Significant Figures Calculator
Counts sig figs digit by digit, rounds correctly and applies the rules to + − × ÷.
Significant figures
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- Significant
- Not significant
- Ambiguous
Why
Rounded
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Steps
| Sig. figs | Standard | Scientific |
|---|
Answer
This is the last answer worked out. Fix the input above to update it.
| Number | Sig. figs | Known to the | Exact? |
|---|
Tick Exact for counted numbers and defined values (12 eggs, 100 cm in 1 m) — they never limit the precision.
Steps
About the Significant Figures Calculator
Type a number to see how many significant figures it has. Every digit is marked significant, not significant or ambiguous, with the rule behind it — leading zeros, zeros between digits, trailing zeros with and without a decimal point, and scientific notation.
Round any number to the significant figures you need, rounding a dropped 5 up or to the even digit as ASTM E29 specifies, and see when standard notation cannot show the result (1.50 × 10³, not 1500). In Calculate mode, type a calculation such as (12.11 + 18.0) × 1.013: all digits are kept until the end, the decimal place is tracked after each addition or subtraction and the significant figures after each multiplication or division, and you get both the unrounded and the correctly rounded answer.
How to use it
- Choose Count, Round or Calculate.
- Count: type a number in standard or scientific notation, such as
0.004500,1500.,4.50e-3or4.50 × 10^-3. Each digit is marked ● significant, ○ not significant or ? ambiguous, and the reasons are listed below. - Round: enter the number and how many significant figures to keep, and choose what happens when the dropped part is exactly 5.
- Calculate: type the calculation with + − × ÷ and brackets. In the table, tick Exact for counted or defined numbers, and pick a reading for numbers with ambiguous zeros such as 1500.
- Copy the answer, or use Copy working to copy every step as text.
Examples
0.004500
4 significant figures
The three leading zeros only place the decimal point; the two trailing zeros count because the number has a decimal point.
1500
2 (could be 3 or 4)
Write 1.5 × 10³, 1.50 × 10³ or 1.500 × 10³ — or 1500. with a decimal point for 4 — so the precision is clear.
0.045678 to 3 significant figures
0.0457
Count from the 4. The digit after 4, 5, 6 is 7, so the 6 rounds up.
0.1245 to 3 significant figures (ASTM E29)
0.124
The dropped part is exactly 5 and the 4 is even, so it stays. Rounding half up gives 0.125.
(12.11 + 18.0) × 1.013
30.5
12.11 + 18.0 = 30.11, good to the tenths place (3 significant figures). Then 30.11 × 1.013 = 30.50143, rounded to 3 significant figures.
Common uses
- Checking chemistry and physics homework where answers must have the right number of significant figures.
- Writing lab results with the precision of the measuring instrument.
- Rounding test data consistently with the ASTM E29 round-half-to-even rule.
The counting rules
- Non-zero digits are always significant: 254 has 3.
- Zeros between non-zero digits are significant: 105 has 3, 4.008 has 4.
- Leading zeros are never significant — they only place the decimal point: 0.0045 has 2.
- Trailing zeros with a decimal point are significant: 2.50 has 3, 1500. has 4, 0.0450 has 3.
- Trailing zeros in a whole number without a decimal point are ambiguous: 1500 has 2 by the usual convention, but could have 3 or 4.
- Scientific notation: only the coefficient counts — 4.50 × 10⁻³ has 3; the power of ten just places the point.
Rules for calculations
- Multiplying and dividing: the answer has as many significant figures as the number with the fewest. 2.5 × 3.42 = 8.55 → 8.6 (2 significant figures).
- Adding and subtracting: round to the last decimal place that every number has. 1.234 + 10.1 = 11.334 → 11.3 (tenths).
- Mixed calculations: keep all digits until the end, but keep track of how precise each intermediate result is. Rounding at every step can change the answer: (1.06 + 0.1) × 2.15 gives 2.5 when you keep the digits, but 2.6 if 1.16 is first rounded to 1.2. The tool keeps digits by default and can round each step if your course asks for it.
- Exact numbers — counts (3 trials) and defined values (1 inch = 2.54 cm exactly) — have unlimited significant figures and never limit the answer.
Rounding a 5: half up or half to even
Most schools round a dropped 5 up: 2.45 → 2.5. ASTM E29 (Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications) rounds an exact 5 — a 5 followed by nothing or only zeros — to the even digit instead, so 2.45 → 2.4 and 2.55 → 2.6. Over many values this avoids a bias upwards. If any non-zero digit follows the 5, both methods round up.
Always round once, from the full value. Rounding 2.4449 to three figures gives 2.44; rounding it first to 2.445 and then again would wrongly give 2.45.
Writing results so the precision is clear
A whole number with trailing zeros, such as 1500, does not show whether the zeros were measured. Scientific notation removes the doubt: 1.5 × 10³ has 2 significant figures and 1.50 × 10³ has 3. A decimal point after the number (1500.) means every digit is significant. The calculator switches to scientific notation, or adds that decimal point, whenever plain digits would hide significant zeros. Very small and very large answers — below 10⁻⁶, or 10¹⁵ and above — are given in scientific notation too, because a long row of zeros is hard to read, and Round keeps a number you typed in scientific notation in scientific notation.
Limitations
- Calculations support + − × ÷ and brackets. Powers, roots and logarithms follow different precision rules and are not supported.
- Numbers can have up to 200 digits (100 inside a calculation) and powers of ten up to ±1000.
- Ambiguous trailing zeros are read as not significant unless you choose otherwise — the tool cannot know how a number was measured.
- Significant figures are a quick way to show precision. For a formal uncertainty analysis use the methods of the GUM (JCGM 100, Guide to the Expression of Uncertainty in Measurement).
Privacy
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Frequently asked questions
How many significant figures does 1500 have?
By the usual convention 2: the trailing zeros of a whole number without a decimal point are treated as placeholders. They might have been measured, though, so 1500 could have 2, 3 or 4. Write 1.5 × 10³, 1.50 × 10³ or 1.500 × 10³ (or 1500. for 4) to remove the doubt.
Are zeros significant?
It depends where they are. Zeros between non-zero digits always count (105 has 3). Leading zeros never count (0.0045 has 2). Trailing zeros count when the number has a decimal point (2.50 has 3, 1500. has 4) and are ambiguous in a whole number without one (1500).
How do I round to 3 significant figures?
Start at the first non-zero digit and count three digits. If the next digit is 5 or more, round the third digit up; otherwise leave it. 0.045678 → 0.0457, and 123,456 → 123,000 — write 1.23 × 10⁵ to show that it has 3 significant figures.
What are the significant figure rules for adding and multiplying?
When multiplying or dividing, keep as many significant figures as the number with the fewest. When adding or subtracting, round to the last decimal place that every number has. So 2.5 × 3.42 = 8.6, but 2.5 + 3.42 = 5.9.
Do exact numbers limit the significant figures?
No. Counted values (12 eggs) and defined conversion factors (1 inch = 2.54 cm exactly) have unlimited significant figures, so only the measured numbers limit the answer. Tick Exact next to such numbers in Calculate mode.
Does the power of ten count in scientific notation?
No. In 4.50 × 10⁻³ only the coefficient 4.50 counts, so it has 3 significant figures; the power of ten only places the decimal point.
Why is the answer shown in scientific notation?
Because plain digits would hide significant zeros. 12.0 × 125 = 1500 to 3 significant figures, but 1500 reads as 2, so the answer is shown as 1.50 × 10³.