How to use the significant figures calculator
- Under What to do, choose Count sig figs to check numbers, or Add, Subtract, Multiply or Divide to calculate with them.
- Type the numbers exactly as written, one per line or separated by spaces or commas. Keep trailing zeros, because they carry information: 2.50 and 2.5 are not the same measurement. Use e notation for powers of ten, such as 6.02e23.
- Read the headline answer, then the exact result before rounding and the number of figures or the decimal place that set the precision.
- The table marks which number limited the answer, so you can see where the precision was lost.
Formula
Count: nonzero digits and zeros between them count; leading zeros do not; trailing zeros count after a decimal point
Add or subtract: round to the coarsest last digit among the numbers
Multiply or divide: round to the fewest significant figures among the numbers
The arithmetic is done exactly with whole-number math, with no floating-point error, and then rounded half up at the end. A plain whole number that ends in zeros, such as 1200, is counted conservatively as 2 figures, with 4 shown as the most it could be.
Counting and calculating
0.00450 has 3 significant figures, because the leading zeros only place the decimal point and the trailing zero after the 5 is a measured digit. 1200 has 2 (and up to 4), 1200. with a decimal point has 4, 1.20e3 has 3, 100.0 has 4 and 0.0030 has 2.
For 12.11 + 18.0 + 1.013 the exact sum is 31.123, but 18.0 is known only to the tenths place, so the answer is 31.1. For 5.3 x 2.14 the exact product is 11.342, and 5.3 has only 2 figures, so the answer is 11. For 6.0e3 / 3.00 the answer keeps 2 figures: 2000, written 2.0 x 10^3. And 9.96 x 1.0 rounds to 10, which has 2 figures, not 10.0.
Why the two rules differ
When you add measurements, what matters is how far out each one is known, in decimal places. A sum cannot be known more finely than its roughest part: adding a length measured to the nearest centimeter to one measured to the nearest millimeter gives a result good only to the centimeter.
When you multiply, the relative size of the uncertainty matters instead, and the number of significant figures is a rough measure of that. The result keeps as many figures as the least precise factor. The rules are shortcuts for propagating uncertainty. For a proper treatment with stated uncertainties, use the Error Propagation Calculator.
Good habits
Round only at the end of a multi-step calculation, carrying extra digits until then. Exact counts (12 eggs) and defined constants (100 centimeters in a meter) have unlimited figures and should not limit the answer. To round a number to a chosen number of figures without a calculation, the Rounding Calculator does that, and the Scientific Notation Converter rewrites numbers in e notation.
More math tools
For complex numbers, polynomials, repeating decimals, differential equations and graphs, try the Complex Number Calculator, Polynomial Arithmetic Calculator and Repeating Decimal to Fraction Converter.
Assumptions and limits
- The numbers are measurements written the way they were measured, with every measured digit shown.
- Rounding is half up (away from zero for negative numbers), the convention taught in most science courses.
- A whole number that ends in zeros and has no decimal point is counted conservatively, with the zeros not counted. Add a decimal point or use e notation if they are measured.
- Zero is counted as one significant figure, at the last place written.
- The example numbers are an illustration. Replace them with your own.
Frequently asked questions
Does 100 have one, two or three significant figures?
It is ambiguous written that way. This page counts it as 1 figure and shows 3 as the most it could be. Write 1.0e2 for two figures, 1.00e2 for three or 100. with a decimal point for three.
Why is the answer to 9.96 x 1.0 shown as 10?
The answer keeps 2 figures, and 9.96 rounds up to 10. Writing 10.0 would claim 3 figures, which the factor 1.0 does not support, so the result is shown as 10.
Do I round each step of a long calculation?
No. Keep extra digits through the steps and round once at the end, or the small rounding errors add up. Use the exact result shown here to check the last step.
Can I mix the two rules in one expression?
Yes, but do each operation in order of the usual rules, keeping extra digits, and apply the right rule at each step of the final answer. This page handles one operation at a time.