Significant Figures - Quick Review
What is a significant figure?
Many numbers used in physics are measured values, or are derived from measured values. Because measuring devices have limitations as to their accuracy, we need a system by which to convey the accuracy of a given value. Significant figures are used to convey the digits in a number that are truly meaningful.
Rules for Determining Number of Significant Figures
The following rules are used to determine the number of significant figures:
All nonzero digits are significant.
5.234 m has 4 significant figures; 1.2 m has 2 significant figures.
Zeroes between nonzero digits are significant.
2005 kg has 4 significant figures; 5.01 kg has 3 significant figures.
Trailing zeroes to the right of the decimal point are significant.
0.0540 ft has 3 significant figures; 2.800 ft has 4 significant figures.
Zeroes to the left of the first nonzero digit are not significant, these are merely place holders.
0.004 s has 1 significant figure; 0.0170 s has 3 significant figures.
Trailing zeroes in numbers with no decimal point are not significant, these are also merely place holders.
120 cm has 2 significant figures; 50,200 cm has 3 significant figures.
Using Scientific Notation to Specify Significant Figures
Scientific notation can be used to specify the number of significant figures when trailing zeros are involved.
Here are some examples:
- 250 kg has 2 significant figures, whereas 2.50 x 102 kg has 3 significant figures.
- 5040 m has 3 significant figure, whereas 5.040 x 103 m has 4 significant figures.
- 5,000,000 N has 1 significant figure, whereas 5.00 x 106 N has 3 significant figures.
Significant Figures with Exact numbers
An exact number is a value that is known with complete certainty. An exact number is considered to have an infinite number of significant figures and does not limit the number of significant figures in a calculation.
Certain unit conversions are considered to be exact (e.g., 12 in = 1 ft; 1 day = 24 h). For those conversions which are not exact, we can generally obtain the necessary number of significant figures so that it does not become a limiting factor in the calculation (e.g., 1 m = 3.2808 ft).
Counting numbers and whole quantities of objects are considered to be exact (e.g., 7 apples; 4 jets).
Fractions are generally considered to be exact (e.g., 1/2 the distance to the goal line).
Certain events with precise names are considered to be exact (e.g., 100 m dash; 5K run).
Rules for Mathematical Operations with Significant Figures
In carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate value involved in the calculation.
The following significant figure rules are used for mathematical operations:
Addition & Subtraction: The answer is limited to the precision of the least precise value. In other words, the number of decimal places retained in the answer should equal the smallest number of decimal places in any of the values used in the calculation. For example: 1.25 m + 100.5 m should be stated as 101.8 m, and not 101.75 m.
Multiplication & Division: The value with the least number of significant figures used in the calculation determines the number of significant figures in the answer. For example: 2.50 kg x 2.0 kg should be stated as 5.0 kg, and not 5.00 kg.
Carrying Significant Figures in Calculations
When solving multiple-step calculations, you should carry at least one extra significant figure for all intermediate calculations and then round the final answer using significant figure rules. If all intermediate calculations are run in succession on a calculator, it automatically keeps track of additional significant figures for each step in the calculation.