\nPlace the decimal point in the quotient above the\r\n<div data-type=\"newline\"><\/div>\r\ndecimal point in the dividend.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829719619\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Write the quotient with the appropriate sign.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836694132\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836628735\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836447446\">\r\n<div data-type=\"problem\" id=\"fs-id1167836692990\">\r\n<p id=\"fs-id1167836692992\">Divide: \\(-23.492\u00f7\\left(-0.04\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836418428\">\r\n<p id=\"fs-id1167836334903\">\\(587.3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836635212\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836585179\">\r\n<div data-type=\"problem\" id=\"fs-id1167836621352\">\r\n<p id=\"fs-id1167836621355\">Divide: \\(-4.11\u00f7\\left(-0.12\\right).\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836407678\">\r\n<p id=\"fs-id1167836800480\">\\(34.25\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836518782\">\r\n<h3 data-type=\"title\">Convert Decimals, Fractions, and Percents<\/h3>\r\n<p id=\"fs-id1167833397224\">In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.<\/p>\r\n<p id=\"fs-id1167829692288\">We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal \\(0.03.\\) the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836509769\" class=\"unnumbered\" data-label=\"\">\\(0.03=\\frac{3}{100}\\)<\/div>\r\n<p id=\"fs-id1167836700940\">The steps to take to convert a decimal to a fraction are summarized in the procedure box.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836700546\" class=\"howto\">\r\n<div data-type=\"title\">Convert a decimal to a proper fraction and a fraction to a decimal.<\/div>\r\n<ol id=\"fs-id1167836509806\" class=\"stepwise\" type=\"1\">\r\n \t<li>To convert a decimal to a proper fraction, determine the place value of the final digit.<\/li>\r\n \t<li>Write the fraction.\r\n<ul id=\"fs-id1167829905871\" data-bullet-style=\"bullet\">\r\n \t<li>numerator\u2014the \u201cnumbers\u201d to the right of the decimal point<\/li>\r\n \t<li>denominator\u2014the place value corresponding to the final digit<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167836533353\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836533355\">\r\n<div data-type=\"problem\" id=\"fs-id1167836684088\">\r\n<p id=\"fs-id1167836684090\">Write: <span class=\"token\">\u24d0<\/span> \\(0.374\\) as a fraction <span class=\"token\">\u24d1<\/span> \\(-\\frac{5}{8}\\) as a decimal.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836700931\">\r\n<p id=\"fs-id1167836628604\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<table id=\"fs-id1167836698909\" class=\"unnumbered unstyled\" summary=\"In the number 0.374, determine the place value of the final digit. The place value of 3 is tenths, that of 7 is hundredths and that of 4 is thousandths. Write the fraction for 0.374. The numerator is 374. The denominator is 1000. Simplify the fraction. 374 is twice 187 and 1000 is twice 500. Dividing out the common factors, we get 187 by 500. So, 0.374 is 187 by 500.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635361\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Determine the place value of the final digit.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836579270\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Write the fraction for 0.374:\r\n<div data-type=\"newline\"><\/div>\r\nThe numerator is 374.\r\n<div data-type=\"newline\"><\/div>\r\nThe denominator is 1,000.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999068\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509683\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Divide out the common factors.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836704756\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836456246\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167829596870\"><span class=\"token\">\u24d1<\/span> Since a fraction bar means division, we begin by writing the fraction \\(\\frac{5}{8}\\) as \\(8\\overline{)5.}\\) Now divide.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836619319\" data-alt=\"The division shows that 5 is divided by 8 to yield 0.625. The result concludes that five eights is equal to negative 0.625.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_010_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The division shows that 5 is divided by 8 to yield 0.625. The result concludes that five eights is equal to negative 0.625.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836699375\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829692448\">\r\n<div data-type=\"problem\" id=\"fs-id1167836704215\">\r\n<p id=\"fs-id1167836704217\">Write: <span class=\"token\">\u24d0<\/span> \\(0.234\\) as a fraction <span class=\"token\">\u24d1<\/span> \\(-\\frac{7}{8}\\) as a decimal.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833058998\">\r\n<p id=\"fs-id1167833059000\"><span class=\"token\">\u24d0<\/span>\\(\\frac{117}{500}\\)<span class=\"token\">\u24d1<\/span>\\(-0.875\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836697042\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836627841\">\r\n<div data-type=\"problem\" id=\"fs-id1167836627843\">\r\n<p id=\"fs-id1167836627845\">Write: <span class=\"token\">\u24d0<\/span> \\(0.024\\) as a fraction <span class=\"token\">\u24d1<\/span> \\(-\\frac{3}{8}\\) as a decimal.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836620435\">\r\n<p id=\"fs-id1167836620437\"><span class=\"token\">\u24d0<\/span>\\(\\frac{3}{125}\\)<span class=\"token\">\u24d1<\/span>\\(-0.375\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829593967\">A <span data-type=\"term\">percent<\/span> is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.<\/p>\r\n<p id=\"fs-id1167836652793\"><em data-effect=\"italics\">To convert a percent number to a decimal number, we move the decimal point two places to the left.<\/em><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836305345\" data-alt=\"Figure shows the value 6 percent. An arrow indicates that the decimal is moved two places to the left. Hence the value is equal to 0.06. Similarly, 78 percent is 0.78, 2.7 percent is 0. 027 and 135 percent is 1.35.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_011_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the value 6 percent. An arrow indicates that the decimal is moved two places to the left. Hence the value is equal to 0.06. Similarly, 78 percent is 0.78, 2.7 percent is 0. 027 and 135 percent is 1.35.\" \/><\/span>\r\n<p id=\"fs-id1167836532690\">To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.<\/p>\r\n<p id=\"fs-id1167836630567\"><em data-effect=\"italics\">To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.<\/em><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836493675\" data-alt=\"Figure shows value 0.05. An arrow indicates that the decimal is moved two places to the right. Hence the value becomes 5 percent. Similarly, 0.83 is 83 percent, 1.05 is 105 percent, 0.075 is 7.5 percent and 0.3 is 30 percent.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_012_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows value 0.05. An arrow indicates that the decimal is moved two places to the right. Hence the value becomes 5 percent. Similarly, 0.83 is 83 percent, 1.05 is 105 percent, 0.075 is 7.5 percent and 0.3 is 30 percent.\" \/><\/span>\r\n<div data-type=\"note\" id=\"fs-id1167836551698\" class=\"howto\">\r\n<div data-type=\"title\">Convert a percent to a decimal and a decimal to a percent.<\/div>\r\n<ol id=\"fs-id1167829683890\" class=\"stepwise\" type=\"1\">\r\n \t<li>To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.<\/li>\r\n \t<li>To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1167829721053\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829721055\">\r\n<div data-type=\"problem\" id=\"fs-id1167829721057\">\r\n<p id=\"fs-id1167836549775\">Convert each:<\/p>\r\n<p id=\"fs-id1167836549778\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 62%, 135%, and 13.7%.<\/p>\r\n<p id=\"fs-id1167833055334\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.51, 1.25, and 0.093.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833224661\">\r\n<p id=\"fs-id1167833224663\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<table id=\"fs-id1167836699430\" class=\"unnumbered unstyled\" summary=\"The given values are 62 percent, 135 percent and 35.7 percent. Moving the decimal point two places to the left in each case, we get 0.62, 1.35 and 0.357.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Move the decimal point two places to the left.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833057802\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167829718691\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<table id=\"fs-id1167836477216\" class=\"unnumbered unstyled\" summary=\"The given values are 0.51, 1.25 and 0.093. Moving the decimal point two places to the right in each case, we get 51 percent, 125 percent and 9.3 percent.\" data-label=\"\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833058770\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"left\">Move the decimal point two places to the right.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635191\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836701289\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836477822\">\r\n<div data-type=\"problem\" id=\"fs-id1167836477824\">\r\n<p id=\"fs-id1167836477826\">Convert each:<\/p>\r\n<p id=\"fs-id1167833224433\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 9%, 87%, and 3.9%.<\/p>\r\n<p id=\"fs-id1167829719411\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.17, 1.75, and 0.0825.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833021569\">\r\n<p id=\"fs-id1167829719350\"><span class=\"token\">\u24d0<\/span> 0.09, 0.87, 0.039 <span class=\"token\">\u24d1<\/span> 17%, 175%, 8.25%<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829716469\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836494036\">\r\n<div data-type=\"problem\" id=\"fs-id1167836494039\">\r\n<p id=\"fs-id1167836494041\">Convert each:<\/p>\r\n<p id=\"fs-id1167836508314\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 3%, 91%, and 8.3%.<\/p>\r\n<p id=\"fs-id1167829628364\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.41, 2.25, and 0.0925.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836417176\">\r\n<p id=\"fs-id1167832982286\"><span class=\"token\">\u24d0<\/span> 0.03, 0.91, 0.083 <span class=\"token\">\u24d1<\/span> 41%, 225%, 9.25%<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836609894\">\r\n<h3 data-type=\"title\">Simplify Expressions with Square Roots<\/h3>\r\n<p id=\"fs-id1167833396879\">Remember that when a number \\(n\\) is multiplied by itself, we write \\({n}^{2}\\) and read it \u201c\\(n\\) squared.\u201d The result is called the <span data-type=\"term\">square of a number<\/span> <em data-effect=\"italics\">n<\/em>. For example, \\({8}^{2}\\) is read \u201c8 squared\u201d and 64 is called the <em data-effect=\"italics\">square<\/em> of 8. Similarly, 121 is the square of 11 because \\({11}^{2}\\) is 121. It will be helpful to learn to recognize the perfect square numbers.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836699183\">\r\n<div data-type=\"title\">Square of a number<\/div>\r\n<p id=\"fs-id1167829579161\">If \\({n}^{2}=m,\\) then <em data-effect=\"italics\">m<\/em> is the <strong data-effect=\"bold\">square<\/strong> of <em data-effect=\"italics\">n<\/em>.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167833061164\">What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836356337\" class=\"unnumbered\" data-label=\"\">\\({\\left(-3\\right)}^{2}=9\\phantom{\\rule{2.5em}{0ex}}{\\left(-8\\right)}^{2}=64\\phantom{\\rule{2.5em}{0ex}}{\\left(-11\\right)}^{2}=121\\phantom{\\rule{2.5em}{0ex}}{\\left(-15\\right)}^{2}=225\\)<\/div>\r\n<p id=\"fs-id1167833054715\">Because \\({10}^{2}=100,\\) we say 100 is the square of 10. We also say that 10 is a <em data-effect=\"italics\">square root<\/em> of 100. A number whose square is <em data-effect=\"italics\">m<\/em> is called a <span data-type=\"term\">square root of a number<\/span> <em data-effect=\"italics\">m<\/em>.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836406664\">\r\n<div data-type=\"title\">Square Root of a Number<\/div>\r\n<p id=\"fs-id1167836530323\">If \\({n}^{2}=m,\\) then <em data-effect=\"italics\">n<\/em> is a <strong data-effect=\"bold\">square root<\/strong> of <em data-effect=\"italics\">m<\/em>.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836295018\">Notice \\({\\left(-10\\right)}^{2}=100\\) also, so \\(-10\\) is also a square root of 100. Therefore, both 10 and \\(-10\\) are square roots of 100. So, every positive number has two square roots\u2014one positive and one negative. The radical sign, \\(\\sqrt{m}\\), denotes the positive square root. The positive square root is called the <span data-type=\"term\">principal square root<\/span>. When we use the radical sign that always means we want the principal square root.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167833022144\">\r\n<div data-type=\"title\">Square Root Notation<\/div>\r\n<p id=\"fs-id1167836707363\">\\(\\sqrt{m}\\) is read \u201cthe square root of \\(m\\).\u201d<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829715458\" data-alt=\"Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_015_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.\" \/><\/span>\r\n<p id=\"fs-id1167836549955\">If \\(m={n}^{2},\\) then \\(\\sqrt{m}=n,\\) for \\(n\\ge 0.\\)<\/p>\r\n<p id=\"fs-id1167836620491\">The square root of <em data-effect=\"italics\">m<\/em>, \\(\\sqrt{m},\\) is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836701072\">We know that every positive number has two square roots and the radical sign indicates the positive one. We write \\(\\sqrt{100}=10.\\) If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \\(\\text{\u2212}\\sqrt{100}=-10.\\) We read \\(\\text{\u2212}\\sqrt{100}\\) as \u201cthe opposite of the principal square root of 10.\u201d<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167833056590\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833056593\">\r\n<div data-type=\"problem\" id=\"fs-id1167833021974\">\r\n<p id=\"fs-id1167833021976\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\sqrt{25}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{121}\\) <span class=\"token\">\u24d2<\/span> \\(\\text{\u2212}\\sqrt{144}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836559725\">\r\n<p id=\"fs-id1167829717722\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\sqrt{25}\\hfill \\\\ \\text{Since}\\phantom{\\rule{0.2em}{0ex}}{5}^{2}=25\\hfill &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}5\\hfill \\end{array}\\)\r\n<p id=\"fs-id1167836550052\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\sqrt{121}\\hfill \\\\ \\text{Since}\\phantom{\\rule{0.2em}{0ex}}{11}^{2}=121\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}11\\hfill \\end{array}\\)\r\n<p id=\"fs-id1167836487508\"><span class=\"token\">\u24d2<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\text{\u2212}\\sqrt{144}\\hfill \\\\ \\text{The negative is in front of}\\hfill &amp; &amp; &amp; -12\\hfill \\\\ \\text{the radical sign.}\\hfill &amp; &amp; &amp; \\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836287231\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836287235\">\r\n<div data-type=\"problem\" id=\"fs-id1167836625114\">\r\n<p id=\"fs-id1167836625116\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\sqrt{36}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{169}\\) <span class=\"token\">\u24d2<\/span> \\(\\text{\u2212}\\sqrt{225}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836551671\">\r\n<p id=\"fs-id1167829720135\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> 13 <span class=\"token\">\u24d2<\/span> \\(-15\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167833175426\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167833175430\">\r\n<div data-type=\"problem\" id=\"fs-id1167833020312\">\r\n<p id=\"fs-id1167833020314\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\sqrt{16}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{196}\\) <span class=\"token\">\u24d2<\/span> \\(\\text{\u2212}\\sqrt{100}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1167836547264\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 14 <span class=\"token\">\u24d2<\/span> \\(-10\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836492536\">\r\n<h3 data-type=\"title\">Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/h3>\r\n<p id=\"fs-id1167833049933\">We have already described numbers as <em data-effect=\"italics\">counting number<\/em><em data-effect=\"italics\">s<\/em>, <em data-effect=\"italics\">whole number<\/em><em data-effect=\"italics\">s<\/em>, and <em data-effect=\"italics\">integers<\/em>. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836800466\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\text{Counting numbers}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1em}{0ex}}1,2,3,4,\\text{\u2026}..\\hfill \\\\ \\text{Whole numbers}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1em}{0ex}}0,1,2,3,4,\\text{\u2026}.\\hfill \\\\ \\text{Integers}\\hfill &amp; &amp; &amp; \\text{\u2026}.-3,-2,-1,0,1,2,3,\\text{\u2026}.\\hfill \\end{array}\\)<\/div>\r\nWhat type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A <span data-type=\"term\">rational number<\/span> is a number that can be written as a ratio of two integers.\r\n<p id=\"fs-id1167829715480\">In general, any decimal that ends after a number of digits (such as 7.3 or \\(-1.2684\\)) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for \\(\\frac{1}{3}\\) is the number \\(0.\\stackrel{\u2013}{3}.\\) The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.<\/p>\r\n<p id=\"fs-id1167829717443\">Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.<\/p>\r\n<p id=\"fs-id1167833022517\"><em data-effect=\"italics\">Every rational number can be written both as a ratio of integers<\/em>\\(\\frac{p}{q},\\)<em data-effect=\"italics\">where p and q are integers and<\/em>\\(q\\ne 0,\\)<em data-effect=\"italics\">and as a decimal that stops or repeats.<\/em><\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167836536548\">\r\n<div data-type=\"title\">Rational Number<\/div>\r\n<p id=\"fs-id1167836536554\">A <strong data-effect=\"bold\">rational number<\/strong> is a number of the form \\(\\frac{p}{q},\\) where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and \\(q\\ne 0.\\)<\/p>\r\n<p id=\"fs-id1167836698900\">Its decimal form stops or repeats.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836698905\">Are there any decimals that do not stop or repeat? Yes! The number \\(\\pi \\) (the Greek letter <em data-effect=\"italics\">pi<\/em>, pronounced \u201cpie\u201d), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (\u2026) to indicate the decimal does not stop or repeat.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167829719570\" class=\"unnumbered\" data-label=\"\">\\(\\pi =3.141592654...\\)<\/div>\r\n<p id=\"fs-id1167836579988\">The square root of a number that is not a perfect square is a decimal that does not stop or repeat.<\/p>\r\n<p id=\"fs-id1167836579992\">A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an <span data-type=\"term\">irrational number<\/span>.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167832981620\">\r\n<div data-type=\"title\">Irrational Number<\/div>\r\n<p id=\"fs-id1167836607383\">An <strong data-effect=\"bold\">irrational number<\/strong> is a number that cannot be written as the ratio of two integers.<\/p>\r\n<p id=\"fs-id1167836530052\">Its decimal form does not stop and does not repeat.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836530057\">Let\u2019s summarize a method we can use to determine whether a number is rational or irrational.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167829715600\">\r\n<div data-type=\"title\">Rational or Irrational<\/div>\r\n<p id=\"fs-id1167836635411\">If the decimal form of a number<\/p>\r\n\r\n<ul id=\"fs-id1167836635415\" data-bullet-style=\"bullet\">\r\n \t<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is a <strong data-effect=\"bold\">rational number<\/strong>.<\/li>\r\n \t<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is an <strong data-effect=\"bold\">irrational number.<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1167833240081\">We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of <span data-type=\"term\">real number<\/span><strong data-effect=\"bold\">s<\/strong>.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1167829719876\">\r\n<div data-type=\"title\">Real Number<\/div>\r\n<p id=\"fs-id1167829717421\">A <strong data-effect=\"bold\">real number<\/strong> is a number that is either rational or irrational.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167836698829\">Later in this course we will introduce numbers beyond the real numbers. <a href=\"#CNX_IntAlg_Figure_01_04_016\" class=\"autogenerated-content\">(Figure)<\/a> illustrates how the number sets we\u2019ve used so far fit together.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_016\">\r\n<div class=\"bc-figcaption figcaption\">This chart shows the number sets that make up the set of real numbers.<\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836447242\" data-alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_016-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\" \/><\/span>\r\n\r\n<\/div>\r\n<p id=\"fs-id1167829624381\">Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal,\u201d and, if so, what could they be? Can we simplify \\(\\sqrt{-25}?\\) Is there a number whose square is \\(-25?\\)<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167832981655\" class=\"unnumbered\" data-label=\"\">\\({\\left(\\phantom{\\rule{0.2em}{0ex}}\\right)}^{2}=-25?\\)<\/div>\r\n<p id=\"fs-id1167833054419\">None of the numbers that we have dealt with so far has a square that is \\(-25.\\) Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to \\(\\sqrt{-25}.\\) The square root of a negative number is not a real number.