Chapter 2. Making Measurements
2.1 Expressing Numbers
Learning Objective
By the end of this section, you will be able to:
- Learn to express numbers properly.
Quantities have two parts: the number and the unit. The number tells “how many.” It is important to be able to express numbers properly so that the quantities can be communicated properly.
Standard notation is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.
Scientific notation is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:
| 100 | = 1 |
| 101 | = 10 |
| 102 | = 100 = 10 × 10 |
| 103 | = 1,000 = 10 × 10 × 10 |
| 104 | = 10,000 = 10 × 10 × 10 × 10 |
and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent. (Scientific notation is sometimes called exponential notation.) The exponent’s value is equal to the number of zeros in the number expressed in standard notation.
Small numbers can also be expressed in scientific notation but with negative exponents:
| 10−1 | = 0.1 = 1/10 |
| 10−2 | = 0.01 = 1/100 |
| 10−3 | = 0.001 = 1/1,000 |
| 10−4 | = 0.0001 = 1/10,000 |
and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.
A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,
79,345 = 7.9345 × 10,000 = 7.9345 × 104
Thus, the number in scientific notation is 7.9345 × 104. For small numbers, the same process is used, but the exponent for the power of 10 is negative:
0.000411 = 4.11 × 1/10,000 = 4.11 × 10−4
Typically, the extra zero digits at the end or the beginning of a number are not included.
Example 1
Express these numbers in scientific notation.
a) 306,000 b) 0.00884 c) 2,760,000 d) 0.000000559
Solution
a) The number 306,000 is 3.06 times 100,000, or 3.06 times 105. In scientific notation, the number is 3.06 × 105.
b) The number 0.00884 is 8.84 times 1/1,000, which is 8.84 times 10−3. In scientific notation, the number is 8.84 × 10−3.
c) The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 106. In scientific notation, the number is written as 2.76 × 106. Note that we omit the zeros at the end of the original number.
d) The number 0.000000559 is 5.59 times 1/10,000,000, which is 5.59 times 10−7. In scientific notation, the number is written as 5.59 × 10−7.
Test Yourself
Express these numbers in scientific notation.
a) 23,070 b) 0.0009706
Answers
a) 2.307 × 104 b) 9.706 × 10−4
Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.
Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.
Example 2
Represent the following numbers in scientific notation.
a) 12,500 b) 1470 c) 0.0024
Solution
a) Move the decimal until you have a number between 1 and 10:
Because the decimal was moved 4 places to the left, the power of ten is 4.
The answer is 25 x 104
b) The decimal was moved 3 places to the left, the answer is 47 x 103
c) The decimal was moved 3 places to the right, the answer is 4 x 10-3
Test Yourself
Represent the following numbers in scientific notation
a) 247 b) 100 c) 0.2089 d) 0.000000003
Answers
a) 2.47 × 102 b) 1 × 102 b) 3 × 10-9
Example 3
Represent the following scientific notation numbers in normal notation
a) 3.06 x 10-2 b) 2.47 x 106
Solution
a) You now have to move the decimal in the opposite direction, thus a negative power of 10
requires a movement to the right (2 spaces): 3.06 → 0.0306
b) The positive exponent requires a movement 6 spaces to the left, adding zeros as required.
2.47 → 2,470,000
Test Yourself
Represent the following scientific notation numbers in normal notation
a) 1.087 x 10-6 b) 4 x 104
Answers
a) 0.000001087 b) 40000
Key Concepts and Summary
Standard notation expresses a number normally. Scientific notation expresses a number as a coefficient times a power of 10. The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.
This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 × 1018, or 3,849,510,000,000,000,000. Source: “Casio”Asim Bijarani is licensed under Creative Commons Attribution 2.0 Generic.
Exercises
1. Express these numbers in scientific notation.
a) 56.9 b) 563,100 c) 0.0804 d) 0.00000667
2. Express these numbers in scientific notation.
a) −890,000 b) 602,000,000,000 c) 0.0000004099 d) 0.000000000000011
3. Express these numbers in scientific notation.
a) 0.00656 b) 65,600 c) 4,567,000 d) 0.000005507
4. Express these numbers in scientific notation.
a) 65 b) −321.09 c) 0.000077099 d) 0.000000000218
5. Express these numbers in standard notation.
a) 1.381 × 105 b) 5.22 × 10−7 c) 9.998 × 104
6. Express these numbers in standard notation.
a) 7.11 × 10−2 b) 9.18 × 102 c) 3.09 × 10−10
7. Express these numbers in standard notation.
a) 8.09 × 100 b) 3.088 × 10−5 c) −4.239 × 102
8. Express these numbers in standard notation.
a) 2.87 × 10−8 b) 1.78 × 1011 c) 1.381 × 10−23
9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
a) 72.44 × 103 b) 9,943 × 10−5 c) 588,399 × 102
10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
a) 0.000077 × 10−7 b) 0.000111 × 108 c) 602,000 × 1018
11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
a) 345.1 × 102 b) 0.234 × 10−3 c) 1,800 × 10−2
12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.
a) 8,099 × 10−8 b) 34.5 × 100 c) 0.000332 × 104
13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
a) 123,456.78 b) 98,490 c) 0.000000445
14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.
a) 0.000552 b) 1,987 c) 0.00000000887
15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
a) 456 × (7.4 × 108) = ? b) (3.02 × 105) ÷ (9.04 × 1015) = ? c) 0.0044 × 0.000833 = ?
16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
a) 98,000 × 23,000 = ? b) 98,000 ÷ 23,000 = ? c) (4.6 × 10−5) × (2.09 × 103) = ?
17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
a) 45 × 132 ÷ 882 = ? b) [(6.37 × 104) × (8.44 × 10−4)] ÷ (3.2209 × 1015) = ?
18. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.
a) (9.09 × 108) ÷ [(6.33 × 109) × (4.066 × 10−7)] = ? b) 9,345 × 34.866 ÷ 0.00665 = ?
Answers
1. a) 5.69 × 101 b) 5.631 × 105 c) 8.04 × 10−2 d) 6.67 × 10−6
2. a) −8.9 × 105 b) 6.02 × 1011 c) 4.099 × 10-7 d) 1.1 × 10-14
3. a) 6.56 × 10−3 b) 6.56 × 104 c) 4.567 × 106 d) 5.507 × 10−6
4. a) 6.5 × 101 b) −3.2109 × 102 c) 7.7099 × 10-5 d) 2.18 × 10-10
5. a) 138,100 b) 0.000000522 c) 99,980
6. a) 0.0711 b) 918 c) 0.000000000309
7. a) 8.09 b) 0.00003088 c) −423.9
8. a) 0.0000000287 b) 178,000,000,000 c) 0.00000000000000000000001381
9. a) 7.244 × 104 b) 9.943 × 10−2 c) 5.88399 × 107
10. a) 7.7 × 10−12 b) 1.11 × 104 c) 6.02000 × 1023
11. a) 3.451 × 104 b) 2.34 × 10−4 c) 1.8 × 101
12. a) 8.099 × 10−5 b) 3.45 × 101 c) 3.32 × 100
13. a) 1.2345678 × 105 b) 9.849 × 104 c) 4.45 × 10−7
14. a) 5.52 × 10-4 b) 1.987 × 103 c) 8.87 × 109
15. a) 3.3744 × 1011 b) 3.3407 × 10−11 c) 3.665 × 10−6
16. a) 2.254 × 109 b) 4.2609 × 100 c) 9.614 × 10-2
17. a) 6.7346 × 100 b) 1.6691 × 10-14
18. a) 3.53177 × 105 b) 4.89959 × 107
