“My lab report was marked down for too many digits.” “I rounded, but my answer still does not match.” — Significant figures come down to three rules:
- Count from the first nonzero digit to the last written digit
- Round once, at the very end
- Multiplication and division use the fewest significant figures; addition and subtraction use the fewest decimal places
This guide works through each rule with examples, then shows a tool walkthrough and the limits of the method.
What significant figures represent
JIS Z 8103:2019 defines significant figures as the meaningful digits of a measured value, excluding zeros that only mark the decimal place. They show how far a measurement can be trusted: 9.0 means 8.95 up to 9.05, while 9 means 8.5 up to 9.5.
Exact values are not limited: counts (3 items), defined conversions (1 in = 25.4 mm) and constants such as π have no measurement error.
Counting digits: four rules
Only zeros need special care.
- All nonzero digits count.
- Zeros between nonzero digits count: 1.02 → 3
- Trailing zeros after a decimal point count: 1.000 → 4, 0.00120 → 3
- Leading zeros do not count: 0.0012 → 2
| Value | Sig figs | Reason |
|---|---|---|
| 0.00120 | 3 | Leading zeros do not count; the final zero does |
| 1.02 | 3 | The zero between digits counts |
| 1.000 | 4 | Trailing zeros after the decimal point count |
| 0.0012 | 2 | Leading zeros mark the decimal place |
| 1200 | Context-dependent; tool uses 2 | The notation alone does not determine whether trailing zeros are significant |
| 1200. | 4 | The decimal point makes the trailing zeros significant |
| 1.200×10³ | 4 | Scientific notation makes the count explicit |
For 1200 the trailing zeros are ambiguous, so two, three or four figures may be intended. Write 1.200×10³ to be explicit; the tool shows the scientific notation with the count.
Rounding: half-up and half-even
Both methods agree unless the next digit is exactly 5 with nothing below.
- Half-up: always round 5 upward
- JIS Z 8401 half-even: keep the retained digit even
Examples: 2.5 to a whole number gives 3 with half-up and 2 with half-even; 3.5 gives 4 either way; 23.45 to three figures gives 23.5 or 23.4. Follow your course convention; JIS Z 8401 specifies half-even.
Round once: to round 7.3453 to two figures, do not go 7.35 → 7.4. Look at all following digits together and write 7.3.
Arithmetic rules: two different targets
- Multiplication and division: fewest significant figures. 1.23 × 4.5 = 5.535 → 5.5; 11.2 × 4.2 = 47.04 → 47
- Addition and subtraction: fewest decimal places. 28.37 + 15.4 = 43.77 → 43.8; 12.34 + 1.2 = 13.54 → 13.5
Matching figure counts in addition is wrong: 12.34 + 1.2 is not 13. Keep extra digits and round only at the end.
Common mistakes
- Counting leading zeros: treating 0.0012 as four figures
- Treating trailing zeros alike: calling 1200 four figures without context, or ignoring the zero in 3.40
- Mixing the rules: figure counts for addition, decimal places for multiplication
- Rounding too early, or applying the rules to exact values
- Adding different units: km with m, or g with kg, hides the coarsest place
Check your work with the tool
The Significant Figures Calculator on Tools Hub has three modes that reproduce the examples above.
Count significant figures
Choose "Count significant figures" and enter 0.00120. The tool shows "Significant figures: 3" with the scientific notation 1.20×10⁻³. Enter 1200 to see the count drop to 2.
Round to a chosen precision
Choose "Round to significant figures", enter 23.45 and set the digits to 3. Switch between half-up (23.5) and half-even (23.4).
Apply the rule to an operation
Choose "Apply the rule to an operation" and enter 1.23 × 4.5. The tool shows the unrounded 5.535 and the result rounded to the fewest significant figures (5.5). With 12.34 + 1.2 it returns 13.5.
Tool used in this guide
Significant Figures Calculator
Count digits, round with half-up or half-even, and apply the arithmetic rules with the working shown. Free, browser-only, no sign-up.
For examples where significant figures matter in practice, see the molarity guide and the recrystallization guide.
When the rules apply, and their limits
- Significant figures are a simple way to express uncertainty. For precision, write it directly: (8.540 ± 0.004) m/s or 8.540(4) m/s.
- The rules do not track accumulated rounding error. Keep guard digits and round once.
- Your course convention takes priority; otherwise match the measured values.
- The tool performs one binary operation, displays up to 10 digits, and does not infer measurement uncertainty.
Summary
- Count from the first nonzero digit to the last written digit
- Integer trailing zeros are placeholders; use scientific notation
- Round once; half-up and half-even differ only for an exact 5
- Multiplication/division uses fewest figures; addition/subtraction uses fewest decimal places
- Exact values are not limited
FAQ
How many significant figures does 1200 have?
The notation alone is ambiguous. The tool uses a convention that ignores integer trailing zeros and reports two. Write 1.200×10³ (or 1200.) to show four.
Should I use half-up or half-even rounding?
Follow your course instructions; JIS Z 8401 specifies half-even. They differ only at an exact 5: 2.5 gives 3 with half-up and 2 with half-even, while 3.5 gives 4 either way.
Why do multiplication and addition use different rules?
In multiplication the relative error follows the figure count. In addition, digits below the coarsest place are unreliable: 12.34 + 1.2 is 13.5, not 13.
Can I round partway through a calculation?
No. Rounding 7.3453 to two figures in two steps gives 7.4; a single step at the end gives the correct 7.3.