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836546317\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836546320\">\r\n<div data-type=\"problem\" id=\"fs-id1167836526165\">\r\n<p id=\"fs-id1167836526168\">Given the numbers \\(-7,\\frac{14}{5},8,\\sqrt{5},5.9,\\text{\u2212}\\sqrt{64},\\) list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers <span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836580197\">\r\n<p id=\"fs-id1167836580199\"><span class=\"token\">\u24d0<\/span> Remember, the whole numbers are \\(0,1,2,3,\\text{\u2026},\\) so 8 is the only whole number given.<\/p>\r\n<p id=\"fs-id1167833018577\"><span class=\"token\">\u24d1<\/span> The integers are the whole numbers and their opposites (which includes 0). So the whole number 8 is an integer, and \\(-7\\) is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so \\(\\text{\u2212}\\sqrt{64}=-8.\\) So the integers are \\(-7,8,\\) and \\(\\text{\u2212}\\sqrt{64}.\\)<\/p>\r\n<p id=\"fs-id1167836700847\"><span class=\"token\">\u24d2<\/span> Since all integers are rational, then \\(-7,8,\\) and \\(\\text{\u2212}\\sqrt{64}\\) are rational. Rational numbers also include fractions and decimals that repeat or stop, so \\(\\frac{14}{5}\\) and \\(5.9\\) are rational. So the list of rational numbers is \\(-7,\\frac{14}{5},8,5.9,\\text{\u200b}\\text{\u200b}\\) and \\(\\text{\u2212}\\sqrt{64}.\\)<\/p>\r\n<p id=\"fs-id1167829721168\"><span class=\"token\">\u24d3<\/span> Remember that 5 is not a perfect square, so \\(\\sqrt{5}\\) is irrational.<\/p>\r\n<p id=\"fs-id1167829717682\"><span class=\"token\">\u24d4<\/span> All the numbers listed are real numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836525716\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836525721\">\r\n<div data-type=\"problem\" id=\"fs-id1167829716771\">\r\n<p id=\"fs-id1167829716773\">Given the numbers \\(-3,\\text{\u2212}\\sqrt{2},0.\\stackrel{\u2013}{3},\\frac{9}{5},4,\\sqrt{49},\\) list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829716001\">\r\n<p id=\"fs-id1167829693412\"><span class=\"token\">\u24d0<\/span>\\(4,\\sqrt{49}\\)<span class=\"token\">\u24d1<\/span>\\(-3,4,\\sqrt{49}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(-3,0.\\stackrel{\u2013}{3},\\frac{9}{5},4,\\sqrt{49}\\)<span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}\\sqrt{2}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d4<\/span>\\(-3,\\text{\u2212}\\sqrt{2},0.\\stackrel{\u2013}{3},\\frac{9}{5},4,\\sqrt{49}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167832981706\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832981709\">\r\n<div data-type=\"problem\" id=\"fs-id1167832981711\">\r\n<p id=\"fs-id1167832981713\">Given numbers \\(\\text{\u2212}\\sqrt{25},-\\frac{3}{8},-1,6,\\sqrt{121},2.041975...,\\) list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers <span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836578852\">\r\n<p id=\"fs-id1167836578854\"><span class=\"token\">\u24d0<\/span>\\(6,\\sqrt{121}\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}\\sqrt{25},-1,6,\\sqrt{121}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(\\text{\u2212}\\sqrt{25},-\\frac{3}{8},-1,6,\\sqrt{121}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span>\\(2.041975...\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d4<\/span>\\(\\text{\u2212}\\sqrt{25},-\\frac{3}{8},-1,6,\\sqrt{121},2.041975...\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836444834\">\r\n<h3 data-type=\"title\">Locate Fractions and Decimals on the Number Line<\/h3>\r\n<p id=\"fs-id1167829627701\">We now want to include fractions and decimals on the number line. Let\u2019s start with fractions and locate \\(\\frac{1}{5},-\\frac{4}{5},3,\\frac{7}{4},-\\frac{9}{2},-5\\) and \\(\\frac{8}{3}\\) on the number line.<\/p>\r\n<p id=\"fs-id1167836521567\">We\u2019ll start with the whole numbers 3 and \\(-5\\) because they are the easiest to plot. See <a href=\"#CNX_IntAlg_Figure_01_04_017\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\r\n<p id=\"fs-id1167836698574\">The proper fractions listed are \\(\\frac{1}{5}\\) and \\(-\\frac{4}{5}.\\) We know the proper fraction \\(\\frac{1}{5}\\) has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts \\(\\frac{1}{5},\\frac{2}{5},\\frac{3}{5},\\frac{4}{5}.\\) We plot \\(\\frac{1}{5}.\\)<\/p>\r\n<p id=\"fs-id1167836607368\">Similarly, \\(-\\frac{4}{5}\\) is between 0 and \\(-1.\\) After dividing the unit into 5 equal parts we plot \\(-\\frac{4}{5}.\\)<\/p>\r\n<p id=\"fs-id1167829717102\">Finally, look at the improper fractions \\(\\frac{7}{4},\\frac{9}{2},\\frac{8}{3}.\\) Locating these points may be easier if you change each of them to a mixed number.<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1167836619278\" class=\"unnumbered\" data-label=\"\">\\(\\frac{7}{4}=1\\frac{3}{4}\\phantom{\\rule{2.5em}{0ex}}-\\frac{9}{2}=-4\\frac{1}{2}\\phantom{\\rule{2.5em}{0ex}}\\frac{8}{3}=2\\frac{2}{3}\\)<\/div>\r\n<p id=\"fs-id1167829714506\"><a href=\"#CNX_IntAlg_Figure_01_04_017\" class=\"autogenerated-content\">(Figure)<\/a> shows the number line with all the points plotted.<\/p>\r\n\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_017\"><span data-type=\"media\" id=\"fs-id1167829714516\" data-alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 5, minus 9 by 2, minus 4 by 5, 1 by 5, 4 by 5, 8 by 3 and 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_017-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 5, minus 9 by 2, minus 4 by 5, 1 by 5, 4 by 5, 8 by 3 and 3.\" \/><\/span><\/div>\r\n<div data-type=\"example\" id=\"fs-id1167832940593\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167832940595\">\r\n<div data-type=\"problem\" id=\"fs-id1167832940597\">\r\n<p id=\"fs-id1167832940599\">Locate and label the following on a number line: \\(4,\\frac{3}{4},-\\frac{1}{4},-3,\\frac{6}{5},-\\frac{5}{2},\\) and \\(\\frac{7}{3}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829720660\">\r\n<p id=\"fs-id1167829720662\">Locate and plot the integers, \\(4,-3.\\)<\/p>\r\n<p id=\"fs-id1167829627594\">Locate the proper fraction \\(\\frac{3}{4}\\) first. The fraction \\(\\frac{3}{4}\\) is between 0 and 1. Divide the distance between 0 and 1 into four equal parts, then we plot \\(\\frac{3}{4}.\\) Similarly plot \\(-\\frac{1}{4}.\\)<\/p>\r\n<p id=\"fs-id1167829627782\">Now locate the improper fractions \\(\\frac{6}{5},-\\frac{5}{2},\\) and \\(\\frac{7}{3}.\\) It is easier to plot them if we convert them to mixed numbers and then plot them as described above: \\(\\frac{6}{5}=1\\frac{1}{5},-\\frac{5}{2}=-2\\frac{1}{2},\\frac{7}{3}=2\\frac{1}{3}.\\)<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836508770\" data-alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 3, minus 5 by 2, minus 1 by 4, 3 by 4, 6 by 5, 7 by 3 and 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_018-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 3, minus 5 by 2, minus 1 by 4, 3 by 4, 6 by 5, 7 by 3 and 4.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836508289\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829719000\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719002\">\r\n<p id=\"fs-id1167829719005\">Locate and label the following on a number line: \\(-1,\\frac{1}{3},\\frac{6}{5},-\\frac{7}{4},\\frac{9}{2},5,-\\frac{8}{3}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836551739\"><span data-type=\"media\" id=\"fs-id1167836551743\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 8 by 3, minus 7 by 4, minus 1, 1 by 3, 6 by 5, 9 by 2 and 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_301_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 8 by 3, minus 7 by 4, minus 1, 1 by 3, 6 by 5, 9 by 2 and 5.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836620594\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836620598\">\r\n<div data-type=\"problem\" id=\"fs-id1167836620600\">\r\n<p id=\"fs-id1167829718132\">Locate and label the following on a number line: \\(-2,\\frac{2}{3},\\frac{7}{5},-\\frac{7}{4},\\frac{7}{2},3,-\\frac{7}{3}.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829579971\"><span data-type=\"media\" id=\"fs-id1167829590318\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 7 by 3, minus 2, minus 7 by 4, 2 by 3, 7 by 5, 3 and 7 by 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_302-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 7 by 3, minus 2, minus 7 by 4, 2 by 3, 7 by 5, 3 and 7 by 2.\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836595922\">Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1167836595926\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836595929\">\r\n<div data-type=\"problem\" id=\"fs-id1167836595931\">\r\n<p id=\"fs-id1167836595933\">Locate on the number line: <span class=\"token\">\u24d0<\/span> 0.4 <span class=\"token\">\u24d1<\/span> \\(-0.74.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833224821\">\r\n<p id=\"fs-id1167833224824\"><span class=\"token\">\u24d0<\/span> The decimal number 0.4 is equivalent to \\(\\frac{4}{10},\\) a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 as 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167836398917\" data-alt=\"Figure shows a number line with numbers ranging from 0.0 to 1. 0.4 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_021_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0.0 to 1. 0.4 is highlighted.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<p id=\"fs-id1167836398929\"><span class=\"token\">\u24d1<\/span> The decimal \\(-0.74\\) is equivalent to \\(-\\frac{74}{100},\\) so it is located between 0 and \\(-1.\\) On a number line, mark off and label the hundredths in the interval between 0 and \\(-1.\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1167829721127\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_022_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836597144\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167836597148\">\r\n<div data-type=\"problem\" id=\"fs-id1167833369022\">\r\n<p id=\"fs-id1167833369024\">Locate on the number line: <span class=\"token\">\u24d0<\/span> \\(0.6\\) <span class=\"token\">\u24d1<\/span> \\(-0.25.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833369143\">\r\n<p id=\"fs-id1167833369145\"><span class=\"token\">\u24d0<\/span><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167833369154\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.6 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_303_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.6 is highlighted.\" \/><\/span>\r\n<p id=\"fs-id1167829719042\"><span class=\"token\">\u24d1<\/span><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829719051\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted, minus 0.25 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_304_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted, minus 0.25 is highlighted.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167829627506\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1167829719437\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719440\">\r\n<p id=\"fs-id1167829719442\">Locate on the number line: <span class=\"token\">\u24d0<\/span> \\(0.9\\) <span class=\"token\">\u24d1<\/span> \\(-0.75.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829714677\">\r\n<p id=\"fs-id1167836613683\"><span class=\"token\">\u24d0<\/span><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167836613691\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.9 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_305_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.9 is highlighted.\" \/><\/span>\r\n<p id=\"fs-id1167829717744\"><span class=\"token\">\u24d1<\/span><\/p>\r\n<span data-type=\"media\" id=\"fs-id1167829717751\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_306_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1167836698527\" class=\"media-2\">\r\n<p id=\"fs-id1167836698531\">Access this online resource for additional instruction and practice with decimals.<\/p>\r\n\r\n<ul id=\"fs-id1167836698534\" data-bullet-style=\"bullet\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/25dividedecim\">Arithmetic Basics: Dividing Decimals<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829714441\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1167836598055\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">How to round decimals.<\/strong>\r\n<ol id=\"fs-id1167833396934\" class=\"stepwise\" type=\"1\">\r\n \t<li>Locate the given place value and mark it with an arrow.<\/li>\r\n \t<li>Underline the digit to the right of the place value.<\/li>\r\n \t<li>Is the underlined digit greater than or equal to \\(5?\\)\r\n<ul id=\"fs-id1167836635513\" data-bullet-style=\"bullet\">\r\n \t<li>Yes: add 1 to the digit in the given place value.<\/li>\r\n \t<li>No: do <u data-effect=\"underline\">not<\/u> change the digit in the given place value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Rewrite the number, deleting all digits to the right of the rounding digit.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to add or subtract decimals.<\/strong>\r\n<ol id=\"fs-id1167829597340\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the sum or difference.<\/li>\r\n \t<li>Write the numbers so the decimal points line up vertically.<\/li>\r\n \t<li>Use zeros as placeholders, as needed.<\/li>\r\n \t<li>Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.<\/li>\r\n \t<li>Write the sum or difference with the appropriate sign<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to multiply decimals.<\/strong>\r\n<ol id=\"fs-id1167836389246\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the product.<\/li>\r\n \t<li>Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\r\n \t<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.<\/li>\r\n \t<li>Write the product with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to multiply a decimal by a power of ten.<\/strong>\r\n<ol id=\"fs-id1167829580349\" class=\"stepwise\" type=\"1\">\r\n \t<li>Move the decimal point to the right the same number of places as the number of zeros in the power of 10.<\/li>\r\n \t<li>Add zeros at the end of the number as needed.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to divide decimals.<\/strong>\r\n<ol id=\"fs-id1167829716804\" class=\"stepwise\" type=\"1\">\r\n \t<li>Determine the sign of the quotient.<\/li>\r\n \t<li>Make the divisor a whole number by \u201cmoving\u201d the decimal point all the way to the right. \u201cMove\u201d the decimal point in the dividend the same number of places\u2014adding zeros as needed.<\/li>\r\n \t<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\r\n \t<li>Write the quotient with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to convert a decimal to a proper fraction and a fraction to a decimal.<\/strong>\r\n<div data-type=\"newline\"><\/div>\r\n<ol id=\"fs-id1167829624407\" class=\"stepwise\" type=\"1\">\r\n \t<li>To convert a decimal to a proper fraction, determine the place value of the final digit.<\/li>\r\n \t<li>Write the fraction.\r\n<ul id=\"fs-id1167836596324\" data-bullet-style=\"bullet\">\r\n \t<li>numerator\u2014the \u201cnumbers\u201d to the right of the decimal point<\/li>\r\n \t<li>denominator\u2014the place value corresponding to the final digit<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">How to convert a percent to a decimal and a decimal to a percent.<\/strong>\r\n<ol id=\"fs-id1167836613313\" class=\"stepwise\" type=\"1\">\r\n \t<li>To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.<\/li>\r\n \t<li>To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Square Root Notation<\/strong>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\sqrt{m}\\) is read \u201cthe square root of <em data-effect=\"italics\">m<\/em>.\u201d\r\n<div data-type=\"newline\"><\/div>\r\nIf \\(m={n}^{2},\\) then \\(\\sqrt{m}=n,\\) for \\(n\\ge 0.\\)\r\n<div data-type=\"newline\"><\/div>\r\nThe square root of <em data-effect=\"italics\">m<\/em>, \\(\\sqrt{m},\\) is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/li>\r\n \t<li><strong data-effect=\"bold\">Rational or Irrational<\/strong>\r\n<div data-type=\"newline\"><\/div>\r\nIf the decimal form of a number\r\n<div data-type=\"newline\"><\/div>\r\n<ul id=\"fs-id1167829717791\" data-bullet-style=\"bullet\">\r\n \t<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is a rational number.<\/li>\r\n \t<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is an irrational number.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong data-effect=\"bold\">Real Numbers<\/strong>\r\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_016a\"><span data-type=\"media\" id=\"fs-id1167829691173\" data-alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_016-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\" \/><\/span><\/div><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836551129\">\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836551133\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1167836551141\"><strong data-effect=\"bold\">Round Decimals<\/strong><\/p>\r\n<p id=\"fs-id1167829692185\">In the following exercises, round each number to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829714550\">\r\n<div data-type=\"problem\" id=\"fs-id1167829714552\">\r\n<p id=\"fs-id1167829714554\">5.781<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829714559\">\r\n<p id=\"fs-id1167829714561\"><span class=\"token\">\u24d0<\/span> 5.78 <span class=\"token\">\u24d1<\/span> 5.8 <span class=\"token\">\u24d2<\/span> 6<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836531237\">\r\n<div data-type=\"problem\" id=\"fs-id1167836531239\">\r\n<p id=\"fs-id1167836531241\">1.638<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833020287\">\r\n<div data-type=\"problem\" id=\"fs-id1167833020290\">\r\n<p id=\"fs-id1167833020292\">0.299<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833020296\">\r\n<p id=\"fs-id1167833020298\"><span class=\"token\">\u24d0<\/span> 0.30 <span class=\"token\">\u24d1<\/span> 0.3 <span class=\"token\">\u24d2<\/span> 0<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833021622\">\r\n<div data-type=\"problem\" id=\"fs-id1167833021624\">\r\n<p id=\"fs-id1167833021626\">0.697<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829627529\">\r\n<div data-type=\"problem\" id=\"fs-id1167829627531\">\r\n<p id=\"fs-id1167836580081\">63.479<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836580085\">\r\n<p id=\"fs-id1167836580087\"><span class=\"token\">\u24d0<\/span> 63.48 <span class=\"token\">\u24d1<\/span> 63.5 <span class=\"token\">\u24d2<\/span> 63<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836699141\">\r\n<div data-type=\"problem\" id=\"fs-id1167836699143\">\r\n<p id=\"fs-id1167836699145\">84.281<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829719115\"><strong data-effect=\"bold\">Add and Subtract Decimals<\/strong><\/p>\r\n<p id=\"fs-id1167829719121\">In the following exercises, add or subtract.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829719124\">\r\n<div data-type=\"problem\" id=\"fs-id1167832936353\">\r\n<p id=\"fs-id1167832936356\">\\(-16.53-24.38\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167832936367\">\r\n<p id=\"fs-id1167832936369\">\\(-40.91\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167832999808\">\r\n<div data-type=\"problem\" id=\"fs-id1167832999810\">\r\n<p id=\"fs-id1167832999812\">\\(-19.47-32.58\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829719888\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719890\">\r\n<p id=\"fs-id1167829719892\">\\(-38.69+31.47\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829719904\">\r\n<p id=\"fs-id1167829719906\">\\(-7.22\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833224567\">\r\n<div data-type=\"problem\" id=\"fs-id1167833224569\">\r\n<p id=\"fs-id1167833224571\">\\(-29.83+19.76\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829720868\">\r\n<div data-type=\"problem\" id=\"fs-id1167829720870\">\r\n<p id=\"fs-id1167829720873\">\\(72.5-100\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829716532\">\r\n<p id=\"fs-id1167829716534\">\\(-27.5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829716543\">\r\n<div data-type=\"problem\" id=\"fs-id1167829716545\">\r\n<p id=\"fs-id1167829716547\">\\(86.2-100\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833240107\">\r\n<div data-type=\"problem\" id=\"fs-id1167833240109\">\r\n<p id=\"fs-id1167833240111\">\\(91.75-\\left(-10.462\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829714180\">\r\n<p id=\"fs-id1167829714182\">\\(02.212\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829690547\">\r\n<div data-type=\"problem\" id=\"fs-id1167829690549\">\r\n<p id=\"fs-id1167829690551\">\\(94.69-\\left(-12.678\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836508374\">\r\n<div data-type=\"problem\" id=\"fs-id1167836508376\">\r\n<p id=\"fs-id1167836508378\">\\(55.01-3.7\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829714589\">\r\n<p id=\"fs-id1167829714592\">\\(51.31\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829714600\">\r\n<div data-type=\"problem\" id=\"fs-id1167829714602\">\r\n<p id=\"fs-id1167829714604\">\\(59.08-4.6\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829624911\">\r\n<div data-type=\"problem\" id=\"fs-id1167829624913\">\r\n<p id=\"fs-id1167829624915\">\\(2.51-7.4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836621826\">\r\n<p id=\"fs-id1167836621828\">\\(-4.89\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836621836\">\r\n<div data-type=\"problem\" id=\"fs-id1167836621838\">\r\n<p id=\"fs-id1167836621840\">\\(3.84-6.1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829721201\"><strong data-effect=\"bold\">Multiply and Divide Decimals<\/strong><\/p>\r\n<p id=\"fs-id1167829721207\">In the following exercises, multiply.