Decimals

Round Decimals

Decimals are another way of writing fractions whose denominators are powers of ten.

Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. (Figure) shows the names of the place values to the left and right of the decimal point.

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.

Round 18.379 to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.

Round

ⓐ to the nearest hundredth

Locate the hundredths place with an arrow.
Underline the digit to the right of the given place value.
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.
Notice that the deleted digits were NOT replaced with zeros.

ⓑ to the nearest tenth

Locate the tenths place with an arrow.
Underline the digit to the right of the given place value.
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.
Notice that the deleted digits were NOT replaced with zeros.

ⓒ to the nearest whole number

Locate the ones place with an arrow.
Underline the digit to the right of the given place value.
Since 3 is not greater than or equal to 5, do not add 1 to the 8.
Rewrite the number, deleting all digits to the right of the rounding digit.

Add and Subtract Decimals

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.

*** QuickLaTeX cannot compile formula:
\begin{array}{cccc}& & & -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 & & & \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 & & & \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 & & & \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 & & & -23.5-41.38=-64.88\hfill \end{array}

*** Error message:
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Missing { inserted.
leading text: ....2em}{0ex}}23.5\\ \underset{______}{+41.38}
Misplaced \crcr. (in macro \endarray)
leading text: ...23.5\\ \underset{______}{+41.38}\end{array}
Missing } inserted.
leading text: ...23.5\\ \underset{______}{+41.38}\end{array}
Missing } inserted.
leading text: ...23.5\\ \underset{______}{+41.38}\end{array}
Missing } inserted.

Multiply and Divide Decimals

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.

Multiply:

The signs are different. The product will be negative.	The product will be negative.
Write in vertical format, lining up the numbers on the right.
Multiply.
Add the number of decimal places in the factors (1 + 3). Place the decimal point 4 places from the right.
The signs are the different, so the product is negative.

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.

Multiply: 5.63 by ⓐ 10 ⓑ 100 ⓒ 1000.

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.

ⓐ

There is 1 zero in 10, so move the decimal point 1 place to the right.

ⓑ

There are 2 zeroes in 100, so move the decimal point 2 places to the right.

ⓒ

There are 3 zeroes in 1,000, so move the decimal point 3 place to the right.
A zero must be added to the end.

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.

We review the notation and vocabulary for division:

We’ll write the steps to take when dividing decimals for easy reference.

Divide:

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property.

The signs are the same.	The quotient is positive.
Make the divisor a whole number by “moving” the decimal point all the way to the right.
“Move” the decimal point in the dividend the same number of places.
Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Write the quotient with the appropriate sign.

Convert Decimals, Fractions, and Percents

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.

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

Write: ⓐ as a fraction ⓑ as a decimal.

ⓐ

Determine the place value of the final digit.
Write the fraction for 0.374: The numerator is 374. The denominator is 1,000.
Simplify the fraction.
Divide out the common factors.

ⓑ Since a fraction bar means division, we begin by writing the fraction as Now divide.

A percent 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.

To convert a percent number to a decimal number, we move the decimal point two places to the left.

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.

To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

Convert each:

ⓐ percent to a decimal: 62%, 135%, and 13.7%.

ⓑ decimal to a percent: 0.51, 1.25, and 0.093.

ⓐ

Move the decimal point two places to the left.

ⓑ

Move the decimal point two places to the right.

Simplify Expressions with Square Roots

Remember that when a number is multiplied by itself, we write and read it “ squared.” The result is called the square of a number n. For example, is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because is 121. It will be helpful to learn to recognize the perfect square numbers.

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.

Because we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m is called a square root of a number m.

Notice also, so is also a square root of 100. Therefore, both 10 and are square roots of 100. So, every positive number has two square roots—one positive and one negative. The radical sign, , denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, We read as “the opposite of the principal square root of 10.”

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. 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?

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 rational number is a number that can be written as a ratio of two integers.

In general, any decimal that ends after a number of digits (such as 7.3 or ) 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 is the number 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.

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.

Every rational number can be written both as a ratio of integerswhere p and q are integers andand as a decimal that stops or repeats.

Are there any decimals that do not stop or repeat? Yes! The number (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat. We use three dots (…) to indicate the decimal does not stop or repeat.

The square root of a number that is not a perfect square is a decimal that does not stop or repeat.

A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

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 real numbers.

Later in this course we will introduce numbers beyond the real numbers. (Figure) illustrates how the number sets we’ve used so far fit together.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? Can we simplify Is there a number whose square is

None of the numbers that we have dealt with so far has a square that is Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to The square root of a negative number is not a real number.

Locate Fractions and Decimals on the Number Line

We now want to include fractions and decimals on the number line. Let’s start with fractions and locate and on the number line.

We’ll start with the whole numbers 3 and because they are the easiest to plot. See (Figure).

The proper fractions listed are and We know the proper fraction 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 We plot

Similarly, is between 0 and After dividing the unit into 5 equal parts we plot

Finally, look at the improper fractions Locating these points may be easier if you change each of them to a mixed number.

(Figure) shows the number line with all the points plotted.

Locate and label the following on a number line:

Locate and label the following on a number line:

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.