Significant figures calculator
Enter a number exactly as it is written — trailing zeros included — and this counts its significant figures, rounds it to whatever precision you need, and shows the scientific-notation form. Writing 2.50 rather than 2.5 is a claim about precision, so the two are counted differently.
Result
- Significant figures in this number
- 5
- Rounded to 3 s.f.
- 0.00457
- Scientific notation
- 4.57e-3
- Decimal places as written
- 7
- Rounded to 1 s.f.
- 0.005
- Rounded to 2 s.f.
- 0.0046
- Rounded to 4 s.f.
- 0.004568
Worked examples
Rounding to three significant figures
0.0045678
- leading zeros are placeholders, so start counting at the 4
- the first three significant digits are 4, 5 and 6
- the next digit is 7, so round the 6 up to 7
0.00457
Where the trailing zero changes the answer
How many significant figures in 2.50?
- 2 and 5 are non-zero, so both count
- the final 0 sits after a decimal point
- a zero after the point is only written to claim precision, so it counts
Three significant figures — 2.5 would be only two.
Multiplication takes the least precise input
4.56 cm × 1.4 cm
- 4.56 has 3 s.f., 1.4 has 2 s.f.
- the raw product is 6.384
- round to the smaller count, 2 s.f.
6.4 cm² — the extra digits the calculator shows are not real.
Addition follows decimal places, not sig figs
12.11 g + 0.3 g
- 12.11 has 2 d.p., 0.3 has 1 d.p.
- the raw sum is 12.41
- round to the fewest decimal places, 1 d.p.
12.4 g — note this has 3 s.f. even though one input had only 1.
What significant figures actually represent
A significant figure is a digit that carries real information about a measured quantity. The idea exists because measurement is never exact: a ruler marked in millimetres cannot justify an answer quoted to the micrometre, and writing one implies a precision the instrument never had.
So significant figures are a form of honesty. The number of digits you write is a statement about how well you know the value. Reporting a length as 2.5 cm says you are confident to the nearest millimetre. Reporting it as 2.500 cm claims confidence to the nearest ten micrometres — a very different claim, made with the same physical object.
This is why the rules seem fussy about zeros. A zero can be doing one of two entirely different jobs: holding a decimal place open, or recording a measured digit that happened to be zero. The rules exist to tell those two jobs apart.
The rules, in full
All non-zero digits are significant. There is no exception to this one — 1, 2, 3 through 9 always count, wherever they appear in the number.
Zeros between non-zero digits are always significant. These are sometimes called captive or sandwiched zeros. In 1.008 all four digits count, because the two zeros are locked in place by digits on both sides and cannot be doing placeholder work.
Leading zeros are never significant. In 0.00450 the three zeros before the 4 only position the decimal point. Writing the number as 4.50 × 10⁻³ makes this obvious: the mantissa is 4.50, three significant figures, and the zeros have vanished into the exponent.
Trailing zeros after a decimal point are significant. This is the rule people most often miss. Nobody writes 2.50 by accident; the zero is there deliberately, to say the measurement was precise to the hundredths place. So 2.50 has three significant figures and 2.5 has two.
Trailing zeros in a whole number are ambiguous. In 1500, it is impossible to tell from the written form whether the value is known to the nearest unit, ten, hundred or thousand. The convention is to treat them as not significant, giving two, but the convention is a fallback rather than a fact.
Resolving the ambiguity in numbers like 1500
Scientific notation removes the problem entirely, which is a large part of why scientists use it. 1.5 × 10³ has two significant figures, 1.50 × 10³ has three, and 1.500 × 10³ has four. Each is unambiguous, and each describes a genuinely different measurement.
A trailing decimal point is an older convention that does the same job for integers: writing 1500. rather than 1500 signals that all four digits are measured. It is easy to miss in print, so scientific notation is generally preferred.
Some fields use an overbar above the last significant digit instead. It is unambiguous but awkward to typeset, and rare outside textbooks.