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836620056\">\r\n<div data-type=\"problem\" id=\"fs-id1167836620058\">\r\n<p id=\"fs-id1167836620061\">\\(94.69-\\left(-12.678\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836620079\">\r\n<p id=\"fs-id1167836620081\">\\(-11.653\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829624935\">\r\n<div data-type=\"problem\" id=\"fs-id1167829624937\">\r\n<p id=\"fs-id1167829624939\">\\(\\left(-8.5\\right)\\left(1.69\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833059335\">\r\n<div data-type=\"problem\" id=\"fs-id1167833059338\">\r\n<p id=\"fs-id1167833059340\">\\(\\left(-5.18\\right)\\left(-65.23\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836331120\">\r\n<p id=\"fs-id1167836331123\">\\(337.8914\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829720751\">\r\n<div data-type=\"problem\" id=\"fs-id1167829720753\">\r\n<p id=\"fs-id1167829720756\">\\(\\left(-9.16\\right)\\left(-68.34\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833054680\">\r\n<div data-type=\"problem\" id=\"fs-id1167833054682\">\r\n<p id=\"fs-id1167833054684\">\\(\\left(0.06\\right)\\left(21.75\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836477505\">\r\n<p id=\"fs-id1167836477507\">\\(1.305\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836477711\">\r\n<div data-type=\"problem\" id=\"fs-id1167836477713\">\r\n<p id=\"fs-id1167836477715\">\\(\\left(0.08\\right)\\left(52.45\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829690917\">\r\n<div data-type=\"problem\" id=\"fs-id1167829690919\">\r\n<p id=\"fs-id1167829690921\">\\(\\left(9.24\\right)\\left(10\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829624428\">\r\n<p id=\"fs-id1167829624430\">\\(92.4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829624439\">\r\n<div data-type=\"problem\" id=\"fs-id1167836698541\">\r\n<p id=\"fs-id1167836698543\">\\(\\left(6.531\\right)\\left(10\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833025125\">\r\n<div data-type=\"problem\" id=\"fs-id1167833025127\">\r\n<p id=\"fs-id1167833025129\">\\(\\left(0.025\\right)\\left(100\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833025626\">\r\n<p id=\"fs-id1167833025628\">2.5<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833025634\">\r\n<div data-type=\"problem\" id=\"fs-id1167833025636\">\r\n<p id=\"fs-id1167833025638\">\\(\\left(0.037\\right)\\left(100\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833025756\">\r\n<div data-type=\"problem\" id=\"fs-id1167833025808\">\r\n<p id=\"fs-id1167833025810\">\\(\\left(55.2\\right)\\left(1000\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833025832\">\r\n<p id=\"fs-id1167836399531\">55200<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836399536\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399538\">\r\n<p id=\"fs-id1167836399541\">\\(\\left(99.4\\right)\\left(1000\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836399835\">In the following exercises, divide. Round money monetary answers to the nearest cent.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836399838\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399840\">\r\n<p id=\"fs-id1167836399843\">\\(\\text{?}117.25\u00f748\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836399945\">\r\n<p id=\"fs-id1167836399947\">\\(\\text{?}2.44\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836399957\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399960\">\r\n<p id=\"fs-id1167836399962\">\\(\\text{?}109.24\u00f736\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829719786\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719819\">\r\n<p id=\"fs-id1167829719822\">\\(1.44\u00f7\\left(-0.3\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829719840\">\r\n<p id=\"fs-id1167829719842\">\\(-4.8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829580307\">\r\n<div data-type=\"problem\" id=\"fs-id1167829580309\">\r\n<p id=\"fs-id1167829580311\">\\(-1.15\u00f7\\left(-0.05\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836597072\">\r\n<div data-type=\"problem\" id=\"fs-id1167836597074\">\r\n\r\n\\(5.2\u00f72.5\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n\r\n\\(2.08\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836542566\">\r\n<div data-type=\"problem\" id=\"fs-id1167836542568\">\r\n<p id=\"fs-id1167836542570\">\\(14\u00f70.35\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836518558\"><strong data-effect=\"bold\">Convert Decimals, Fractions and Percents<\/strong><\/p>\r\n<p id=\"fs-id1167836518563\">In the following exercises, write each decimal as a fraction.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836518566\">\r\n<div data-type=\"problem\" id=\"fs-id1167836518568\">\r\n<p id=\"fs-id1167836518570\">\\(0.04\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829716255\">\r\n<p id=\"fs-id1167829716258\">\\(\\frac{1}{25}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829716270\">\r\n<div data-type=\"problem\" id=\"fs-id1167829716272\">\r\n<p id=\"fs-id1167829716275\">1.464<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829715938\">\r\n<div data-type=\"problem\" id=\"fs-id1167829715940\">\r\n<p id=\"fs-id1167829715942\">\\(0.095\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829715950\">\r\n<p id=\"fs-id1167829715952\">\\(\\frac{19}{200}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829717177\">\r\n<div data-type=\"problem\" id=\"fs-id1167829717180\">\r\n<p id=\"fs-id1167829717182\">\\(-0.375\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836484387\">In the following exercises, convert each fraction to a decimal.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167836484390\">\r\n<div data-type=\"problem\" id=\"fs-id1167836484392\">\r\n<p id=\"fs-id1167836484394\">\\(\\frac{17}{20}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836484407\">\r\n<p id=\"fs-id1167836484814\">\\(0.85\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836484822\">\r\n<div data-type=\"problem\" id=\"fs-id1167836484824\">\r\n<p id=\"fs-id1167836484827\">\\(\\frac{17}{4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836684117\">\r\n<div data-type=\"problem\" id=\"fs-id1167836684119\">\r\n<p id=\"fs-id1167836684121\">\\(-\\frac{310}{25}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836684136\">\r\n<p id=\"fs-id1167836684138\">\\(-12.4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829589886\">\r\n<div data-type=\"problem\" id=\"fs-id1167829589888\">\r\n<p id=\"fs-id1167829589890\">\\(-\\frac{18}{11}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167829719662\">In the following exercises, convert each percent to a decimal.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829719665\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719667\">\r\n<p id=\"fs-id1167829719669\">\\(71%\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829719679\">\r\n<p id=\"fs-id1167833025497\">\\(0.71\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833025505\">\r\n<div data-type=\"problem\" id=\"fs-id1167833025507\">\r\n<p id=\"fs-id1167833025510\">\\(150%\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836542526\">\r\n<div data-type=\"problem\" id=\"fs-id1167836542528\">\r\n<p id=\"fs-id1167836542530\">\\(39.3%\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836542540\">\r\n<p id=\"fs-id1167836542542\">\\(0.393\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836542551\">\r\n<div data-type=\"problem\" id=\"fs-id1167836542553\">\r\n<p id=\"fs-id1167836542555\">\\(7.8%\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833053575\">In the following exercises, convert each decimal to a percent.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167833053578\">\r\n<div data-type=\"problem\" id=\"fs-id1167833053580\">\r\n<p id=\"fs-id1167833053582\">\\(1.56\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836621850\">\r\n<p id=\"fs-id1167836621853\">\\(156%\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836621863\">\r\n<div data-type=\"problem\" id=\"fs-id1167836621865\">\r\n<p id=\"fs-id1167836621868\">3<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836508489\">\r\n<div data-type=\"problem\" id=\"fs-id1167836508492\">\r\n<p id=\"fs-id1167836508494\">\\(0.0625\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836508501\">\r\n<p id=\"fs-id1167836508503\">\\(6.25%\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836508514\">\r\n<div data-type=\"problem\" id=\"fs-id1167836508516\">\r\n<p id=\"fs-id1167833158153\">\\(2.254\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833158174\"><strong data-effect=\"bold\">Simplify Expressions with Square Roots<\/strong><\/p>\r\n<p id=\"fs-id1167833158180\">In the following exercises, simplify.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167833158183\">\r\n<div data-type=\"problem\" id=\"fs-id1167829717857\">\r\n<p id=\"fs-id1167829717859\">\\(\\sqrt{64}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829717869\">\r\n<p id=\"fs-id1167829717871\">8<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829717876\">\r\n<div data-type=\"problem\" id=\"fs-id1167829717878\">\r\n<p id=\"fs-id1167829717880\">\\(\\sqrt{169}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833397155\">\r\n<div data-type=\"problem\" id=\"fs-id1167833397157\">\r\n<p id=\"fs-id1167833397160\">\\(\\sqrt{144}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833397169\">\r\n<p id=\"fs-id1167833397171\">12<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833397177\">\r\n<div data-type=\"problem\" id=\"fs-id1167833397179\">\r\n<p id=\"fs-id1167836477531\">\\(\\text{\u2212}\\sqrt{4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836477552\">\r\n<div data-type=\"problem\" id=\"fs-id1167836477554\">\r\n<p id=\"fs-id1167836477557\">\\(\\text{\u2212}\\sqrt{100}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836398828\">\r\n<p id=\"fs-id1167836398830\">\\(-10\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836398839\">\r\n<div data-type=\"problem\" id=\"fs-id1167836398841\">\r\n<p id=\"fs-id1167836398843\">\\(\\text{\u2212}\\sqrt{121}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167836456213\"><strong data-effect=\"bold\">Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/strong><\/p>\r\n<p id=\"fs-id1167836456219\">In the following exercises, list the <span class=\"token\">\u24d0<\/span> whole numbers, <span class=\"token\">\u24d1<\/span> integers, <span class=\"token\">\u24d2<\/span> rational numbers, <span class=\"token\">\u24d3<\/span> irrational numbers, <span class=\"token\">\u24d4<\/span> real numbers for each set of numbers.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167829720690\">\r\n<div data-type=\"problem\" id=\"fs-id1167829720693\">\r\n<p id=\"fs-id1167829720695\">\\(-8,0,1.95286...,\\frac{12}{5},\\sqrt{36},9\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829690686\">\r\n<p id=\"fs-id1167829690688\"><span class=\"token\">\u24d0<\/span>\\(0,\\sqrt{36},9\\)<span class=\"token\">\u24d1<\/span>\\(-8,0,\\sqrt{36},9\\)<span class=\"token\">\u24d2<\/span>\\(-8,0,\\sqrt{36},9\\)<span class=\"token\">\u24d3<\/span>\\(1.95286...,\\)<span class=\"token\">\u24d4<\/span>\\(-8,0,1.95286...,\\frac{12}{5},\\sqrt{36},9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833025160\">\r\n<div data-type=\"problem\" id=\"fs-id1167833025162\">\r\n<p id=\"fs-id1167833025164\">\\(-9,-3\\frac{4}{9},\\text{\u2212}\\sqrt{9},0.4\\stackrel{\u2014}{09},\\frac{11}{6},7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836398857\">\r\n<div data-type=\"problem\" id=\"fs-id1167836398859\">\r\n<p id=\"fs-id1167836398861\">\\(\\text{\u2212}\\sqrt{100},-7,-\\frac{8}{3},-1,0.77,3\\frac{1}{4}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836700865\">\r\n<p id=\"fs-id1167836700867\"><span class=\"token\">\u24d0<\/span> none <span class=\"token\">\u24d1<\/span> \\(\\text{\u2212}\\sqrt{100},-7,-1\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\text{\u2212}\\sqrt{100},-7,-\\frac{8}{3},-1,0.77,3\\frac{1}{4}\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span> none\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d4<\/span> \\(\\text{\u2212}\\sqrt{100},-7,-\\frac{8}{3},-1,0.77,3\\frac{1}{4}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836484440\">\r\n<div data-type=\"problem\" id=\"fs-id1167836484442\">\r\n<p id=\"fs-id1167836484444\">\\(-6,-\\frac{5}{2},0,0.\\stackrel{\u2014\u2014\u2014}{714285},2\\frac{1}{5},\\sqrt{14}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167833368929\"><strong data-effect=\"bold\">Locate Fractions and Decimals on the Number Line<\/strong><\/p>\r\n<p id=\"fs-id1167833368936\">In the following exercises, locate the numbers on a number line.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1167833368939\">\r\n<div data-type=\"problem\" id=\"fs-id1167833368941\">\r\n<p id=\"fs-id1167833368943\">\\(\\frac{3}{10},\\frac{7}{2},\\frac{11}{6},4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167833412462\"><span data-type=\"media\" id=\"fs-id1167833412467\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 6. Some values are highlighted. From left to right, these are: 3 by 10, 11 by 6, 7 by 2 and 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_307_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 6. Some values are highlighted. From left to right, these are: 3 by 10, 11 by 6, 7 by 2 and 4.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167833412483\">\r\n<div data-type=\"problem\" id=\"fs-id1167833412485\">\r\n<p id=\"fs-id1167833369747\">\\(\\frac{7}{10},\\frac{5}{2},\\frac{13}{8},3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836410212\">\r\n<div data-type=\"problem\" id=\"fs-id1167836410214\">\r\n<p id=\"fs-id1167836410216\">\\(\\frac{3}{4},-\\frac{3}{4},1\\frac{2}{3},-1\\frac{2}{3},\\frac{5}{2},-\\frac{5}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829719936\"><span data-type=\"media\" id=\"fs-id1167829719939\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted. From left to right, these are: minus 5 by 2, minus 1 and two thirds, minus 3 by 4, 3 by 4, 1 and two thirds, and 5 by 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_309_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted. From left to right, these are: minus 5 by 2, minus 1 and two thirds, minus 3 by 4, 3 by 4, 1 and two thirds, and 5 by 2.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829719955\">\r\n<div data-type=\"problem\" id=\"fs-id1167829719957\">\r\n<p id=\"fs-id1167829719959\">\\(\\frac{2}{5},-\\frac{2}{5},1\\frac{3}{4},-1\\frac{3}{4},\\frac{8}{3},-\\frac{8}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829580180\">\r\n<div data-type=\"problem\" id=\"fs-id1167829580182\">\r\n<p id=\"fs-id1167829580184\"><span class=\"token\">\u24d0<\/span>\\(0.8\\)<span class=\"token\">\u24d1<\/span>\\(-1.25\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836399691\"><span data-type=\"media\" id=\"fs-id1167836399694\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 0 and 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_311_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 0 and 1.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836399709\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399711\">\r\n<p id=\"fs-id1167836399714\"><span class=\"token\">\u24d0<\/span>\\(-0.9\\)<span class=\"token\">\u24d1<\/span>\\(-2.75\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167829692233\">\r\n<div data-type=\"problem\" id=\"fs-id1167829692235\">\r\n<p id=\"fs-id1167829692237\"><span class=\"token\">\u24d0<\/span>\\(-1.6\\)<span class=\"token\">\u24d1<\/span>\\(3.25\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836399887\"><span data-type=\"media\" id=\"fs-id1167836399890\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 3 and 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_313_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 3 and 4.\" \/><\/span><\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836399904\">\r\n<div data-type=\"problem\" id=\"fs-id1167836399906\">\r\n<p id=\"fs-id1167836399908\"><span class=\"token\">\u24d0<\/span>\\(3.1\\)<span class=\"token\">\u24d1<\/span>\\(-3.65\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829627750\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1167829627757\">\r\n<div data-type=\"problem\" id=\"fs-id1167829627759\">\r\n<p id=\"fs-id1167829627761\">How does knowing about U.S. money help you learn about decimals?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167829627766\">\r\n<p id=\"fs-id1167829627768\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836538664\">\r\n<div data-type=\"problem\" id=\"fs-id1167836538666\">\r\n<p id=\"fs-id1167836538668\">When the Szetos sold their home, the selling price was 500% of what they had paid for the house 30 years ago. Explain what 500% means in this context.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836538683\">\r\n<div data-type=\"problem\" id=\"fs-id1167836538685\">\r\n<p id=\"fs-id1167836538687\">In your own words, explain the difference between a rational number and an irrational number.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1167836538692\">\r\n<p id=\"fs-id1167836538694\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1167836538700\">\r\n<div data-type=\"problem\" id=\"fs-id1167836538702\">\r\n<p id=\"fs-id1167836538704\">Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833240026\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1167833240032\"><span class=\"token\">\u24d0<\/span> Use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1167833240046\" data-alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The statements in the first column are: round decimals, add and subtract decimals, multiply and divide decimals, convert decimals, fractions and percents, simplify expressions with square roots, identify integers, rational numbers, irrational numbers and real numbers, locate fractions and decimals on the number line. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_215_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The statements in the first column are: round decimals, add and subtract decimals, multiply and divide decimals, convert decimals, fractions and percents, simplify expressions with square roots, identify integers, rational numbers, irrational numbers and real numbers, locate fractions and decimals on the number line. The remaining columns are blank.\" \/><\/span>\r\n<p id=\"fs-id1167833240055\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl id=\"fs-id1167833240069\">\r\n \t<dt>irrational number<\/dt>\r\n \t<dd id=\"fs-id1167833240074\">An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167836398934\">\r\n \t<dt>percent<\/dt>\r\n \t<dd id=\"fs-id1167836398939\">A percent is a ratio whose denominator is 100.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167836398944\">\r\n \t<dt>principal square root<\/dt>\r\n \t<dd id=\"fs-id1167836398949\">The positive square root is called the principal square root.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167836398953\">\r\n \t<dt>rational number<\/dt>\r\n \t<dd id=\"fs-id1167836398959\">A rational number is a number of the form \\(\\frac{p}{q},\\) where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and \\(q\\ne 0.\\) Its decimal form stops or repeats.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167829690613\">\r\n \t<dt>real number<\/dt>\r\n \t<dd id=\"fs-id1167829690618\">A real number is a number that is either rational or irrational.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167829690622\">\r\n \t<dt>square of a number<\/dt>\r\n \t<dd id=\"fs-id1167829690628\">If \\({n}^{2}=m,\\) then <em data-effect=\"italics\">m<\/em> is the square of <em data-effect=\"italics\">n<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167829690657\">\r\n \t<dt>square root of a number<\/dt>\r\n \t<dd id=\"fs-id1167836700140\">If \\({n}^{2}=m,\\) then <em data-effect=\"italics\">n<\/em> is a square root of <em data-effect=\"italics\">m<\/em>.<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li>Round decimals<\/li>\n<li>Add and subtract decimals<\/li>\n<li>Multiply and divide decimals<\/li>\n<li>Convert decimals, fractions, and percents<\/li>\n<li>Simplify expressions with square roots<\/li>\n<li>Identify integers, rational numbers, irrational numbers, and real numbers<\/li>\n<li>Locate fractions and decimals on the number line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833056324\" class=\"be-prepared\">\n<p id=\"fs-id1167833047251\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Round Decimals<\/h3>\n<p id=\"fs-id1167836545783\">Decimals are another way of writing fractions whose denominators are powers of ten.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836608668\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd31a123e7534c92f0cf8097bd3e2dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#46;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#96;&#96;&#111;&#110;&#101;&#32;&#116;&#101;&#110;&#116;&#104;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#46;&#48;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#96;&#96;&#111;&#110;&#101;&#32;&#104;&#117;&#110;&#100;&#114;&#101;&#100;&#116;&#104;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#46;&#48;&#48;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#96;&#96;&#111;&#110;&#101;&#32;&#116;&#104;&#111;&#117;&#115;&#97;&#110;&#100;&#116;&#104;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#46;&#48;&#48;&#48;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#44;&#48;&#48;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#96;&#96;&#111;&#110;&#101;&#32;&#116;&#101;&#110;&#45;&#116;&#104;&#111;&#117;&#115;&#97;&#110;&#100;&#116;&#104;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"372\" style=\"vertical-align: -41px;\" \/><\/div>\n<p id=\"fs-id1167836334129\">Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. <a href=\"#CNX_IntAlg_Figure_01_04_023\" class=\"autogenerated-content\">(Figure)<\/a> shows the names of the place values to the left and right of the decimal point.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_023\"><span data-type=\"media\" id=\"fs-id1167829692275\" data-alt=\"This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_023-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table is labeled place value and has 12 columns. The seventh column is blank. Starting from here and going left the columns are labeled: ones, tens, hundreds, thousands, ten thousands, hundred thousands. Starting from the blank column and going right the columns are labeled: tenths, hundredths, thousandths, ten thousandths hundred thousandths. There is a dot under the blank column.\" \/><\/span><\/div>\n<p id=\"fs-id1167836578683\">When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836543296\" class=\"howto\">\n<div data-type=\"title\">Round decimals.<\/div>\n<ol id=\"fs-id1167836323236\" class=\"stepwise\" type=\"1\">\n<li>Locate the given place value and mark it with an arrow.<\/li>\n<li>Underline the digit to the right of the place value.<\/li>\n<li>Is the underlined digit greater than or equal to 5?\n<ul data-bullet-style=\"bullet\">\n<li>Yes: add 1 to the digit in the given place value.<\/li>\n<li>No: do <u data-effect=\"underline\">not<\/u> change the digit in the given place value<\/li>\n<\/ul>\n<\/li>\n<li>Rewrite the number, deleting all digits to the right of the rounding digit.