How to count and round significant figures
Significant figures show how precisely a number was measured. This guide gives the counting rules with examples, explains the tricky case of trailing zeros, and shows how to round and calculate with measurements.
The short version
- Find the first non-zero digit. Everything to its left is a leading zero and never counts.
- Count from there to the last digit. Zeros between non-zero digits count; trailing zeros count only if there is a decimal point.
- To round, keep that many digits and round up if the next digit is 5 or more.
- In calculations, keep the fewest significant figures when multiplying or dividing, and the fewest decimal places when adding or subtracting.
Skip the arithmetic. The free Significant Figures Calculator does this for you, shows the formula and the working with your own numbers, and runs in your browser - nothing you type is uploaded. Also useful: Long Division Calculator.
The counting rules
| Rule | Example | Sig figs |
|---|---|---|
| Non-zero digits always count | 4.53 | 3 |
| Zeros between non-zero digits count | 1007 | 4 |
| Leading zeros never count | 0.0045 | 2 |
| Trailing zeros after a decimal point count | 0.004500 | 4 |
| Trailing zeros in a whole number are ambiguous | 1200 | 2 to 4 |
| Scientific notation removes the doubt | 1.20 × 10³ | 3 |
For a whole number like 1200, write 1200. (with a point) to show all four digits are measured, or 1.2 × 10³ to show only two.
Rounding to significant figures
Count from the first non-zero digit, keep that many, and look at the next digit. 0.012345 to 3 significant figures is 0.0123 (the next digit is 4). 2.996 to 3 significant figures is 3.00, because rounding up carries through the nines and the zeros are kept to show the precision. 123,456 to 2 significant figures is 120,000, which is best written 1.2 × 10⁵.
Calculating with measurements
Multiplying or dividing: the answer has as many significant figures as the input with the fewest. 2.5 × 3.42 = 8.55 on a calculator; 2.5 has 2 significant figures, so the answer is 8.6.
Adding or subtracting: the answer has as many decimal places as the input with the fewest. 12.11 + 18.0 = 30.11; 18.0 has one decimal place, so the answer is 30.1.
Keep all the digits in the middle of a multi-step calculation and round only the final answer.
Exact numbers do not limit precision
Counted or defined values - 12 eggs, 100 centimetres in a metre, a factor of 2 in a formula - are exact and never limit the answer. Only measured values count.
Mistakes to avoid
- Counting leading zeros as significant.
- Dropping trailing zeros after a decimal point: 2.50 and 2.5 are different precisions.
- Rounding at every step instead of at the end.
- Applying the multiplication rule to addition, or the reverse.