Exact numbers have unlimited significant figures
Not every number in a calculation is a measurement. If you count twelve items, that twelve is exact — there is no uncertainty in it, and it never limits the precision of a result. The same applies to defined conversion factors: there are exactly 12 inches in a foot and exactly 2.54 centimetres in an inch, both by definition rather than measurement.
Mathematical constants behave the same way. π is known to as many digits as you care to use, so it never constrains an answer. When a calculation gives a disappointing number of significant figures, it is worth checking whether the limiting input was genuinely measured or merely defined.
Arithmetic: two different rules for two different situations
Multiplication and division are governed by significant figures. The result takes the count of the least precise input. Multiplying a 3-figure value by a 2-figure value gives a 2-figure answer, however many digits appear on the display.
Addition and subtraction are governed by decimal places instead. Adding 12.11 to 0.3 gives 12.4, because 0.3 is only known to one decimal place. Note that the answer has three significant figures even though one input had only one — this is the clearest demonstration that the two rules are genuinely different and not interchangeable.
The reason for the difference is that addition aligns numbers by place value while multiplication scales them. When you add, uncertainty in the tenths column swamps everything to its right regardless of how large the numbers are. When you multiply, relative uncertainty is what propagates, and that is what significant figures measure.
Subtraction of two close values deserves particular care. 10.01 minus 10.00 leaves 0.01 — one significant figure remaining from inputs that each had four. This is catastrophic cancellation, and it is a common source of nonsense results in numerical work.
Round once, at the end
Carry full precision through intermediate steps and round only the final answer. Rounding at each stage compounds the error, sometimes badly.
Take 2.46 rounded to one significant figure. Done directly, the answer is 2, because the digit after the 2 is 4. Done in stages — first to two figures, giving 2.5, then to one — you get 3, which is wrong. The intermediate rounding pushed the value across a boundary it should never have crossed.
In practice this means keeping the calculator's full display through a multi-step calculation and applying the significant-figure rule once, to the result you actually report.
Rounding half: the tie-breaking problem
When the part being discarded is exactly half, there is a genuine choice. The rule taught in most schools is round half up: 2.5 becomes 3. It is simple and predictable, but it biases results upward, because ties always move in one direction.
Scientific and financial work often uses round half to even instead, also called banker's rounding. Ties go to whichever neighbour is even, so 2.5 becomes 2 and 3.5 becomes 4. Over many values the upward and downward moves cancel, which keeps sums unbiased.
This calculator uses the standard JavaScript rounding behaviour, which rounds half away from zero. For most everyday work the difference is invisible; for large aggregations of rounded values it is not.
Formula
Count from the first non-zero digit. Leading zeros never count; trailing zeros count only after a decimal point.
Significant figure count for common numbers
Every case people usually get wrong, in one place. Note that 100 and 100.00 are different claims about precision, which is why they are counted differently.
| Number | Sig figs | Why |
|---|---|---|
| 100 | 1 | Trailing zeros in a plain integer are ambiguous and conventionally not counted |
| 100. | 3 | The trailing decimal point signals that both zeros are measured |
| 100.0 | 4 | Any zero after a decimal point is significant |
| 100.00 | 5 | All five digits are measured — this is a more precise claim than 100 |
| 0.01 | 1 | Leading zeros only place the decimal point |
| 0.010 | 2 | The final zero is after the point, so it counts |
| 0.00208 | 3 | 2, 0 and 8 — the captive zero counts, the leading ones do not |
| 1.008 | 4 | Zeros between non-zero digits always count |
| 2.50 | 3 | Writing the zero claims precision to the hundredths |
| 2648 | 4 | No zeros involved, so every digit counts |
| 1.2 × 10⁵ | 2 | In scientific notation only the mantissa carries significance |
| 0 | 1 | Zero itself is conventionally treated as one significant figure |
Rounding the same number to different precisions
Rounding is applied once, to the original value. Rounding in stages compounds error — 2.46 to two figures is 2.5, but rounding first to 2.5 and then to one figure gives 3, where the correct answer is 2.