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836309901\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836525533\">\n<div data-type=\"problem\" id=\"fs-id1167836622858\">\n<p id=\"fs-id1167836699130\">Round 18.379 to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829697149\">\n<p id=\"fs-id1167836506417\">Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd11de43d54273c6b39f3505e8d30d22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#46;&#51;&#55;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/p>\n<p><span class=\"token\">\u24d0<\/span> to the nearest hundredth<\/p>\n<table class=\"unnumbered unstyled\" summary=\"Locate the hundredths place with an arrow. In the number 18.379, the arrow points to 7. Underline the digit to the right of the given place value. Underline 9. Because 9 is greater than or equal to 5, add 1 to the 7. Rewrite the number, deleting all digits to the right of the rounding digit. Now we have 18.38. Notice that the deleted digits were NOT replaced with zeros. So, 18.379 rounded to the nearest hundredth is 18.38.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate the hundredths place with an arrow.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836550242\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the given<\/p>\n<div data-type=\"newline\"><\/div>\n<p>place value.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289202\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Because 9 is greater than or equal to 5, add 1 to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the 7.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836392196\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>right of the rounding digit.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515429\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Notice that the deleted digits were NOT<\/p>\n<div data-type=\"newline\"><\/div>\n<p>replaced with zeros.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836706605\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_001e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836416714\"><span class=\"token\">\u24d1<\/span> to the nearest tenth<\/p>\n<table id=\"fs-id1167829691732\" class=\"unnumbered unstyled can-break\" summary=\"Locate the tenths place with an arrow. In the number 18.379, the arrow points to 3. Underline the digit to the right of the given place value. Underline 7. Because 7 is greater than or equal to 5, add 1 to the 3. Rewrite the number, deleting all digits to the right of the rounding digit. Now we have 18.4. Notice that the deleted digits were NOT replaced with zeros. So, 18.379 rounded to the nearest tenth is 18.4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate the tenths place with an arrow.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836334185\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>given place value.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833022051\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Because 7 is greater than or equal to 5,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>add 1 to the 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829598222\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the right of the rounding digit.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Notice that the deleted digits were NOT<\/p>\n<div data-type=\"newline\"><\/div>\n<p>replaced with zeros.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836319802\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_002e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836620890\"><span class=\"token\">\u24d2<\/span> to the nearest whole number<\/p>\n<table id=\"fs-id1167836294764\" class=\"unnumbered unstyled\" summary=\"Locate the ones place with an arrow. In the number 18.379, the arrow points to 8. Underline the digit to the right of the given place value. Underline 3. Since 3 is not greater than or equal to 5, do not add 1 to 8. Rewrite the number, deleting all digits to the right of the rounding digit. So, 18.379 rounded to the nearest whole number is 18.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate the ones place with an arrow.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836376014\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Underline the digit to the right of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>given place value.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since 3 is not greater than or equal to 5,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>do not add 1 to the 8.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836329514\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the number, deleting all digits to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the right of the rounding digit.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836296579\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836407551\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_003e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836414960\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836515968\">\n<div data-type=\"problem\" id=\"fs-id1167836530505\">\n<p id=\"fs-id1167836512053\">Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7195cbc57fc89ea93f94d2f085cf2aac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#53;&#56;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836624764\">\n<p id=\"fs-id1167836526857\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3b1e65b3a4c13960fc407b60ec561bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#53;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40a011ebaaf254bf7bb08f5797139d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span> 7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836415391\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829694242\">\n<div data-type=\"problem\" id=\"fs-id1167836554945\">\n<p id=\"fs-id1167836367818\">Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c2555a3ee9b66fe21410404e0578ddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#46;&#50;&#49;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/> to the nearest <span class=\"token\">\u24d0<\/span> thousandth <span class=\"token\">\u24d1<\/span> hundredth <span class=\"token\">\u24d2<\/span> tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836626479\">\n<p id=\"fs-id1167833023132\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfb8e79b1051ec6c68c5f536805f71df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#46;&#50;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-704acad8d99e905819ce0cc442773ab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#46;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b6e389d07179c8315c2bb62b4456ffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836392732\">\n<h3 data-type=\"title\">Add and Subtract Decimals<\/h3>\n<p id=\"fs-id1167836597180\">To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836334612\" class=\"howto\">\n<div data-type=\"title\">Add or subtract decimals.<\/div>\n<ol id=\"fs-id1167829590511\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the sum or difference.<\/li>\n<li>Write the numbers so the decimal points line up vertically.<\/li>\n<li>Use zeros as placeholders, as needed.<\/li>\n<li>Add or subtract the numbers as if they were whole numbers. Then place the\n<div data-type=\"newline\"><\/div>\n<p>decimal point in the answer under the decimal points in the given numbers.<\/li>\n<li>Write the sum or difference with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167833059005\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836537217\">\n<div data-type=\"problem\" id=\"fs-id1167836366635\">\n<p id=\"fs-id1167836605234\">Add or subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ecf40eb2e50d4234b45394b0f524ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#51;&#46;&#53;&#45;&#52;&#49;&#46;&#51;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"107\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae814b914dba9e1a9640b0d771cccebd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#46;&#54;&#53;&#45;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836556022\">\n<p id=\"fs-id1167836606910\"><span class=\"token\">\u24d0<\/span><\/p>\n<\/p>\n<div data-type=\"newline\"><\/div>\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#50;&#51;&#46;&#53;&#45;&#52;&#49;&#46;&#51;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#46;&#32;&#84;&#111;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#44;&#32;&#119;&#101;&#32;&#97;&#100;&#100;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#117;&#109;&#101;&#114;&#97;&#108;&#115;&#46;&#32;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#115;&#111;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#32;&#108;&#105;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#112;&#32;&#118;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#117;&#116;&#32;&#48;&#32;&#97;&#115;&#32;&#97;&#32;&#112;&#108;&#97;&#99;&#101;&#104;&#111;&#108;&#100;&#101;&#114;&#32;&#97;&#102;&#116;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#53;&#32;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#101;&#109;&#98;&#101;&#114;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#48;&#125;&#123;&#49;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#53;&#61;&#48;&#46;&#53;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#48;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#97;&#115;&#32;&#105;&#102;&#32;&#116;&#104;&#101;&#121;&#32;&#119;&#101;&#114;&#101;&#32;&#119;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#110;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#48;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#92;&#32;&#54;&#52;&#46;&#56;&#56;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#115;&#117;&#108;&#116;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#114;&#114;&#101;&#99;&#116;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#50;&#51;&#46;&#53;&#45;&#52;&#49;&#46;&#51;&#56;&#61;&#45;&#54;&#52;&#46;&#56;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;\r\n&#77;&#105;&#115;&#112;&#108;&#97;&#99;&#101;&#100;&#32;&#92;&#99;&#114;&#99;&#114;&#46;&#32;&#40;&#105;&#110;&#32;&#109;&#97;&#99;&#114;&#111;&#32;&#92;&#101;&#110;&#100;&#97;&#114;&#114;&#97;&#121;&#41;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#50;&#51;&#46;&#53;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#43;&#52;&#49;&#46;&#51;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n\n<\/pre>\n<p id=\"fs-id1167836312509\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8873b576f03064ac10970b078de036aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#52;&#46;&#54;&#53;&#45;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#46;&#32;&#84;&#111;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#44;&#32;&#119;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#49;&#52;&#46;&#54;&#53;&#32;&#102;&#114;&#111;&#109;&#32;&#50;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#115;&#111;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#32;&#108;&#105;&#110;&#101;&#32;&#117;&#112;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#45;&#49;&#52;&#46;&#54;&#53;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#101;&#109;&#98;&#101;&#114;&#44;&#32;&#50;&#48;&#32;&#105;&#115;&#32;&#97;&#32;&#119;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#44;&#32;&#115;&#111;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#32;&#97;&#102;&#116;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#117;&#116;&#32;&#105;&#110;&#32;&#122;&#101;&#114;&#111;&#115;&#32;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#114;&#105;&#103;&#104;&#116;&#32;&#97;&#115;&#32;&#112;&#108;&#97;&#99;&#101;&#104;&#111;&#108;&#100;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#46;&#48;&#48;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#45;&#49;&#52;&#46;&#54;&#53;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#97;&#110;&#100;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#49;&#48;&#125;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#49;&#48;&#125;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#115;&#117;&#108;&#116;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#114;&#114;&#101;&#99;&#116;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#52;&#46;&#54;&#53;&#45;&#50;&#48;&#61;&#45;&#53;&#46;&#51;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833057966\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836547290\">\n<div data-type=\"problem\" id=\"fs-id1167836620098\">\n<p id=\"fs-id1167836484787\">Add or subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe3ed54eb88b262f9bf8db601ef7bae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;&#56;&#45;&#49;&#49;&#46;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"98\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9665ff5faf6cf233757b7b4a9d48902_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#46;&#53;&#56;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"75\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836622008\">\n<p id=\"fs-id1167836283439\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cba5e14d6f74e54e0190450aa199b9ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#46;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0118a6b8720328f568719bce00b8edf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836627381\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836558181\">\n<div data-type=\"problem\" id=\"fs-id1167836391088\">\n<p id=\"fs-id1167833018605\">Add or subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a24fa40b314efa8bdbaebd3bf027e82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#46;&#49;&#50;&#51;&#45;&#49;&#56;&#46;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"116\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24dc8832e691f03e9e4461c81c17aa3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#46;&#52;&#50;&#45;&#53;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833051205\">\n<p id=\"fs-id1167836597977\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47ecd757bce98c368644c1644a1c76b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#51;&#46;&#53;&#57;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-392b7755d550eaac4f03730bf5d8e2aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#46;&#53;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829691951\">\n<h3 data-type=\"title\">Multiply and Divide Decimals<\/h3>\n<p id=\"fs-id1167836580114\">When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833019249\" class=\"howto\">\n<div data-type=\"title\">Multiply decimals.<\/div>\n<ol id=\"fs-id1167836688746\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the product.<\/li>\n<li>Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\n<li>Place the decimal point. The number of decimal places in the product is the sum of\n<div data-type=\"newline\"><\/div>\n<p>the number of decimal places in the factors.<\/li>\n<li>Write the product with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836524294\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836367150\">\n<div data-type=\"problem\" id=\"fs-id1167836484476\">\n<p id=\"fs-id1167836598179\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc36f19d89a064056c74456a700bee8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#46;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#48;&#55;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829718181\">\n<table id=\"fs-id1167836333804\" class=\"unnumbered unstyled\" summary=\"Multiply minus 3.9 with 4.075. The signs are different. The product will be negative. Write in vertical format, lining up the numbers on the right. Multiply to get 158925. Add the number of decimal places in the factors, 1 plus 3 equals 4. Place the decimal point 4 places from the right. The number becomes 15.8925. The signs of the factors are different, so the product is negative. The answer is minus 15.8925.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f32062cf05fccef29f13ce095a5bca24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#46;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#48;&#55;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The signs are different. The product<\/p>\n<div data-type=\"newline\"><\/div>\n<p>will be negative.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/td>\n<td data-valign=\"top\" data-align=\"left\">The product will be negative.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write in vertical format, lining up the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>numbers on the right.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507262\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836520918\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the number of decimal places in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the factors (1 + 3).<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Place the decimal point 4 places from the right.<\/td>\n<td rowspan=\"2\" data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836533297\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836507277\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_004a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The signs are the different, so the product is negative.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce96da5917974d3102b60233628d5d6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#46;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#48;&#55;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;&#46;&#56;&#57;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"202\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836406682\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836526133\">\n<div data-type=\"problem\" id=\"fs-id1167836521632\">\n<p id=\"fs-id1167829645141\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1537b26ba002db472eb63f72ba18cf45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#49;&#48;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836511260\">\n<p id=\"fs-id1167836507336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9357cbeb9d25deb71ef36193f8a83c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&#46;&#52;&#56;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836556209\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829716991\">\n<div data-type=\"problem\" id=\"fs-id1167836320388\">\n<p id=\"fs-id1167836704688\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dce5698f902e15858c0e78b7945ca71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#46;&#55;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#46;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836340427\">\n<p id=\"fs-id1167836693316\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0348acef69a998c00d746ef2d450247a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#55;&#46;&#54;&#49;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836359860\">Often, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836628738\" class=\"howto\">\n<div data-type=\"title\">Multiply a decimal by a power of ten.<\/div>\n<ol id=\"fs-id1167832940646\" class=\"stepwise\" type=\"1\">\n<li>Move the decimal point to the right the same number of places as the\n<div data-type=\"newline\"><\/div>\n<p>number of zeros in the power of 10.<\/li>\n<li>Add zeros at the end of the number as needed.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836408406\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833058013\">\n<div data-type=\"problem\" id=\"fs-id1167836321760\">\n<p id=\"fs-id1167833019684\">Multiply: 5.63 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836532215\">\n<p id=\"fs-id1167836688719\">By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.<\/p>\n<p id=\"fs-id1167836546356\"><span class=\"token\">\u24d0<\/span><\/p>\n<table id=\"fs-id1167829721047\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 10. There is 1 zero in 10. So move the decimal point 1 place to the right, to get 56.3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829693057\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">There is 1 zero in 10, so move the decimal point 1 place to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836513010\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833025149\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_005c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836399856\"><span class=\"token\">\u24d1<\/span><\/p>\n<table id=\"fs-id1167836525944\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 100. There are 2 zeroes in 100. So move the decimal point 2 places to the right, to get 563.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509774\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">There are 2 zeroes in 100, so move the decimal point 2 places to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836693882\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524191\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_006c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836309296\"><span class=\"token\">\u24d2<\/span><\/p>\n<table id=\"fs-id1167836408573\" class=\"unnumbered unstyled\" summary=\"Multiply 5.63 by 1000. There are 3 zeroes in 100. So move the decimal point 2 places to the right. A zero must be added to the end to get 5630.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836531255\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">There are 3 zeroes in 1,000, so move the decimal point 3 place to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836522133\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">A zero must be added to the end.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833060124\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829597736\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836547160\">\n<div data-type=\"problem\" id=\"fs-id1167836547163\">\n<p id=\"fs-id1167836313834\">Multiply 2.58 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836510292\">\n<p id=\"fs-id1167833056159\"><span class=\"token\">\u24d0<\/span> 25.8 <span class=\"token\">\u24d1<\/span> 258 <span class=\"token\">\u24d2<\/span> 2,580<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836545249\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836522719\">\n<div data-type=\"problem\" id=\"fs-id1167836626486\">\n<p id=\"fs-id1167836626488\">Multiply 14.2 by <span class=\"token\">\u24d0<\/span> 10 <span class=\"token\">\u24d1<\/span> 100 <span class=\"token\">\u24d2<\/span> 1000.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518934\">\n<p id=\"fs-id1167836518936\"><span class=\"token\">\u24d0<\/span> 142 <span class=\"token\">\u24d1<\/span> 1,420 <span class=\"token\">\u24d2<\/span> 14,200<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836546040\">Just as with multiplication, division of signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.