Arithmetic
What Are Significant Figures?
Significant figures communicate the precision of a measured quantity. They include all known digits plus the final estimated digit, so they describe information in the measurement rather than the size of the number.
Reviewed
Which digits count
Every nonzero digit is significant. Zeros between nonzero digits are significant, and leading zeros are not because they only locate the decimal point. Trailing zeros after a decimal point are significant when they were recorded as part of the measurement.
A whole number such as 1200 can be ambiguous. Scientific notation removes that ambiguity: 1.2 x 10^3 has two significant figures, while 1.200 x 10^3 has four.
- 0.00450 has three significant figures: 4, 5, and the trailing 0.
- 1002 has four significant figures because the zeros lie between nonzero digits.
- 7.00 has three significant figures; the zeros record measured precision.
Rounding a reported value
Keep one extra guard digit while calculating, then round once at the end. If the first discarded digit is greater than 5, round up; if it is less than 5, leave the retained digit unchanged.
When the discarded part is exactly halfway, some scientific standards use round-to-even to avoid a long-run upward bias. Follow the convention required by the course, laboratory, or standard you are using.
Different rules for different operations
For multiplication and division, the result normally has the same number of significant figures as the least precise input. For addition and subtraction, precision is controlled by decimal place, not by the total count of significant figures.
Exact counts and defined conversion factors do not limit significant figures. Twelve objects is an exact count; 1 inch = 2.54 centimeters is defined exactly.
multiplication/division: fewest significant figures; addition/subtraction: least precise decimal placeCommon mistakes
- Counting leading zeros as measured digits.
- Rounding every intermediate step and accumulating avoidable error.
- Applying the multiplication rule to addition or treating exact definitions as measurements.
Worked examples
Multiply measured values
Calculate 4.56 x 1.4.
The calculator value is 6.3844.56 has three significant figures1.4 has two significant figures
Result: Report 6.4 to two significant figures
Add measured values
Calculate 12.11 + 0.3 + 1.245.
The unrounded sum is 13.655The least precise input is measured to the tenths place
Result: Report 13.7
Where this idea is used
- Reporting laboratory measurements and derived quantities.
- Choosing a useful display precision for scientific notation.
- Distinguishing measurement uncertainty from ordinary calculator precision.
Questions about significant figures
Are significant figures the same as decimal places?
No. Significant figures begin at the first nonzero measured digit, while decimal places count positions to the right of the decimal point.
Is every digit shown by a calculator significant?
No. A calculator displays arithmetic precision; the inputs determine how much measurement precision the result can claim.
Why can 1200 be ambiguous?
Without a decimal point, uncertainty notation, or scientific notation, the written number does not reveal whether one or both trailing zeros were measured.