The precision of a measurement is indicated by the way in which the answer is reported. The number of Significant Figures is the number of digits known with certainty to be correct. The number 1.23 has three Significant Figures. The first two, 1 and 2 are known to be correct and the 3 has an uncertainty. Determining the correct number of Significant Figures depends on how zeros are counted.
| A. | If a zero is between other digits, it is a Significant Figure. Both 1,001 and 10.01 have four Significant Figures (NOTE the difference: one has a comma, the other a decimal point). |
| B. | Zeros located to the left of a decimal point are not Significant Figures. These zeros are used to locate the decimal point and give the size of the number. They do not indicate the precision of the measurement. Numbers such as 0.011, 0.0021, 0.00012 all havetwo Significant Figures. |
| C. | Zeros to the right of the decimal point are always Significant. Zeros at the end of a number are trailing zeros. In the number 3.07000 has five and 123,000 has six Significant Figures. In both cases, three are trailing zeros to the right of the decimal point. |
| D. | Trailing zeros that are to the left of a decimal point are Significantt only if noted as such by the author of the book or article. For example, in the measurement 1,000,000, the trailing zeros may be precise to 7, 6, 5, 4, 3, 2, Significant Figures or only 1. Most scientists will write this number in scientific notation so that any trailing zeros can be placed after the decimal point. The example would be precise to seven Significantt figure when written as 1.0 x 106 or three Significant Figures if expressed as 1.00 x 10 3. |
| Addition and Subraction: |
| The result of adding or subtracting numbers should contain no more digits to the right of the decimal point than are in the measurement which has the least number of digits to the right of the decimal point. |
| Addition Example: |
| 24.372 |
| 72.21 |
| + 6.1488 | |
| 102.7308 |
| Since 72.21 has four digits, the sum can have no more than four Significant figures. Therefore, the answer is correctly reported as 102.73. |
| Subtraction Example: |
| 23.56 |
| - 9.16 |
| |
| 14.4 |
| Since 9.16 only has three Signicficant Figures, the answer will have three Significant Figures. Since the answer has three Significant Figures it is not necessary to change it. |
| Multiplication and Division: |
| When multiplying or dividing, the answer should have no more Significant Figures than the measurement with the least Significant Figures. The placement of the decimal point is not dependent on the number of Significant Figures. |
| Multiplication Example: |
| 31.416 | X 9.66 | |
| 303.47856 |
| (a.k.a., "Calculator Answer") |
| Since 9.66 has the least number of Significant Figures (3), the answer should have no more than three Significantct Figures. Therefore, the answer is reported as 303. |
| Division Example: |
| 716.88 ÷ 581.5 = 1.238117 |
| (a.k.a., "Calculator Answer") | Since 581.5 has the least number of Significant Figures (4), the answer should have no more than FOUR Significantct Figures. Therefore, the answer is reported as 1.233. |
| Some Numbers are Infinitely Significantt | |
| Example 1: | 100 centimeters = 1 meter (by definition) |
| Example 2: | 1 Liter = 1000 milliLiters (by definition of milliLiters) |
| The reason for an infinite number of Significant Figures is to avoid such numbers as 1 in the 1 L when, by definition, the 1 is meant to be an exact number. |
| Numbers resulting from Multiplicaton or Division | |
| In the Division Example of the previous section the answer was 1.238117. The reported value was 1.24. The reason for the change from 1.23 to 1.24 is as follows: | |
| A. | If the first digit to be rounded is greater than 5, increase the digit to the left by one. If it is less than 5, the digit on the left remains as is. If it is exactly 5, the digit on the left is adjusted to make it an even number. |
| B. | Exact numbers are treated as having an infinite number of Significant Figures. As an example, 100 centimeters = 1 meter, the numbers 100 and 1 are exact and, for purposes of rounding answers, they will have an infinite number of Significant Figures. |
| Example of Multiplication: |
| A board is 5.08 cm long X 10.16 cm wide. What is the area when the answer is correctly rounded? |
| Area = Length) x Width) = 5.08 cm X 10.16 cm = 51.6128 cm2 |
| Area = 51.6 cm2 (rounded correctly) |
| The measured length, 5.08 cm, has only three Significant Figures, while the width, 10.16 cm, has four. As a result, the calculated area should contain the smallest number, three, of Significant Figures. |
| Example of Division: |
| The area of a board is 51.6 cm2 and the length is 5.08 cm. What is the width when the answer is correctly rounded? |
| Width = Area ÷ Length) = 51.6 cm2 ÷ 5.08 cm = 10.15748 cm |
| The measured length, 5.08 cm, has only three Significant Figures as does the Area. Therefore, the calculated width should contain the smallest number, three, of Significant figures. The answer, correctly rounded, is 10.2 cm. |
| Numbers resulting from Addition or Subtraction |
| Example of Addition: |
| A geologist collected a three rock samples that weigh 11.132 kg, 25.1 kg, and 1.1666 kg. What is the sum of the weights of all three rocks when the answer is correctly rounded? |
| Addition Answer |
| 11.132 kg | 25.1 kg | 1.1666 kg | |
37.3986 kg |
| One measurement only has three Significant Figures (25.1 kg). Therefore the answer should contain only three Significant Figures. The correctly rounded answer is 37.4 kg. |
| Example of Subtraction |
| The same geologist in the Addition Example had trouble carrying the rocks. He decided it was time to "get in shape." He weighs 101 kg and is five-foot-four tall. He wants to be a trim 75.44 kg. How much weight does he need to lose and what is the answer correctly rounded? |
| Subtraction Answer |
| 101 kg | - 75.44 kg |
| |
| 25.56 kg. |
| Since his original weight is known to three Sigificant Figures, his corrected weight loss is 25.6 kg. |