<\/p>\n<p id=\"fs-id1167836518923\">We review the notation and vocabulary for division:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836292139\" data-alt=\"In the expression a divided by b equals c, a is the dividend, b is the divisor and c is the quotient. This can be written as b right parentheses a overbar, with c on top of the bar. In this case too, a is the dividend, b is the divisor and c is the quotient.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_024_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the expression a divided by b equals c, a is the dividend, b is the divisor and c is the quotient. This can be written as b right parentheses a overbar, with c on top of the bar. In this case too, a is the dividend, b is the divisor and c is the quotient.\" \/><\/span><\/p>\n<p id=\"fs-id1167836288209\">We\u2019ll write the steps to take when dividing decimals for easy reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836521247\" class=\"howto\">\n<div data-type=\"title\">Divide decimals.<\/div>\n<ol id=\"fs-id1167836389529\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the quotient.<\/li>\n<li>Make the divisor a whole number by \u201cmoving\u201d the decimal point all the way to the right. \u201cMove\u201d the decimal point in the dividend the same number of places\u2014adding zeros as needed.<\/li>\n<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\n<li>Write the quotient with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836544303\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836504089\">\n<div data-type=\"problem\" id=\"fs-id1167836492443\">\n<p id=\"fs-id1167836492445\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d7eaf1224472c91482d783d15e59e9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#46;&#54;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#48;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417299\">\n<p id=\"fs-id1167836521102\">Remember, you can \u201cmove\u201d the decimals in the divisor and dividend because of the Equivalent Fractions Property.<\/p>\n<table id=\"fs-id1167829717572\" class=\"unnumbered unstyled\" summary=\"Minus 25.65 divided by minus 0.06. The signs are the same, so the quotient is positive. Make the divisor a whole number by moving the decimal all the way to the right. Move the decimal point in the dividend the same number of places. Hence the divisor is 6 and the dividend is 2565. Divide, placing the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. The quotient is 427.5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836551421\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The signs are the same.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The quotient is positive.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Make the divisor a whole number by \u201cmoving\u201d the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>decimal point all the way to the right.<\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u201cMove\u201d the decimal point in the dividend the same<\/p>\n<div data-type=\"newline\"><\/div>\n<p>number of places.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836515519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Place the decimal point in the quotient above the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>decimal point in the dividend.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829719619\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the quotient with the appropriate sign.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836694132\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_008d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836628735\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836447446\">\n<div data-type=\"problem\" id=\"fs-id1167836692990\">\n<p id=\"fs-id1167836692992\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09c428c58e778efc58141d2f8281bde2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#51;&#46;&#52;&#57;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#48;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836418428\">\n<p id=\"fs-id1167836334903\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-250ae9ecdbd0c83dedf6537b1f54885a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#56;&#55;&#46;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836635212\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836585179\">\n<div data-type=\"problem\" id=\"fs-id1167836621352\">\n<p id=\"fs-id1167836621355\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-462e0477e60a55141de4076659ec808e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;&#49;&#49;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836407678\">\n<p id=\"fs-id1167836800480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09a61c663feec433590d173c7a60a99f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#52;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836518782\">\n<h3 data-type=\"title\">Convert Decimals, Fractions, and Percents<\/h3>\n<p id=\"fs-id1167833397224\">In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.<\/p>\n<p id=\"fs-id1167829692288\">We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80f06cf0cd3c7b48a1ebefb8894b3fcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/> the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836509769\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95fef59c7b1c6ba0022941b7c9763bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1167836700940\">The steps to take to convert a decimal to a fraction are summarized in the procedure box.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836700546\" class=\"howto\">\n<div data-type=\"title\">Convert a decimal to a proper fraction and a fraction to a decimal.<\/div>\n<ol id=\"fs-id1167836509806\" class=\"stepwise\" type=\"1\">\n<li>To convert a decimal to a proper fraction, determine the place value of the final digit.<\/li>\n<li>Write the fraction.\n<ul id=\"fs-id1167829905871\" data-bullet-style=\"bullet\">\n<li>numerator\u2014the \u201cnumbers\u201d to the right of the decimal point<\/li>\n<li>denominator\u2014the place value corresponding to the final digit<\/li>\n<\/ul>\n<\/li>\n<li>To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836533353\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836533355\">\n<div data-type=\"problem\" id=\"fs-id1167836684088\">\n<p id=\"fs-id1167836684090\">Write: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fb72f5a72eca5b99b44cb6e7adebb70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#51;&#55;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -1px;\" \/> as a fraction <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f8db5feff6785204c5d807666d4af85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> as a decimal.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836700931\">\n<p id=\"fs-id1167836628604\"><span class=\"token\">\u24d0<\/span><\/p>\n<table id=\"fs-id1167836698909\" class=\"unnumbered unstyled\" summary=\"In the number 0.374, determine the place value of the final digit. The place value of 3 is tenths, that of 7 is hundredths and that of 4 is thousandths. Write the fraction for 0.374. The numerator is 374. The denominator is 1000. Simplify the fraction. 374 is twice 187 and 1000 is twice 500. Dividing out the common factors, we get 187 by 500. So, 0.374 is 187 by 500.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635361\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Determine the place value of the final digit.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836579270\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the fraction for 0.374:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The numerator is 374.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The denominator is 1,000.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832999068\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509683\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide out the common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836704756\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836456246\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_009f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829596870\"><span class=\"token\">\u24d1<\/span> Since a fraction bar means division, we begin by writing the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44cf49c06a351926440bb2ad833776c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ad7454fda2130450973ed30f5379acc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#53;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"30\" style=\"vertical-align: -4px;\" \/> Now divide.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836619319\" data-alt=\"The division shows that 5 is divided by 8 to yield 0.625. The result concludes that five eights is equal to negative 0.625.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_010_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The division shows that 5 is divided by 8 to yield 0.625. The result concludes that five eights is equal to negative 0.625.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836699375\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829692448\">\n<div data-type=\"problem\" id=\"fs-id1167836704215\">\n<p id=\"fs-id1167836704217\">Write: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a91f5940295de948d0cff9dd2af28179_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#51;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> as a fraction <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43467322c65033e36c92ea885a66c85c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> as a decimal.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833058998\">\n<p id=\"fs-id1167833059000\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fea94e81c9c8d72801aa04b0d626041_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#55;&#125;&#123;&#53;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d079f44bac785047ece87b816dc283c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#56;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836697042\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836627841\">\n<div data-type=\"problem\" id=\"fs-id1167836627843\">\n<p id=\"fs-id1167836627845\">Write: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a957ff7487844045e77da7df98ec0fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> as a fraction <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6227e12992966bd4415b4db6f7a63a93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> as a decimal.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620435\">\n<p id=\"fs-id1167836620437\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38e14a0a2ab5f89a362377f0548e4022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a4e94c522264239aef6473626e5cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#51;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829593967\">A <span data-type=\"term\">percent<\/span> is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent. Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator. After doing this many times, you may see the pattern.<\/p>\n<p id=\"fs-id1167836652793\"><em data-effect=\"italics\">To convert a percent number to a decimal number, we move the decimal point two places to the left.<\/em><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836305345\" data-alt=\"Figure shows the value 6 percent. An arrow indicates that the decimal is moved two places to the left. Hence the value is equal to 0.06. Similarly, 78 percent is 0.78, 2.7 percent is 0. 027 and 135 percent is 1.35.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_011_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the value 6 percent. An arrow indicates that the decimal is moved two places to the left. Hence the value is equal to 0.06. Similarly, 78 percent is 0.78, 2.7 percent is 0. 027 and 135 percent is 1.35.\" \/><\/span><\/p>\n<p id=\"fs-id1167836532690\">To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent. After many conversions, you may recognize the pattern.<\/p>\n<p id=\"fs-id1167836630567\"><em data-effect=\"italics\">To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.<\/em><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836493675\" data-alt=\"Figure shows value 0.05. An arrow indicates that the decimal is moved two places to the right. Hence the value becomes 5 percent. Similarly, 0.83 is 83 percent, 1.05 is 105 percent, 0.075 is 7.5 percent and 0.3 is 30 percent.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_012_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows value 0.05. An arrow indicates that the decimal is moved two places to the right. Hence the value becomes 5 percent. Similarly, 0.83 is 83 percent, 1.05 is 105 percent, 0.075 is 7.5 percent and 0.3 is 30 percent.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167836551698\" class=\"howto\">\n<div data-type=\"title\">Convert a percent to a decimal and a decimal to a percent.<\/div>\n<ol id=\"fs-id1167829683890\" class=\"stepwise\" type=\"1\">\n<li>To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.<\/li>\n<li>To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829721053\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829721055\">\n<div data-type=\"problem\" id=\"fs-id1167829721057\">\n<p id=\"fs-id1167836549775\">Convert each:<\/p>\n<p id=\"fs-id1167836549778\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 62%, 135%, and 13.7%.<\/p>\n<p id=\"fs-id1167833055334\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.51, 1.25, and 0.093.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833224661\">\n<p id=\"fs-id1167833224663\"><span class=\"token\">\u24d0<\/span><\/p>\n<table id=\"fs-id1167836699430\" class=\"unnumbered unstyled\" summary=\"The given values are 62 percent, 135 percent and 35.7 percent. Moving the decimal point two places to the left in each case, we get 0.62, 1.35 and 0.357.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836620094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Move the decimal point two places to the left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833057802\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829718691\"><span class=\"token\">\u24d1<\/span><\/p>\n<table id=\"fs-id1167836477216\" class=\"unnumbered unstyled\" summary=\"The given values are 0.51, 1.25 and 0.093. Moving the decimal point two places to the right in each case, we get 51 percent, 125 percent and 9.3 percent.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833058770\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Move the decimal point two places to the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836635191\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836701289\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836477822\">\n<div data-type=\"problem\" id=\"fs-id1167836477824\">\n<p id=\"fs-id1167836477826\">Convert each:<\/p>\n<p id=\"fs-id1167833224433\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 9%, 87%, and 3.9%.<\/p>\n<p id=\"fs-id1167829719411\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.17, 1.75, and 0.0825.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833021569\">\n<p id=\"fs-id1167829719350\"><span class=\"token\">\u24d0<\/span> 0.09, 0.87, 0.039 <span class=\"token\">\u24d1<\/span> 17%, 175%, 8.25%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829716469\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836494036\">\n<div data-type=\"problem\" id=\"fs-id1167836494039\">\n<p id=\"fs-id1167836494041\">Convert each:<\/p>\n<p id=\"fs-id1167836508314\"><span class=\"token\">\u24d0<\/span> percent to a decimal: 3%, 91%, and 8.3%.<\/p>\n<p id=\"fs-id1167829628364\"><span class=\"token\">\u24d1<\/span> decimal to a percent: 0.41, 2.25, and 0.0925.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417176\">\n<p id=\"fs-id1167832982286\"><span class=\"token\">\u24d0<\/span> 0.03, 0.91, 0.083 <span class=\"token\">\u24d1<\/span> 41%, 225%, 9.25%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836609894\">\n<h3 data-type=\"title\">Simplify Expressions with Square Roots<\/h3>\n<p id=\"fs-id1167833396879\">Remember that when a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is multiplied by itself, we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-444500c4e43e38956dd248f19f2f7129_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/> and read it \u201c<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> squared.\u201d The result is called the <span data-type=\"term\">square of a number<\/span> <em data-effect=\"italics\">n<\/em>. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c07e84fa47c71df6d6c4bc1c2208e8e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#56;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> is read \u201c8 squared\u201d and 64 is called the <em data-effect=\"italics\">square<\/em> of 8. Similarly, 121 is the square of 11 because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c98915ee12cb78b8c6e7c5c2b937fb3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#49;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -1px;\" \/> is 121. It will be helpful to learn to recognize the perfect square numbers.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836699183\">\n<div data-type=\"title\">Square of a number<\/div>\n<p id=\"fs-id1167829579161\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">m<\/em> is the <strong data-effect=\"bold\">square<\/strong> of <em data-effect=\"italics\">n<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167833061164\">What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836356337\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbd08566828af26b028852bf23f197c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"500\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167833054715\">Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c465a5fd7db52cdd10635f242e06d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -4px;\" \/> we say 100 is the square of 10. We also say that 10 is a <em data-effect=\"italics\">square root<\/em> of 100. A number whose square is <em data-effect=\"italics\">m<\/em> is called a <span data-type=\"term\">square root of a number<\/span> <em data-effect=\"italics\">m<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836406664\">\n<div data-type=\"title\">Square Root of a Number<\/div>\n<p id=\"fs-id1167836530323\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">n<\/em> is a <strong data-effect=\"bold\">square root<\/strong> of <em data-effect=\"italics\">m<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167836295018\">Notice <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd3ccc26a4a88f5a84134b6afc2774d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/> also, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> is also a square root of 100. Therefore, both 10 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> are square roots of 100. So, every positive number has two square roots\u2014one positive and one negative. The radical sign, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/>, denotes the positive square root. The positive square root is called the <span data-type=\"term\">principal square root<\/span>. When we use the radical sign that always means we want the principal square root.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833022144\">\n<div data-type=\"title\">Square Root Notation<\/div>\n<p id=\"fs-id1167836707363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/> is read \u201cthe square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/>.\u201d<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829715458\" data-alt=\"Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_015_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows the expression square root of m. The square root sign is labeled radical sign and m is labeled radicand.\" \/><\/span><\/p>\n<p id=\"fs-id1167836549955\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c7c8991cf54ee68b9a5ca8f8014af5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144c0e1f4a31eb91b31a23122c558c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#110;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de0b3f268c78cbcbba48c42081539c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/p>\n<p id=\"fs-id1167836620491\">The square root of <em data-effect=\"italics\">m<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2447480eb926a3c6e489624b0a8579e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -4px;\" \/> is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/p>\n<\/div>\n<p id=\"fs-id1167836701072\">We know that every positive number has two square roots and the radical sign indicates the positive one. We write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5687e7adbc139c31409a60b7c14f986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -2px;\" \/> If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-798ebec064d3bb415cb2ad8c894da954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -2px;\" \/> We read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0e99e0fb77048431e44e24f06f05afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> as \u201cthe opposite of the principal square root of 10.