Significant Figures Calculator
Type a number to see how many significant figures it has, with every digit explained, and round it to any number of sig figs. Sig fig maths applies the rules for + − × ÷ to a whole sum. Advanced adds 10 rounding methods and rounding to decimal places or the nearest 10, 0.5 or any step.
Scientific notation works too: 6.02e23, 6.02 × 10^23 or 6.02×10²³.
Use + − × ÷ and brackets. Adding and subtracting keep the fewest decimal places; multiplying and dividing keep the fewest significant figures.
Any step works: 0.05 for 5 cents, 0.25 for quarters, 15 for quarter hours in minutes.
The half methods only differ when the dropped part is exactly half, like 2.5.
With a decimal point (1500.) they always count.
How your answer is worked out
Your number, digit by digit: skip the leading zeros, check the other zeros, then round.
- 1
Read the number
0.0040507 digits
- 2
Skip leading zeros
3 leading zerosnever count4 left
- 3
Check other zeros
1 in between: yes1 trailing zero: yes= 4 sig figs
- 4
Round to 2 s.f.
next digit: 5so round up= 0.0041
Read more: the rules, the formula, worked examples and rounding methods
Significant figures (sig figs) are the digits in a number that carry real meaning about how precisely it was measured. A ruler marked in millimetres can give 12.3 cm (3 significant figures), but not 12.34 cm. Zeros are the tricky part: some only hold the decimal point in place, others are real measured digits.
The rules for counting
- Non-zero digits always count. 1.234 has 4.
- Zeros between non-zero digits always count. 10.05 has 4.
- Leading zeros never count. 0.0045 has 2. They only show where the decimal point is.
- Trailing zeros count if there is a decimal point. 2.50 has 3, 1500. has 4 and 1500.0 has 5.
- Trailing zeros in a whole number with no decimal point usually don’t count. 1500 has 2. In Advanced you can choose to count them if you know they were measured.
- In scientific notation, every digit in the first part counts and the power of ten never does. 6.020 × 10²³ has 4.
- Exact numbers (counted things like 3 people, or defined values like 100 cm in a metre) have unlimited significant figures and never limit an answer.
The formula
significant figures = the digits from the first non-zero digit to the last digit that counts
rounded to n sig figs = round(x ÷ 10^(p − n + 1)) × 10^(p − n + 1), where p = the power of ten of the first significant digit
rounded to d decimal places = round(x × 10^d) ÷ 10^d
rounded to the nearest m = round(x ÷ m) × m
a + b or a − b: keep the decimal places of the least precise number
a × b or a ÷ b: keep the significant figures of the number with the fewest
Worked example
Counting 0.004050. The 0 before the point and the two 0s after it are leading zeros, so they don’t count. The 4 counts, the 0 between the 4 and the 5 counts, the 5 counts, and the last 0 counts because there is a decimal point. That is 4 significant figures, or 4.050 × 10⁻³ in scientific notation.
Rounding it to 2 significant figures. The first significant digit, 4, is in the thousandths, so p = −3 and p − n + 1 = −3 − 2 + 1 = −4. Then 0.004050 ÷ 10⁻⁴ = 40.50. The dropped part, 0.50, is exactly half, and the school rule (half away from zero) rounds it up to 41. Finally 41 × 10⁻⁴ = 0.0041, or 4.1 × 10⁻³. With banker’s rounding (half to even) the 40 is already even, so the answer would be 0.0040.
Adding 12.11 + 18.0 + 1.013. The exact sum is 31.123. The numbers have 2, 1 and 3 decimal places, and 18.0 has the fewest (1), so the answer is rounded to 1 decimal place: 31.1.
Multiplying 4.56 × 1.4. The exact product is 6.384. 4.56 has 3 significant figures and 1.4 has 2, so the answer keeps 2: 6.4.
Mixed sums
Apply the rules one step at a time, in the usual order (brackets, then × and ÷, then + and −). Keep the extra digits for the next step and only round the final answer, or rounding errors build up. For (2.0 + 3.456) × 1.25: 2.0 + 3.456 = 5.456, which is good to 1 decimal place, so it has 2 significant figures (5.5). Then 5.456 × 1.25 = 6.82, and the fewest significant figures is 2, so the answer is 6.8. In Advanced you can round every step instead, which some textbooks ask for.