| Original | 1 s.f. | 2 s.f. | 3 s.f. | 4 s.f. |
|---|---|---|---|---|
| 2648 | 3000 | 2600 | 2650 | 2648 |
| 0.0045678 | 0.005 | 0.0046 | 0.00457 | 0.004568 |
| 1.008 | 1 | 1.0 | 1.01 | 1.008 |
| 98765 | 100000 | 99000 | 98800 | 98770 |
| 0.09999 | 0.1 | 0.10 | 0.100 | 0.09999 |
Which rule applies to which operation
| Operation | Governed by | Example |
|---|---|---|
| Multiplication | Fewest significant figures | 4.56 × 1.4 = 6.4 (2 s.f.) |
| Division | Fewest significant figures | 100.0 ÷ 3.0 = 33 (2 s.f.) |
| Addition | Fewest decimal places | 12.11 + 0.3 = 12.4 (1 d.p.) |
| Subtraction | Fewest decimal places | 9.87 − 1.2 = 8.7 (1 d.p.) |
| Logarithms | Sig figs of the input become decimal places of the result | log(2.0 × 10³) = 3.30 |
| Exact counts and constants | Unlimited — they never limit the result | 12 items, or π |
Frequently asked questions
How many sig figs in 100?
One, by the usual convention. Trailing zeros in a plain whole number are ambiguous, and the convention is not to count them. If all three digits are measured, write 100. with a trailing decimal point, or 1.00 × 10², which gives three unambiguously.
How many sig figs in 100.00?
Five. Every digit after a decimal point counts, including zeros, so all five digits are significant. This is a more precise claim than 100 — it says the value is known to the nearest hundredth.
How many sig figs in 0.01?
One. The zeros before the 1 are leading zeros that only position the decimal point. Written as 1 × 10⁻² it is obviously a single significant figure.
How many significant figures in 0.00208?
Three. The leading zeros do not count, but the zero between the 2 and the 8 is a captive zero and always does. The significant digits are 2, 0 and 8.
How many significant figures in 100.10?
Five. The captive zeros count because they sit between non-zero digits, and the final zero counts because it follows the decimal point.
What is 2648 to three significant figures?
2650. The first three significant digits are 2, 6 and 4, and the next digit is 8, so the 4 rounds up to 5.
What is 2648 to two significant figures?
2600. The first two significant digits are 2 and 6, and the next digit is 4, so the 6 stays as it is.
Are trailing zeros significant?
After a decimal point, always — 2.50 has three significant figures. In a whole number like 1500 they are ambiguous and conventionally not counted, giving two. Scientific notation removes the doubt.
Are leading zeros ever significant?
No. A leading zero only places the decimal point and never carries measured information. 0.0045 has two significant figures, not six.
What is the difference between significant figures and decimal places?
Decimal places count digits after the point regardless of value; significant figures count meaningful digits starting from the first non-zero one. 0.0045 has four decimal places but only two significant figures.
How many significant figures should my answer have?
For multiplication and division, match the input with the fewest significant figures. For addition and subtraction, match the input with the fewest decimal places. These are different rules and applying the wrong one is the most common mistake.
Do exact numbers affect significant figures?
No. Counted quantities and defined constants — twelve items, 2.54 cm per inch, π — carry unlimited precision and never limit a result. Only measured values do.
Should I round at each step of a calculation?
No. Round only the final answer. Rounding intermediate values compounds error: 2.46 to one significant figure is 2, but rounding to 2.5 first and then to one figure wrongly gives 3.
Why does 100 have fewer significant figures than 100.00?
Because they are different claims. 100 might be rounded from anywhere between 50 and 150; 100.00 says the value is known to within half a hundredth. The written form is how precision gets communicated.
How do significant figures work in scientific notation?
Only the mantissa counts. In 1.2 × 10⁵ there are two significant figures; in 1.200 × 10⁵ there are four. The exponent only sets the scale and never contributes.