\u201d<\/p>\n<div data-type=\"example\" id=\"fs-id1167833056590\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833056593\">\n<div data-type=\"problem\" id=\"fs-id1167833021974\">\n<p id=\"fs-id1167833021976\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c6a36f543576cfd4fa34a645e3eff65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7472a15648ee154629be2bf540389b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-116046b7e27ef833675d1aa25d82a6ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836559725\">\n<p id=\"fs-id1167829717722\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdce8c3b13f044d0b87a0194913682ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"272\" style=\"vertical-align: -11px;\" \/><\/p>\n<p id=\"fs-id1167836550052\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c524cd0f838cf34e827bff3d0592b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#49;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"281\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1167836487508\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4490bfb68d6c76ef36ad56509c9aed38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#32;&#105;&#115;&#32;&#105;&#110;&#32;&#102;&#114;&#111;&#110;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"293\" style=\"vertical-align: -25px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836287231\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836287235\">\n<div data-type=\"problem\" id=\"fs-id1167836625114\">\n<p id=\"fs-id1167836625116\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98077b985c4d3e263c7a4b085c65329f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd4c05e7550847a7adda1c07adcd62f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d6067ddb8f55315b083735f09fc51f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836551671\">\n<p id=\"fs-id1167829720135\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> 13 <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833175426\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833175430\">\n<div data-type=\"problem\" id=\"fs-id1167833020312\">\n<p id=\"fs-id1167833020314\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a782559bab50d90947de7be011d6866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-326c826152ee55e8686d89ebc6a491e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-281bbffdc5cab7d5b04d65f4bede8037_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836547264\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 14 <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836492536\">\n<h3 data-type=\"title\">Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/h3>\n<p id=\"fs-id1167833049933\">We have already described numbers as <em data-effect=\"italics\">counting number<\/em><em data-effect=\"italics\">s<\/em>, <em data-effect=\"italics\">whole number<\/em><em data-effect=\"italics\">s<\/em>, and <em data-effect=\"italics\">integers<\/em>. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish between these types of numbers?<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836800466\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fe5c952508acda6dc99a4ec1647994a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#117;&#110;&#116;&#105;&#110;&#103;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#46;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"367\" style=\"vertical-align: -26px;\" \/><\/div>\n<p>What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A <span data-type=\"term\">rational number<\/span> is a number that can be written as a ratio of two integers.<\/p>\n<p id=\"fs-id1167829715480\">In general, any decimal that ends after a number of digits (such as 7.3 or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfc628917a3e1eb86891becfb753a5ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#50;&#54;&#56;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: -1px;\" \/>) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. The decimal for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is the number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45f5a975ac4244253251b41a5e17efec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -1px;\" \/> The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.<\/p>\n<p id=\"fs-id1167829717443\">Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.<\/p>\n<p id=\"fs-id1167833022517\"><em data-effect=\"italics\">Every rational number can be written both as a ratio of integers<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2234e67f0d02bee8d24190a2ade2e6be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -9px;\" \/><em data-effect=\"italics\">where p and q are integers and<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7926ff049b2f8d8ca2771a6474084a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">and as a decimal that stops or repeats.<\/em><\/p>\n<div data-type=\"note\" id=\"fs-id1167836536548\">\n<div data-type=\"title\">Rational Number<\/div>\n<p id=\"fs-id1167836536554\">A <strong data-effect=\"bold\">rational number<\/strong> is a number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2234e67f0d02bee8d24190a2ade2e6be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -9px;\" \/> where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e417a0769b70c865e6dde403ebb92108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836698900\">Its decimal form stops or repeats.<\/p>\n<\/div>\n<p id=\"fs-id1167836698905\">Are there any decimals that do not stop or repeat? Yes! The number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26d6788550ffd50fe94542bb3e8ee615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> (the Greek letter <em data-effect=\"italics\">pi<\/em>, pronounced \u201cpie\u201d), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (\u2026) to indicate the decimal does not stop or repeat.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829719570\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5843d8baec38be0666138aeff0da0ffd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;&#61;&#51;&#46;&#49;&#52;&#49;&#53;&#57;&#50;&#54;&#53;&#52;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"143\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"fs-id1167836579988\">The square root of a number that is not a perfect square is a decimal that does not stop or repeat.<\/p>\n<p id=\"fs-id1167836579992\">A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an <span data-type=\"term\">irrational number<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167832981620\">\n<div data-type=\"title\">Irrational Number<\/div>\n<p id=\"fs-id1167836607383\">An <strong data-effect=\"bold\">irrational number<\/strong> is a number that cannot be written as the ratio of two integers.<\/p>\n<p id=\"fs-id1167836530052\">Its decimal form does not stop and does not repeat.<\/p>\n<\/div>\n<p id=\"fs-id1167836530057\">Let\u2019s summarize a method we can use to determine whether a number is rational or irrational.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829715600\">\n<div data-type=\"title\">Rational or Irrational<\/div>\n<p id=\"fs-id1167836635411\">If the decimal form of a number<\/p>\n<ul id=\"fs-id1167836635415\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is a <strong data-effect=\"bold\">rational number<\/strong>.<\/li>\n<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is an <strong data-effect=\"bold\">irrational number.<\/strong><\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167833240081\">We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of <span data-type=\"term\">real number<\/span><strong data-effect=\"bold\">s<\/strong>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829719876\">\n<div data-type=\"title\">Real Number<\/div>\n<p id=\"fs-id1167829717421\">A <strong data-effect=\"bold\">real number<\/strong> is a number that is either rational or irrational.<\/p>\n<\/div>\n<p id=\"fs-id1167836698829\">Later in this course we will introduce numbers beyond the real numbers. <a href=\"#CNX_IntAlg_Figure_01_04_016\" class=\"autogenerated-content\">(Figure)<\/a> illustrates how the number sets we\u2019ve used so far fit together.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_016\">\n<div class=\"bc-figcaption figcaption\">This chart shows the number sets that make up the set of real numbers.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836447242\" data-alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_016-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167829624381\">Does the term \u201creal numbers\u201d seem strange to you? Are there any numbers that are not \u201creal,\u201d and, if so, what could they be? Can we simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-106759f3f92dcbbd85e8e38a55b5c608_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -3px;\" \/> Is there a number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61e2fb1fc3a33032fc37d9990d7d438c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167832981655\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac0afda07e9c59744283048b0a226380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#50;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167833054419\">None of the numbers that we have dealt with so far has a square that is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e362d973b8b62a09dead6c6e5f8812d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/> Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-423095b79830dafcaa8969f96f2894f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/> The square root of a negative number is not a real number.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836546317\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836546320\">\n<div data-type=\"problem\" id=\"fs-id1167836526165\">\n<p id=\"fs-id1167836526168\">Given the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d21edc29ff6bf3d6a7a2bb950c01eb97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#44;&#56;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;&#53;&#46;&#57;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"170\" style=\"vertical-align: -6px;\" \/> list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers <span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836580197\">\n<p id=\"fs-id1167836580199\"><span class=\"token\">\u24d0<\/span> Remember, the whole numbers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34860bfcc45d45dcb9b93dfc8b54f0ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -4px;\" \/> so 8 is the only whole number given.<\/p>\n<p id=\"fs-id1167833018577\"><span class=\"token\">\u24d1<\/span> The integers are the whole numbers and their opposites (which includes 0). So the whole number 8 is an integer, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa35c0a4f962b2e1496c031076421eaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#61;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -2px;\" \/> So the integers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fac71070730e0ef690fb3b0398db7f57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1690bafb5e6a0ecda7d31800ddceeeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167836700847\"><span class=\"token\">\u24d2<\/span> Since all integers are rational, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fac71070730e0ef690fb3b0398db7f57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d128c7fbe583eaf26031f2f9eccf6630_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> are rational. Rational numbers also include fractions and decimals that repeat or stop, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d18a4913fb16a1657114075f3b808607_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e7f0f1cca05a76bd49bd118365ff6b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> are rational. So the list of rational numbers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ca0be316c8d115735cd402d9bbfab0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#44;&#56;&#44;&#53;&#46;&#57;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1690bafb5e6a0ecda7d31800ddceeeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167829721168\"><span class=\"token\">\u24d3<\/span> Remember that 5 is not a perfect square, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ba329d3f2f86c0fb47d848202e0ee7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> is irrational.<\/p>\n<p id=\"fs-id1167829717682\"><span class=\"token\">\u24d4<\/span> All the numbers listed are real numbers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836525716\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836525721\">\n<div data-type=\"problem\" id=\"fs-id1167829716771\">\n<p id=\"fs-id1167829716773\">Given the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37a286193bb6712fb6114aa3bf664008_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" style=\"vertical-align: -6px;\" \/> list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829716001\">\n<p id=\"fs-id1167829693412\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acbce00f4d6bdd2ab7e2f9f29f41aa81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"50\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78feb015bf7d488e9d493ee526306cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9d07149b8fa7b95bfec6633e8f83caa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"135\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75f09696be431c541397f32b7d6eb36a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9787a326696d2bbdbb55fe16450059_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;&#44;&#52;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"167\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832981706\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832981709\">\n<div data-type=\"problem\" id=\"fs-id1167832981711\">\n<p id=\"fs-id1167832981713\">Given numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23541efc8ad5a3165bb5fd319f28c91f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#44;&#45;&#49;&#44;&#54;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;&#44;&#50;&#46;&#48;&#52;&#49;&#57;&#55;&#53;&#46;&#46;&#46;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"255\" style=\"vertical-align: -6px;\" \/> list the <span class=\"token\">\u24d0<\/span> whole numbers <span class=\"token\">\u24d1<\/span> integers <span class=\"token\">\u24d2<\/span> rational numbers <span class=\"token\">\u24d3<\/span> irrational numbers <span class=\"token\">\u24d4<\/span> real numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836578852\">\n<p id=\"fs-id1167836578854\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31ed32024b40aa7576b7a725e4594525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78ab186522367628a4a383e40898d551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#44;&#45;&#49;&#44;&#54;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1317ef28791b4192e574027211f267a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#44;&#45;&#49;&#44;&#54;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d04fe6788f80f8e0378695ded700bc80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#48;&#52;&#49;&#57;&#55;&#53;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7578a01ffe75fb28b1eaf83f2f6d2a88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#44;&#45;&#49;&#44;&#54;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;&#44;&#50;&#46;&#48;&#52;&#49;&#57;&#55;&#53;&#46;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"250\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836444834\">\n<h3 data-type=\"title\">Locate Fractions and Decimals on the Number Line<\/h3>\n<p id=\"fs-id1167829627701\">We now want to include fractions and decimals on the number line. Let\u2019s start with fractions and locate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-209395f4d39d67d0378618c684cb5ef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#44;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44def33f7275fe3dbc5caccd0de620c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> on the number line.<\/p>\n<p id=\"fs-id1167836521567\">We\u2019ll start with the whole numbers 3 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/> because they are the easiest to plot. See <a href=\"#CNX_IntAlg_Figure_01_04_017\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1167836698574\">The proper fractions listed are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a832bf5ca042b668a46de6257a63089a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/> We know the proper fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad4fa46969309535c076089d99ae8f9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -6px;\" \/> We plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6902a28e3644036efecf42c67afdc3f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167836607368\">Similarly, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e0151b434578c78a09b0cbb9194484e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> is between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> After dividing the unit into 5 equal parts we plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a832bf5ca042b668a46de6257a63089a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167829717102\">Finally, look at the improper fractions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a74374863e0980302ca1c4105ff8389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/> Locating these points may be easier if you change each of them to a mixed number.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836619278\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-933f0aa968e4a254ff7775a0910c5be0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#61;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#61;&#45;&#52;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"282\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167829714506\"><a href=\"#CNX_IntAlg_Figure_01_04_017\" class=\"autogenerated-content\">(Figure)<\/a> shows the number line with all the points plotted.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_017\"><span data-type=\"media\" id=\"fs-id1167829714516\" data-alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 5, minus 9 by 2, minus 4 by 5, 1 by 5, 4 by 5, 8 by 3 and 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_017-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 5, minus 9 by 2, minus 4 by 5, 1 by 5, 4 by 5, 8 by 3 and 3.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1167832940593\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832940595\">\n<div data-type=\"problem\" id=\"fs-id1167832940597\">\n<p id=\"fs-id1167832940599\">Locate and label the following on a number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c623b56baa8acb67756eeac6add606b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#45;&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"146\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-532b9597c46141b8d6e744b8a4d8e48e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720660\">\n<p id=\"fs-id1167829720662\">Locate and plot the integers, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e3c62f43075293ef1c882f6a261abba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167829627594\">Locate the proper fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> first. The fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> is between 0 and 1. Divide the distance between 0 and 1 into four equal parts, then we plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbe55600b16679e403bd6ae9e9efc7b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/> Similarly plot <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c09cf4f91687a00995a8b66829a5ecdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167829627782\">Now locate the improper fractions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6692f6a38baca610fbdf5bcb1c1183c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"45\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-532b9597c46141b8d6e744b8a4d8e48e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/> It is easier to plot them if we convert them to mixed numbers and then plot them as described above: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06de453e9e47448456a2d37d7b430082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#61;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#61;&#45;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"207\" style=\"vertical-align: -6px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836508770\" data-alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 3, minus 5 by 2, minus 1 by 4, 3 by 4, 6 by 5, 7 by 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_018-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 6 to 6. Various points on the line are highlighted. From left to right, these are: minus 3, minus 5 by 2, minus 1 by 4, 3 by 4, 6 by 5, 7 by 3 and 4.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836508289\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829719000\">\n<div data-type=\"problem\" id=\"fs-id1167829719002\">\n<p id=\"fs-id1167829719005\">Locate and label the following on a number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd9d5034ee4068a60b0ffb32f57610f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#53;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836551739\"><span data-type=\"media\" id=\"fs-id1167836551743\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 8 by 3, minus 7 by 4, minus 1, 1 by 3, 6 by 5, 9 by 2 and 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_301_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 8 by 3, minus 7 by 4, minus 1, 1 by 3, 6 by 5, 9 by 2 and 5.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836620594\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836620598\">\n<div data-type=\"problem\" id=\"fs-id1167836620600\">\n<p id=\"fs-id1167829718132\">Locate and label the following on a number line: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cfa218b5819d90b0cecdd49e8bedd45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#44;&#51;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829579971\"><span data-type=\"media\" id=\"fs-id1167829590318\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 7 by 3, minus 2, minus 7 by 4, 2 by 3, 7 by 5, 3 and 7 by 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_302-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 5. Various points on the line are highlighted. From left to right, these are: minus 7 by 3, minus 2, minus 7 by 4, 2 by 3, 7 by 5, 3 and 7 by 2.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836595922\">Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836595926\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836595929\">\n<div data-type=\"problem\" id=\"fs-id1167836595931\">\n<p id=\"fs-id1167836595933\">Locate on the number line: <span class=\"token\">\u24d0<\/span> 0.4 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48faf8ab3274b73c98b4d4a4c52dd83c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#55;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833224821\">\n<p id=\"fs-id1167833224824\"><span class=\"token\">\u24d0<\/span> The decimal number 0.4 is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a85ae032f12e6a331be819c8f22d018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"20\" style=\"vertical-align: -7px;\" \/> a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 as 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836398917\" data-alt=\"Figure shows a number line with numbers ranging from 0.0 to 1. 0.4 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_021_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0.0 to 1. 