Rounding methods
| Method | 2.5 | −2.5 | 2.4 | 2.6 |
|---|---|---|---|---|
| Half away from zero (school rule) | 3 | −3 | 2 | 3 |
| Half up (towards +∞) | 3 | −2 | 2 | 3 |
| Half down (towards −∞) | 2 | −3 | 2 | 3 |
| Half towards zero | 2 | −2 | 2 | 3 |
| Half to even (banker’s) | 2 | −2 | 2 | 3 |
| Half to odd | 3 | −3 | 2 | 3 |
| Always away from zero | 3 | −3 | 3 | 3 |
| Truncate (towards zero) | 2 | −2 | 2 | 2 |
| Ceiling (towards +∞) | 3 | −2 | 3 | 3 |
| Floor (towards −∞) | 2 | −3 | 2 | 2 |
Writing hidden significant zeros
Rounding 1499 to 3 significant figures gives 1500, but 1500 on its own reads as 2 significant figures. Write it as 1.50 × 10³, or put a bar over the last zero that counts (150). When the zeros reach the ones place, a decimal point does the job: 99.96 to 3 significant figures is written 100. (with the point). The calculator shows the clear way to write any answer like this.
Notation
- Scientific notation: one digit before the point, times a power of ten: 0.004050 = 4.050 × 10⁻³.
- E notation is the same thing typed on a calculator or in a spreadsheet: 4.050e−3.
- Engineering notation uses powers of ten that are multiples of 3, matching milli, kilo and mega: 0.004050 = 4.050 × 10⁻³ and 45,600 = 45.6 × 10³.
All answers are exact: the calculator works with the decimal digits you typed (up to 200 digits and powers of ten up to ±400), not with binary floating point, so 1.005 rounds to 1.01 and 0.1 + 0.2 is exactly 0.3. Long divisions are worked to 60 digits before rounding.
Common questions
How many significant figures does 0.004050 have?
It has 4. The zero before the decimal point and the two zeros after it are leading zeros, which never count. The 4 counts, the 0 between 4 and 5 counts, the 5 counts, and the last 0 counts because the number has a decimal point. In scientific notation it is 4.050 × 10⁻³, which shows the four figures clearly.
Do trailing zeros count as significant figures?
Only when the number has a decimal point. 1500.0 has 5 significant figures and 1500. (with a point) has 4. In 1500 with no point the zeros usually do not count, so it has 2, because they may only be placeholders. To show 3 figures, write 1.50 × 10³ or put a bar over the first zero.
How do you round to significant figures?
Count from the first non-zero digit and keep that many digits. Then look at the next digit: 5 or more rounds up, less than 5 leaves it. For 0.004050 to 2 significant figures, keep 0.0040, the next part is 50, so it rounds up to 0.0041. Zeros needed to hold the place stay, so 1499 to 2 significant figures is 1500.
What are the sig fig rules for adding and multiplying?
When adding or subtracting, the answer keeps as many decimal places as the number with the fewest: 12.11 + 18.0 + 1.013 = 31.123, which becomes 31.1. When multiplying or dividing, the answer keeps as many significant figures as the number with the fewest: 4.56 × 1.4 = 6.384, which becomes 6.4.
What is banker’s rounding?
Banker’s rounding, or round half to even, only changes what happens when the dropped part is exactly half. Instead of always going up, it goes to the even neighbour, so 2.345 to 2 decimal places is 2.34, while 2.355 is 2.36. Over many numbers the ups and downs balance out, so totals do not drift upwards.
Why do some calculators round 1.005 to 1.00?
Most computers store numbers in binary, and 1.005 cannot be stored exactly, so it is really saved as 1.00499999… which rounds down. This calculator works with the decimal digits you typed, so 1.005 to 2 decimal places is 1.01, and 0.1 + 0.2 is exactly 0.3.
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