2.5. Significant Figures#
Learning Objectives#
By the end of this section, you will be able to:
Explain the purpose of significiant figures to engineers and scientists.
Determine the correct number of significant figures present in a measured or calculated value.
Determine the correct number of significant figures for the result of a computation.
Why Significant Figures?#
Why should engineers learn about and use significant figures?
An important factor in any measurement is the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter whereas calipers can measure length to the nearest 0.01 mm. A caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise the measurements.
When we express measured values, we can only list as many digits (called significant figures) as we measured initially with our measuring tool. For example, if we use a standard ruler to measure the length of a stick, we may measure it to be 36.7 cm. We can’t express this value as 36.71 cm because our measuring tool is not precise enough to measure a hundredth of a centimeter.
It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices the stick length seems to be somewhere in between 36.6 cm and 36.7 cm, and they must estimate the value of the last digit.
Note
Significant figures are one way that engineers and scientists communicate the precision of a measured or calculated value.
2.3 cm and 2.30000 cm are equal values. However, 2.30000 cm demonstates a value with greater precision than 2.3 cm.
Using a method of significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty. To determine the number of significant digits in a value, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 36.7 cm has three digits, or three significant figures.
Tip
Significant figures indicate the precision of the measuring tool used to measure a value.
Significant figures indicates the certainty or confidenece in a calculated value.
Counting Significant Figures#
Significant figures are critical because they indicate how precisely you can actually measure/report your measured data or calculated values. Below are the guidelines for counting the number of significant figures displayed in a measured or calculated value.
ALL non-zero digits (1,2,3,4,5,6,7,8,9) are significant
ALL zeroes between non-zero digits are significant
ALL zeroes which are SIMULTANEOUSLY to the right of the decimal point AND at the end of the number are significant
ALL zeroes which are to the left of a written decimal point and are in a number >= 10 are significant
Examples of Significant Figures
Below is a chart of example measured and calculated values and the corresponding number of significant figures represented by each value. Note the abreviation sig figs stands for significant figures.
Value |
Number of Significant Figures |
|---|---|
48,923 people |
5 sig figs |
3.967 mm |
4 sig figs |
900.06 kg |
5 sig figs |
0.0004 mW |
1 sig fig |
8.1000 MPa |
5 sig figs |
501.040 A |
6 sig figs |
10.0 cm |
3 sig figs |
130. in/in |
3 sig figs |
Significant Figures in Calculations#
The following specifies the rules for using significant figures in calculations. Remember the purpose of significant figures is to show other engineers and scientists how precisely you are certain about a measured or calculated value. Any calculation made with a measured value can not be more precise (or be known with more certainty) than the measured values that go into the calculation.
Multiply/Divide: Count the NUMBER OF SIGNIFICANT FIGURES. The answer cannot CONTAIN MORE SIGNIFICANT FIGURES THAN THE NUMBER BEING MULTIPLIED OR DIVIDED with the LEAST NUMBER OF SIGNIFICANT FIGURES.
Add/Subtract: Count the NUMBER OF DECIMAL PLACES. The answer cannot CONTAIN MORE PLACES AFTER THE DECIMAL POINT THAN THE SMALLEST NUMBER OF DECIMAL PLACES in the numbers being added or subtracted.
Exact Numbers: The numbers \(\pi\) and \(e\) are derived from mathematical relationships and are precise to an infinite number of significant figures. Exact unit conversions or exact counted quantities are precise to an infinite number of significant figures.
Rounding Error: Be aware of rounding error. Keep +2 more significant figures in all intermediate caclulations. Only round the final calculated value.
Rounding Numbers#
After completing a calculation, the final caclualted value needs to be rounded to the appropriate number of significant figures. The number of significant figures or decimal place to round to was specificed in the previous section. In this section, we discuss how to round a numerical value given the number of significant figures or decimal place has already been determined.
Replace a number having a given number of digits with a number (called the rounded number) having a smaller number of digits, apply the following rules:
Rule 1. If the digits to be discarded begin with a digit less than 5, the digit preceding the 5 is not changed.
Example: 6.9749515 rounded to 3 digits is 6.97
Rule 2. If the digits to be discarded begin with a 5 and at least one of the following digits is greater than 0, the digit preceding the 5 is increased by 1.
Examples: 6.9749515 rounded to 2 digits is 7.0
6.9749515 rounded to 5 digits is 6.9750
Rule 3. If the digits to be discarded begin with a 5 and all of the following digits are 0, the digit preceding the 5 is unchanged if it is even and increased by 1 if it is odd. (Note that this means that the final digit is always even.)
Examples: 6.9749515 rounded to 7 digits is 6.974952
6.9749505 rounded to 7 digits is 6.974950
Section Summary#
Significant figures are important for engineers and scientists to convey the precision of measurements and calculations. Significant figures reflect the precision of the measuring tools used; only as many digits can be reported as can be confidently measured.
Round the final calculated value in a given problem to the smallest number of significant figures provided as input to the problem. In all intermediate calculations, two additional significant figures are carried out that are more than the number of significant figures supplied as input.