0.4 is highlighted.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p id=\"fs-id1167836398929\"><span class=\"token\">\u24d1<\/span> The decimal <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d30ea149e810e5e4675dea74e975f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#55;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/> is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d771a23d373dcc9e56a3ba91f1436fa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#52;&#125;&#123;&#49;&#48;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"41\" style=\"vertical-align: -7px;\" \/> so it is located between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> On a number line, mark off and label the hundredths in the interval between 0 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829721127\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_022_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836597144\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836597148\">\n<div data-type=\"problem\" id=\"fs-id1167833369022\">\n<p id=\"fs-id1167833369024\">Locate on the number line: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6467b5d4c60ec605876c9aa484e54ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df5b9943f5e922b533c445a8d2576927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833369143\">\n<p id=\"fs-id1167833369145\"><span class=\"token\">\u24d0<\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833369154\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.6 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_303_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.6 is highlighted.\" \/><\/span><\/p>\n<p id=\"fs-id1167829719042\"><span class=\"token\">\u24d1<\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829719051\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted, minus 0.25 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_304_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted, minus 0.25 is highlighted.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829627506\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829719437\">\n<div data-type=\"problem\" id=\"fs-id1167829719440\">\n<p id=\"fs-id1167829719442\">Locate on the number line: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3815411102441e95e9321a770366833a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e59f9cd2102b9d4259a73d55966ec2a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#55;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714677\">\n<p id=\"fs-id1167836613683\"><span class=\"token\">\u24d0<\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836613691\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.9 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_305_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 1. 0.9 is highlighted.\" \/><\/span><\/p>\n<p id=\"fs-id1167829717744\"><span class=\"token\">\u24d1<\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829717751\" data-alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_306_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836698527\" class=\"media-2\">\n<p id=\"fs-id1167836698531\">Access this online resource for additional instruction and practice with decimals.<\/p>\n<ul id=\"fs-id1167836698534\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/25dividedecim\">Arithmetic Basics: Dividing Decimals<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829714441\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836598055\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to round decimals.<\/strong>\n<ol id=\"fs-id1167833396934\" class=\"stepwise\" type=\"1\">\n<li>Locate the given place value and mark it with an arrow.<\/li>\n<li>Underline the digit to the right of the place value.<\/li>\n<li>Is the underlined digit greater than or equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d256ac5eaf9e7cc901831170e187c577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/>\n<ul id=\"fs-id1167836635513\" data-bullet-style=\"bullet\">\n<li>Yes: add 1 to the digit in the given place value.<\/li>\n<li>No: do <u data-effect=\"underline\">not<\/u> change the digit in the given place value<\/li>\n<\/ul>\n<\/li>\n<li>Rewrite the number, deleting all digits to the right of the rounding digit.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to add or subtract decimals.<\/strong>\n<ol id=\"fs-id1167829597340\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the sum or difference.<\/li>\n<li>Write the numbers so the decimal points line up vertically.<\/li>\n<li>Use zeros as placeholders, as needed.<\/li>\n<li>Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.<\/li>\n<li>Write the sum or difference with the appropriate sign<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to multiply decimals.<\/strong>\n<ol id=\"fs-id1167836389246\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the product.<\/li>\n<li>Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\n<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.<\/li>\n<li>Write the product with the appropriate sign.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to multiply a decimal by a power of ten.<\/strong>\n<ol id=\"fs-id1167829580349\" class=\"stepwise\" type=\"1\">\n<li>Move the decimal point to the right the same number of places as the number of zeros in the power of 10.<\/li>\n<li>Add zeros at the end of the number as needed.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to divide decimals.<\/strong>\n<ol id=\"fs-id1167829716804\" class=\"stepwise\" type=\"1\">\n<li>Determine the sign of the quotient.<\/li>\n<li>Make the divisor a whole number by \u201cmoving\u201d the decimal point all the way to the right. \u201cMove\u201d the decimal point in the dividend the same number of places\u2014adding zeros as needed.<\/li>\n<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\n<li>Write the quotient with the appropriate sign.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to convert a decimal to a proper fraction and a fraction to a decimal.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167829624407\" class=\"stepwise\" type=\"1\">\n<li>To convert a decimal to a proper fraction, determine the place value of the final digit.<\/li>\n<li>Write the fraction.\n<ul id=\"fs-id1167836596324\" data-bullet-style=\"bullet\">\n<li>numerator\u2014the \u201cnumbers\u201d to the right of the decimal point<\/li>\n<li>denominator\u2014the place value corresponding to the final digit<\/li>\n<\/ul>\n<\/li>\n<li>To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to convert a percent to a decimal and a decimal to a percent.<\/strong>\n<ol id=\"fs-id1167836613313\" class=\"stepwise\" type=\"1\">\n<li>To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.<\/li>\n<li>To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Square Root Notation<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/> is read \u201cthe square root of <em data-effect=\"italics\">m<\/em>.\u201d<\/p>\n<div data-type=\"newline\"><\/div>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c7c8991cf54ee68b9a5ca8f8014af5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1144c0e1f4a31eb91b31a23122c558c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#110;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de0b3f268c78cbcbba48c42081539c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>The square root of <em data-effect=\"italics\">m<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2447480eb926a3c6e489624b0a8579e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -4px;\" \/> is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/li>\n<li><strong data-effect=\"bold\">Rational or Irrational<\/strong>\n<div data-type=\"newline\"><\/div>\n<p>If the decimal form of a number<\/p>\n<div data-type=\"newline\"><\/div>\n<ul id=\"fs-id1167829717791\" data-bullet-style=\"bullet\">\n<li><em data-effect=\"italics\">repeats or stops<\/em>, the number is a rational number.<\/li>\n<li><em data-effect=\"italics\">does not repeat and does not stop<\/em>, the number is an irrational number.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Real Numbers<\/strong>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_04_016a\"><span data-type=\"media\" id=\"fs-id1167829691173\" data-alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_016-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.\" \/><\/span><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836551129\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836551133\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836551141\"><strong data-effect=\"bold\">Round Decimals<\/strong><\/p>\n<p id=\"fs-id1167829692185\">In the following exercises, round each number to the nearest <span class=\"token\">\u24d0<\/span> hundredth <span class=\"token\">\u24d1<\/span> tenth <span class=\"token\">\u24d2<\/span> whole number.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829714550\">\n<div data-type=\"problem\" id=\"fs-id1167829714552\">\n<p id=\"fs-id1167829714554\">5.781<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714559\">\n<p id=\"fs-id1167829714561\"><span class=\"token\">\u24d0<\/span> 5.78 <span class=\"token\">\u24d1<\/span> 5.8 <span class=\"token\">\u24d2<\/span> 6<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836531237\">\n<div data-type=\"problem\" id=\"fs-id1167836531239\">\n<p id=\"fs-id1167836531241\">1.638<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833020287\">\n<div data-type=\"problem\" id=\"fs-id1167833020290\">\n<p id=\"fs-id1167833020292\">0.299<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833020296\">\n<p id=\"fs-id1167833020298\"><span class=\"token\">\u24d0<\/span> 0.30 <span class=\"token\">\u24d1<\/span> 0.3 <span class=\"token\">\u24d2<\/span> 0<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833021622\">\n<div data-type=\"problem\" id=\"fs-id1167833021624\">\n<p id=\"fs-id1167833021626\">0.697<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829627529\">\n<div data-type=\"problem\" id=\"fs-id1167829627531\">\n<p id=\"fs-id1167836580081\">63.479<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836580085\">\n<p id=\"fs-id1167836580087\"><span class=\"token\">\u24d0<\/span> 63.48 <span class=\"token\">\u24d1<\/span> 63.5 <span class=\"token\">\u24d2<\/span> 63<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836699141\">\n<div data-type=\"problem\" id=\"fs-id1167836699143\">\n<p id=\"fs-id1167836699145\">84.281<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829719115\"><strong data-effect=\"bold\">Add and Subtract Decimals<\/strong><\/p>\n<p id=\"fs-id1167829719121\">In the following exercises, add or subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829719124\">\n<div data-type=\"problem\" id=\"fs-id1167832936353\">\n<p id=\"fs-id1167832936356\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc56dae7896fffb50553b1752c74ad9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#46;&#53;&#51;&#45;&#50;&#52;&#46;&#51;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832936367\">\n<p id=\"fs-id1167832936369\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-393972d398a9ba2e7a4b7370eadfbcb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#48;&#46;&#57;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832999808\">\n<div data-type=\"problem\" id=\"fs-id1167832999810\">\n<p id=\"fs-id1167832999812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07715f644cf92d325033dd2d50b19046_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;&#46;&#52;&#55;&#45;&#51;&#50;&#46;&#53;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719888\">\n<div data-type=\"problem\" id=\"fs-id1167829719890\">\n<p id=\"fs-id1167829719892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b667f13c951d748d4441ed01c01ab17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#56;&#46;&#54;&#57;&#43;&#51;&#49;&#46;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719904\">\n<p id=\"fs-id1167829719906\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fdfa2b092f8324f50610d7e9d34726d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#46;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833224567\">\n<div data-type=\"problem\" id=\"fs-id1167833224569\">\n<p id=\"fs-id1167833224571\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6542b4c0f640701a74f5cc67df18041a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#57;&#46;&#56;&#51;&#43;&#49;&#57;&#46;&#55;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720868\">\n<div data-type=\"problem\" id=\"fs-id1167829720870\">\n<p id=\"fs-id1167829720873\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9446c4d52ab3e0238d118948713366ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#50;&#46;&#53;&#45;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"80\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829716532\">\n<p id=\"fs-id1167829716534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b950529f706a9a7e0ba550d56bfaab44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829716543\">\n<div data-type=\"problem\" id=\"fs-id1167829716545\">\n<p id=\"fs-id1167829716547\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdb1f85f6e9f3f3b02c993291ce7e3cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#54;&#46;&#50;&#45;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"80\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833240107\">\n<div data-type=\"problem\" id=\"fs-id1167833240109\">\n<p id=\"fs-id1167833240111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a0c7bbb56d74ea3a9bd750f3feb4264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#49;&#46;&#55;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#46;&#52;&#54;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714180\">\n<p id=\"fs-id1167829714182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a7a6a1b1edb6521c073286b0bc22208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#50;&#46;&#50;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829690547\">\n<div data-type=\"problem\" id=\"fs-id1167829690549\">\n<p id=\"fs-id1167829690551\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09195eb25217376eb13574fd67a7fd77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#52;&#46;&#54;&#57;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#46;&#54;&#55;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836508374\">\n<div data-type=\"problem\" id=\"fs-id1167836508376\">\n<p id=\"fs-id1167836508378\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04329a849fd13da72e55d368855df813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#53;&#46;&#48;&#49;&#45;&#51;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714589\">\n<p id=\"fs-id1167829714592\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-564daeecdd4ea32db12aaa227fed4cb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#49;&#46;&#51;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829714600\">\n<div data-type=\"problem\" id=\"fs-id1167829714602\">\n<p id=\"fs-id1167829714604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56cfa80c61e697a7fadb27b27c30a8c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#57;&#46;&#48;&#56;&#45;&#52;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624911\">\n<div data-type=\"problem\" id=\"fs-id1167829624913\">\n<p id=\"fs-id1167829624915\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ebd7337a99ff13161d28ddc0c6418ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&#49;&#45;&#55;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621826\">\n<p id=\"fs-id1167836621828\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cadc76228d39486be78625fcc9adc6c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;&#56;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836621836\">\n<div data-type=\"problem\" id=\"fs-id1167836621838\">\n<p id=\"fs-id1167836621840\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b8601dc20ab1fea93bd2d1b64e28ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#56;&#52;&#45;&#54;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829721201\"><strong data-effect=\"bold\">Multiply and Divide Decimals<\/strong><\/p>\n<p id=\"fs-id1167829721207\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836620056\">\n<div data-type=\"problem\" id=\"fs-id1167836620058\">\n<p id=\"fs-id1167836620061\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09195eb25217376eb13574fd67a7fd77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#52;&#46;&#54;&#57;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#46;&#54;&#55;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836620079\">\n<p id=\"fs-id1167836620081\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b02d2500eeec3c4c7683d4665c8b797c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#46;&#54;&#53;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624935\">\n<div data-type=\"problem\" id=\"fs-id1167829624937\">\n<p id=\"fs-id1167829624939\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07733ce4f658001e97c55a2ab7be6211_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#46;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#54;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833059335\">\n<div data-type=\"problem\" id=\"fs-id1167833059338\">\n<p id=\"fs-id1167833059340\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20721ef49950b8a59f7ee1a2908e9edc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#46;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#53;&#46;&#50;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836331120\">\n<p id=\"fs-id1167836331123\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5e558cd1783b830c0e09d0c6e295c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#51;&#55;&#46;&#56;&#57;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"68\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720751\">\n<div data-type=\"problem\" id=\"fs-id1167829720753\">\n<p id=\"fs-id1167829720756\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79bf99462ddafef638a9460e7c94f76d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#46;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#56;&#46;&#51;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833054680\">\n<div data-type=\"problem\" id=\"fs-id1167833054682\">\n<p id=\"fs-id1167833054684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0be1395202e50fdf220fd5823d8fb02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#48;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#49;&#46;&#55;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836477505\">\n<p id=\"fs-id1167836477507\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf8e819210ff402e2a63e79347f57b3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#51;&#48;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477711\">\n<div data-type=\"problem\" id=\"fs-id1167836477713\">\n<p id=\"fs-id1167836477715\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bcaba92dcdfde0414620087234d7d8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#48;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#50;&#46;&#52;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829690917\">\n<div data-type=\"problem\" id=\"fs-id1167829690919\">\n<p id=\"fs-id1167829690921\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27ac039f6cc1c41adbac82ac99f94c0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#46;&#50;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829624428\">\n<p id=\"fs-id1167829624430\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b57e9a1b379ae157e04a7d0ca51055d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#50;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624439\">\n<div data-type=\"problem\" id=\"fs-id1167836698541\">\n<p id=\"fs-id1167836698543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f85dc9fdc74886a32839a5dca5d34492_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#53;&#51;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025125\">\n<div data-type=\"problem\" id=\"fs-id1167833025127\">\n<p id=\"fs-id1167833025129\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b524d22494fec8496c1bd9a687ec6d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#48;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025626\">\n<p id=\"fs-id1167833025628\">2.5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025634\">\n<div data-type=\"problem\" id=\"fs-id1167833025636\">\n<p id=\"fs-id1167833025638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c9d0a565b3bf1c3ad22252a03e159a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#48;&#51;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025756\">\n<div data-type=\"problem\" id=\"fs-id1167833025808\">\n<p id=\"fs-id1167833025810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0d251ec8ec732db09c90b7f1d9418a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#53;&#46;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025832\">\n<p id=\"fs-id1167836399531\">55200<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399536\">\n<div data-type=\"problem\" id=\"fs-id1167836399538\">\n<p id=\"fs-id1167836399541\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e594ab215afd26c285b72b9710ba2176_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#57;&#46;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836399835\">In the following exercises, divide. Round money monetary answers to the nearest cent.