Significant figures are the digits in a measurement that convey meaningful information about its precision. They provide a way to communicate how precisely a quantity has been measured and help prevent reporting more certainty than is justified by the measuring process.
Because no measurement is perfectly exact, the number of digits reported in a measurement is important. Significant figures indicate which digits are known with confidence and which digit has been estimated.
Why Significant Figures Matter
Consider two measurements of the same length:
12 cm
12.000 cm
Although both measurements have the same numerical value, they do not imply the same level of precision. The measurement 12.000 cm suggests a much more precise measurement than 12 cm.
Significant figures help communicate the quality of a measurement and prevent the reporting of unnecessary or misleading digits.
In science, engineering, and manufacturing, reported values should reflect the capability of the measuring instrument and the measurement process.
Reading Measurements
When using an analog measuring instrument, all digits that can be read directly from the scale are recorded, along with one estimated digit.
For example, suppose a ruler is graduated in millimeters. If an object appears to measure slightly more than 25 mm, the measurement might be recorded as:
25.3 mm
The final digit is an estimate made by the observer. Although estimated, it is still considered significant because it represents the best available interpretation of the measurement.
This practice allows measurements to convey both the measured value and the precision of the measurement process.
Identifying Significant Figures
Several general rules are commonly used to determine the number of significant figures in a measurement.
All nonzero digits are significant.
Examples:
25.4 has three significant figures.
1738 has four significant figures.
Zeros between nonzero digits are significant.
Examples:
101 has three significant figures.
20.05 has four significant figures.
Leading zeros are not significant. They merely locate the decimal point.
Examples:
0.25 has two significant figures.
0.0034 has two significant figures.
Trailing zeros to the right of a decimal point are significant.
Examples:
12.0 has three significant figures.
12.00 has four significant figures.
12.000 has five significant figures.
Significant Figures and Measurement
The number of significant figures reported should be consistent with the precision of the measuring instrument.
Suppose a steel rule is graduated in millimeters. Reporting a measurement as:
25.37284 mm
would imply a level of precision far beyond the capability of the instrument.
A more realistic measurement might be:
25.4 mm
Similarly, a digital caliper displaying measurements to 0.01 mm should generally not be reported as:
25.400000 mm
unless additional information justifies those extra digits.
The goal is to report enough digits to reflect the measurement without implying unwarranted accuracy.
Significant Figures in Calculations
When measurements are used in calculations, the result should not imply greater precision than the measurements used to obtain it.
For example, suppose two lengths are measured as:
12.3 cm
4.56 cm
The calculated area is:
12.3 × 4.56 = 56.088 cm²
Reporting the result as 56.088 cm² suggests a level of precision that does not exist in the original measurements. The result should instead be rounded to an appropriate number of significant figures:
56.1 cm²
Rules for determining the number of significant figures in calculated results vary depending on the mathematical operation being performed, but the underlying principle remains the same: the final answer should reflect the precision of the measurements used.
Significant Figures and Modern Instruments
Digital instruments often display more digits than are truly meaningful. The presence of additional digits on a display does not necessarily mean that all of those digits are significant.
The practical significance of a digit depends on factors such as instrument calibration, resolution, measurement uncertainty, and the measurement procedure itself.
For this reason, engineers and technicians must use judgment when recording and reporting measurement results.
Summary
Significant figures communicate the precision of a measurement. They indicate which digits are known with confidence and which digit has been estimated. Proper use of significant figures helps ensure that reported measurements and calculated results accurately reflect the quality of the measurement process without implying greater certainty than actually exists.