Significant Figures Calculator
Add, subtract, multiply and divide measured values with the right number of significant figures. Every step shows which rule applies — the least precise decimal place for + and −, the fewest significant figures for × and ÷ — and the answer is rounded only at the end.
Why Two Different Rules?
A calculated answer cannot be more precise than the measurements it comes from. How that works out depends on the operation.
Adding and subtracting combine absolute uncertainties: if one length is known to the tenth of a centimetre, the sum is too, however precise the other length is. So the answer keeps the least precise decimal place: 12.11 + 18.0 = 30.11 → 30.1. Subtracting nearly equal numbers can lose most of the significant figures: 100.0 − 99.5 = 0.5 has only one.
Multiplying and dividing combine relative uncertainties, so the answer keeps the fewest significant figures: 4.56 × 1.4 = 6.384 → 6.4. A power is repeated multiplication and keeps the significant figures of its base.
Digit by Digit
digitsStep-by-Step Solution
Worked Techniques
For + and −, write the numbers under each other: the column where the first number ends early (18.0 stops at the tenths) is where the answer must stop.
For × and ÷, count the significant figures of every number first; the smallest count is the number of figures in the answer.
Work step by step: the sum is known to the tenths (7.3, 2 significant figures), and the product keeps 2. Keep the extra digits until the end so rounding errors do not pile up.
Sources & Verification
- J. R. Taylor, An Introduction to Error Analysis, 2nd ed., University Science Books, 1997 — Ch. 2 and 3 (significant figures; how uncertainties combine in sums and in products).
- A. Thompson & B. N. Taylor, Guide for the Use of the International System of Units (SI), NIST Special Publication 811, 2008 — Appendix B (rounding numbers).
Frequently Asked Questions
What is the rule for adding and subtracting with significant figures?
The answer keeps the least precise decimal place of the numbers you add or subtract. Example: 12.11 + 18.0 + 1.013.
1. Add as usual: 12.11 + 18.0 + 1.013 = 31.123.
2. 12.11 is known to the hundredths, 18.0 only to the tenths, 1.013 to the thousandths. The least precise place is the tenths.
3. Round the answer to the tenths: 31.1 (3 significant figures).
What is the rule for multiplying and dividing with significant figures?
The answer has as many significant figures as the number with the fewest. 4.56 × 1.4 = 6.384: 4.56 has 3 significant figures and 1.4 has 2, so the answer is 6.4. Dividing works the same way: 2.00 ÷ 3.00 = 0.666…, which becomes 0.667 (3 significant figures). A power is repeated multiplication: 2.0³ = 8.0 keeps the 2 significant figures of 2.0.
How do you handle a calculation with both + and ×?
Work step by step with the right rule for each step, keep the unrounded values, and round only at the end. (3.2 + 4.11) × 2.0: the sum 7.31 is known to the tenths, so it has 2 significant figures; then 7.31 × 2.0 = 14.62 keeps 2 significant figures: 15. Rounding in between would let rounding errors pile up.
Why can subtracting lose significant figures?
Subtracting keeps the decimal place, not the number of significant figures. 100.0 − 99.5 = 0.5: 100.0 has 4 significant figures and 99.5 has 3, but the difference is only known to the tenths, which leaves 1 significant figure. Taking the difference of two nearly equal measurements is therefore much less precise than the measurements themselves.
Does dividing by a counted number limit the significant figures?
No, a counted or defined number is exact. The average of 12.1 and 12.3 is (12.1 + 12.3) ÷ 2 = 12.2, with 3 significant figures, because the 2 is a count. If the 2 were treated as a measurement with 1 significant figure, the answer would be rounded to 1 × 10¹. Choose “Exact” for whole numbers that are counts or definitions.