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836399838\">\n<div data-type=\"problem\" id=\"fs-id1167836399840\">\n<p id=\"fs-id1167836399843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53e5a350af51d052ec7bb6ae01c409e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#49;&#55;&#46;&#50;&#53;&divide;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399945\">\n<p id=\"fs-id1167836399947\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93fc8acf520abf1995d7d7e1c459dc68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#50;&#46;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399957\">\n<div data-type=\"problem\" id=\"fs-id1167836399960\">\n<p id=\"fs-id1167836399962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e37234d165a426194edb557854762eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#48;&#57;&#46;&#50;&#52;&divide;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719786\">\n<div data-type=\"problem\" id=\"fs-id1167829719819\">\n<p id=\"fs-id1167829719822\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc62cff48ae3a2d472150adee41aed61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#52;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719840\">\n<p id=\"fs-id1167829719842\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c89e7492512d2df59c5e900b950453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829580307\">\n<div data-type=\"problem\" id=\"fs-id1167829580309\">\n<p id=\"fs-id1167829580311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0fd62c248455182309528a519885917_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#49;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#46;&#48;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836597072\">\n<div data-type=\"problem\" id=\"fs-id1167836597074\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1003c619f1fa5d15defac6cc6613197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#50;&divide;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c193ecb6a8f79b1604351d8fa4be92d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#48;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836542566\">\n<div data-type=\"problem\" id=\"fs-id1167836542568\">\n<p id=\"fs-id1167836542570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95c84179d37b528953a5b4c886cc3c96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&divide;&#48;&#46;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836518558\"><strong data-effect=\"bold\">Convert Decimals, Fractions and Percents<\/strong><\/p>\n<p id=\"fs-id1167836518563\">In the following exercises, write each decimal as a fraction.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836518566\">\n<div data-type=\"problem\" id=\"fs-id1167836518568\">\n<p id=\"fs-id1167836518570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d9a18298d71046d4938c101e1367c19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829716255\">\n<p id=\"fs-id1167829716258\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-987c8c247ee1bfbcaa09ef7b7ce42fbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829716270\">\n<div data-type=\"problem\" id=\"fs-id1167829716272\">\n<p id=\"fs-id1167829716275\">1.464<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829715938\">\n<div data-type=\"problem\" id=\"fs-id1167829715940\">\n<p id=\"fs-id1167829715942\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dda88cfa505474dd5132ab197bfef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#57;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829715950\">\n<p id=\"fs-id1167829715952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-597bcb3c2f10c32b9c574b0e300dfb5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#125;&#123;&#50;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829717177\">\n<div data-type=\"problem\" id=\"fs-id1167829717180\">\n<p id=\"fs-id1167829717182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a4e94c522264239aef6473626e5cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#51;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836484387\">In the following exercises, convert each fraction to a decimal.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836484390\">\n<div data-type=\"problem\" id=\"fs-id1167836484392\">\n<p id=\"fs-id1167836484394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dec0bfaac7a24643c8e5fb816c534b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836484407\">\n<p id=\"fs-id1167836484814\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf1b857ee4fa9a5f760e68199d008cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836484822\">\n<div data-type=\"problem\" id=\"fs-id1167836484824\">\n<p id=\"fs-id1167836484827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c52635531dfb283ca461a9ffc685ada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836684117\">\n<div data-type=\"problem\" id=\"fs-id1167836684119\">\n<p id=\"fs-id1167836684121\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd44f265426e0c6c8f1cdd41bdf5145a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#48;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836684136\">\n<p id=\"fs-id1167836684138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01027fb3f026eb695283721aac4e9e8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829589886\">\n<div data-type=\"problem\" id=\"fs-id1167829589888\">\n<p id=\"fs-id1167829589890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aac643350db5829a3559643d62767685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"29\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829719662\">In the following exercises, convert each percent to a decimal.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829719665\">\n<div data-type=\"problem\" id=\"fs-id1167829719667\">\n<p id=\"fs-id1167829719669\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2ed1625b5129553f589f4e79e89cf1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#49;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719679\">\n<p id=\"fs-id1167833025497\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8033e7f94a1fa979d0fdc28a3a2e679f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#55;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025505\">\n<div data-type=\"problem\" id=\"fs-id1167833025507\">\n<p id=\"fs-id1167833025510\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49a8f43581708e5aeffc109458be1dbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#48;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836542526\">\n<div data-type=\"problem\" id=\"fs-id1167836542528\">\n<p id=\"fs-id1167836542530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc78be026c5c90fabff5ee5457c1bf06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#57;&#46;&#51;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542540\">\n<p id=\"fs-id1167836542542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d6197f6f43ca8fb698bc18f11cf41a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#51;&#57;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836542551\">\n<div data-type=\"problem\" id=\"fs-id1167836542553\">\n<p id=\"fs-id1167836542555\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e550eaeafb14bb77e0622783ac76558_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#46;&#56;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833053575\">In the following exercises, convert each decimal to a percent.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833053578\">\n<div data-type=\"problem\" id=\"fs-id1167833053580\">\n<p id=\"fs-id1167833053582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfdc502238b1625e7c2a9304d22e8769_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#53;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621850\">\n<p id=\"fs-id1167836621853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8345d773b9b2fd8d6ce58ff34179e52e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#54;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836621863\">\n<div data-type=\"problem\" id=\"fs-id1167836621865\">\n<p id=\"fs-id1167836621868\">3<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836508489\">\n<div data-type=\"problem\" id=\"fs-id1167836508492\">\n<p id=\"fs-id1167836508494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf64c24f183de783abb0dbb3a89118de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#54;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836508501\">\n<p id=\"fs-id1167836508503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c770c48e7e64d248e34c132f9ecefbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#50;&#53;&#37;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836508514\">\n<div data-type=\"problem\" id=\"fs-id1167836508516\">\n<p id=\"fs-id1167833158153\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-483d8c1cdd5b8f9e414ba45a7df4ec55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#50;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833158174\"><strong data-effect=\"bold\">Simplify Expressions with Square Roots<\/strong><\/p>\n<p id=\"fs-id1167833158180\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833158183\">\n<div data-type=\"problem\" id=\"fs-id1167829717857\">\n<p id=\"fs-id1167829717859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d5fdaeda697f10142dbe9236e272283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717869\">\n<p id=\"fs-id1167829717871\">8<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829717876\">\n<div data-type=\"problem\" id=\"fs-id1167829717878\">\n<p id=\"fs-id1167829717880\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd4c05e7550847a7adda1c07adcd62f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833397155\">\n<div data-type=\"problem\" id=\"fs-id1167833397157\">\n<p id=\"fs-id1167833397160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb4b7e7ac9715b068342f7e4ccdd555c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833397169\">\n<p id=\"fs-id1167833397171\">12<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833397177\">\n<div data-type=\"problem\" id=\"fs-id1167833397179\">\n<p id=\"fs-id1167836477531\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-615d679daccd625cc0819d01a9d4de82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477552\">\n<div data-type=\"problem\" id=\"fs-id1167836477554\">\n<p id=\"fs-id1167836477557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0e99e0fb77048431e44e24f06f05afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836398828\">\n<p id=\"fs-id1167836398830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836398839\">\n<div data-type=\"problem\" id=\"fs-id1167836398841\">\n<p id=\"fs-id1167836398843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1cdc30fd002524bb45c411f5ce2b5f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836456213\"><strong data-effect=\"bold\">Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers<\/strong><\/p>\n<p id=\"fs-id1167836456219\">In the following exercises, list the <span class=\"token\">\u24d0<\/span> whole numbers, <span class=\"token\">\u24d1<\/span> integers, <span class=\"token\">\u24d2<\/span> rational numbers, <span class=\"token\">\u24d3<\/span> irrational numbers, <span class=\"token\">\u24d4<\/span> real numbers for each set of numbers.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829720690\">\n<div data-type=\"problem\" id=\"fs-id1167829720693\">\n<p id=\"fs-id1167829720695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-798d8222eea854afafac429b1219a97b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#46;&#46;&#46;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"202\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829690686\">\n<p id=\"fs-id1167829690688\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45541897b6113e04601cfa0911e1817d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"66\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-519e787c6ab416b7dd43c24053b63023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-519e787c6ab416b7dd43c24053b63023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56544c259a22cb7f53ae8d4ffdc57ac5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#46;&#46;&#46;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d4<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-798d8222eea854afafac429b1219a97b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;&#48;&#44;&#49;&#46;&#57;&#53;&#50;&#56;&#54;&#46;&#46;&#46;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"202\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025160\">\n<div data-type=\"problem\" id=\"fs-id1167833025162\">\n<p id=\"fs-id1167833025164\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f123e57a7bbd31c9f437a16667f46b18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#44;&#45;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;&#44;&#48;&#46;&#52;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#8212;&#125;&#123;&#48;&#57;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;&#44;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"190\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836398857\">\n<div data-type=\"problem\" id=\"fs-id1167836398859\">\n<p id=\"fs-id1167836398861\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f70c12228d4e80a955c8d0685582f280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#44;&#45;&#55;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#44;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836700865\">\n<p id=\"fs-id1167836700867\"><span class=\"token\">\u24d0<\/span> none <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7edbc683e14f885b2ea96d606535895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#44;&#45;&#55;&#44;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f70c12228d4e80a955c8d0685582f280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#44;&#45;&#55;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#44;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> none<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f70c12228d4e80a955c8d0685582f280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#44;&#45;&#55;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#44;&#48;&#46;&#55;&#55;&#44;&#51;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836484440\">\n<div data-type=\"problem\" id=\"fs-id1167836484442\">\n<p id=\"fs-id1167836484444\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26d1a65e6ba8dcf44df6823c2229af98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#48;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#8212;&#8212;&#8212;&#125;&#123;&#55;&#49;&#52;&#50;&#56;&#53;&#125;&#44;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"219\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833368929\"><strong data-effect=\"bold\">Locate Fractions and Decimals on the Number Line<\/strong><\/p>\n<p id=\"fs-id1167833368936\">In the following exercises, locate the numbers on a number line.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833368939\">\n<div data-type=\"problem\" id=\"fs-id1167833368941\">\n<p id=\"fs-id1167833368943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b5ff7f9a409125784cafca354daa19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;&#44;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"77\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833412462\"><span data-type=\"media\" id=\"fs-id1167833412467\" data-alt=\"Figure shows a number line with numbers ranging from 0 to 6. Some values are highlighted. From left to right, these are: 3 by 10, 11 by 6, 7 by 2 and 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_307_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from 0 to 6. Some values are highlighted. From left to right, these are: 3 by 10, 11 by 6, 7 by 2 and 4.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833412483\">\n<div data-type=\"problem\" id=\"fs-id1167833412485\">\n<p id=\"fs-id1167833369747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-998ef95d2bb26df987039782017fa621_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#56;&#125;&#44;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"77\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836410212\">\n<div data-type=\"problem\" id=\"fs-id1167836410214\">\n<p id=\"fs-id1167836410216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98cb68153ae225f7300229b12ddfe0b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#45;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"159\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719936\"><span data-type=\"media\" id=\"fs-id1167829719939\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted. From left to right, these are: minus 5 by 2, minus 1 and two thirds, minus 3 by 4, 3 by 4, 1 and two thirds, and 5 by 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_309_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted. From left to right, these are: minus 5 by 2, minus 1 and two thirds, minus 3 by 4, 3 by 4, 1 and two thirds, and 5 by 2.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719955\">\n<div data-type=\"problem\" id=\"fs-id1167829719957\">\n<p id=\"fs-id1167829719959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd9e3b12d149a34d605f18c23a14b341_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"159\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829580180\">\n<div data-type=\"problem\" id=\"fs-id1167829580182\">\n<p id=\"fs-id1167829580184\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e4419032e9b2a7c997243dc95aced4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-041b53cfa49d2fe629b3dc28a28b6ccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399691\"><span data-type=\"media\" id=\"fs-id1167836399694\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 0 and 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_311_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 0 and 1.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399709\">\n<div data-type=\"problem\" id=\"fs-id1167836399711\">\n<p id=\"fs-id1167836399714\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5857539969c55edce3a6888e199b957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b97f3917928f9008dd7816f185d1f7df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829692233\">\n<div data-type=\"problem\" id=\"fs-id1167829692235\">\n<p id=\"fs-id1167829692237\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dcf50e0b47145a50e261490f61cf521_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"36\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a005928d9ea75d0170b694f5cffc8c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399887\"><span data-type=\"media\" id=\"fs-id1167836399890\" data-alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_313_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 3 and 4.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836399904\">\n<div data-type=\"problem\" id=\"fs-id1167836399906\">\n<p id=\"fs-id1167836399908\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b12efa8606060b4386c4bb17d4d2732_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd75a77253884aec70b32a2e5e022aec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829627750\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829627757\">\n<div data-type=\"problem\" id=\"fs-id1167829627759\">\n<p id=\"fs-id1167829627761\">How does knowing about U.S. money help you learn about decimals?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829627766\">\n<p id=\"fs-id1167829627768\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836538664\">\n<div data-type=\"problem\" id=\"fs-id1167836538666\">\n<p id=\"fs-id1167836538668\">When the Szetos sold their home, the selling price was 500% of what they had paid for the house 30 years ago. Explain what 500% means in this context.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836538683\">\n<div data-type=\"problem\" id=\"fs-id1167836538685\">\n<p id=\"fs-id1167836538687\">In your own words, explain the difference between a rational number and an irrational number.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836538692\">\n<p id=\"fs-id1167836538694\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836538700\">\n<div data-type=\"problem\" id=\"fs-id1167836538702\">\n<p id=\"fs-id1167836538704\">Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833240026\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167833240032\"><span class=\"token\">\u24d0<\/span> Use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833240046\" data-alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The statements in the first column are: round decimals, add and subtract decimals, multiply and divide decimals, convert decimals, fractions and percents, simplify expressions with square roots, identify integers, rational numbers, irrational numbers and real numbers, locate fractions and decimals on the number line. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_04_215_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The statements in the first column are: round decimals, add and subtract decimals, multiply and divide decimals, convert decimals, fractions and percents, simplify expressions with square roots, identify integers, rational numbers, irrational numbers and real numbers, locate fractions and decimals on the number line. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167833240055\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167833240069\">\n<dt>irrational number<\/dt>\n<dd id=\"fs-id1167833240074\">An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836398934\">\n<dt>percent<\/dt>\n<dd id=\"fs-id1167836398939\">A percent is a ratio whose denominator is 100.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836398944\">\n<dt>principal square root<\/dt>\n<dd id=\"fs-id1167836398949\">The positive square root is called the principal square root.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836398953\">\n<dt>rational number<\/dt>\n<dd id=\"fs-id1167836398959\">A rational number is a number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2234e67f0d02bee8d24190a2ade2e6be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -9px;\" \/> where <em data-effect=\"italics\">p<\/em> and <em data-effect=\"italics\">q<\/em> are integers and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e417a0769b70c865e6dde403ebb92108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> Its decimal form stops or repeats.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829690613\">\n<dt>real number<\/dt>\n<dd id=\"fs-id1167829690618\">A real number is a number that is either rational or irrational.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829690622\">\n<dt>square of a number<\/dt>\n<dd id=\"fs-id1167829690628\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">m<\/em> is the square of <em data-effect=\"italics\">n<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829690657\">\n<dt>square root of a number<\/dt>\n<dd id=\"fs-id1167836700140\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">n<\/em> is a square root of <em data-effect=\"italics\">m<\/em>.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-976","chapter","type-chapter","status-publish","hentry"],"part":762,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":5,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/976\/revisions"}],"predecessor-version":[{"id":14498,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/976\/revisions\/14498"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/762"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/976\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=976"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=976"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=976"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}