Significant Figures Calculator

Count the significant figures in any number, round to any number of sig figs, or run a whole calculation and see which input limited the answer.

Plain decimals, integers, 1.20e3 or 1.20 × 10³ all work.
Significant figures in 0.0045604Declared by the notation
0 (leading zero — not significant). (decimal point)0 (leading zero — not significant)0 (leading zero — not significant)4 (significant)5 (significant)6 (significant)0 (significant)
  • Significant
  • Not significant
  • Not declared
Significant digits
4560
First significant digit
4
Last significant digit
0
Last significant place
millionths (10⁻⁶)
Scientific notation
4.560 × 10⁻³
E notation
4.560e-3
Notation read
decimal
Exact value
0.00456

Why

  1. Every non-zero digit is significant, starting at the leading 4.
  2. The 3 zeros before the first non-zero digit only place the decimal point, so they are not significant.
  3. The 1 trailing zero is significant because the notation declares it — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
  4. The last significant digit sits in the millionths place (10⁻⁶).

How to Use the Significant Figures Calculator

Count

Type a number and the calculator reads it as written before it reads it as a value. It reports how many significant figures the notation declares, marks each digit as significant, a leading zero or an undeclared trailing zero, names the place of the last significant digit, and restates the value in scientific notation. If the number is a count or a definition rather than a measurement, tick exact and it will carry unlimited precision.

Round

Enter a number and a target number of significant figures. The result shows where the number is cut, what the first discarded digit is, whether the discarded part was an exact tie, and which rounding policy decided the last kept digit. If the plain decimal answer would hide the precision — as 1000 does for three significant figures — the answer is reported in scientific notation instead.

Calculate

Type a whole expression. Each number becomes an operand card you can switch between Measured and Exact, and any operand whose notation is ambiguous gets a control for declaring what you meant. The answer card names the limiting operand (for × and ÷) or the limiting decimal place (for + and −), shows the unrounded result alongside the reported one, and walks through the steps.

What Are Significant Figures?

The significant figures of a measurement are the digits that carry real information about how precisely it was measured. They are also called significant digits, or just sig figs. In 4.56 g there are three; in 0.00456 g there are still three, because the zeros only locate the decimal point.

The idea exists because a written number makes a claim. Reporting a mass as 4.5 g says you know it to about a tenth of a gram; reporting it as 4.500 g says you know it to about a thousandth. Writing more digits than your instrument supports is a false claim of precision, and dropping digits your instrument earned throws away information. Significant figures are the bookkeeping convention that keeps the written number honest — a fast, approximate stand-in for a full uncertainty statement, not a replacement for one.

Significant Figures Rules

Six rules cover every case. The calculator applies exactly these, and its explanation panel names the one it used for each digit.

The significant-figure counting rules with examples
RuleExampleSig figsWhy
Non-zero digits are always significant123.455Every non-zero digit was measured; nothing else could have put it there.
Leading zeros are never significant0.00452They only place the decimal point. Writing the value as 4.5 × 10⁻³ makes them disappear without changing anything.
Captive zeros are always significant10024A zero trapped between significant digits cannot be a placeholder — it is a measured value of zero.
Trailing zeros after a decimal point are significant4.5004Nobody writes those zeros unless they mean them; they exist only to declare precision.
Trailing zeros in an integer with no decimal point are ambiguous12002, 3 or 4The notation cannot tell you whether the zeros are measured or placeholders. See the next section.
In scientific notation, every mantissa digit is significant1.200 × 10³4The exponent carries the magnitude, so the mantissa carries nothing but precision.

A sign is not a digit: −0.0320 and 0.0320 both have three significant figures. An exact number — a count, a defined conversion factor or a mathematical constant — is outside the system entirely and has unlimited significant figures.

Why Are 100, 1000 and 1200 Ambiguous?

Because plain integer notation has no way to say where the measurement stopped and the placeholders began. If you measure a distance as “about a hundred metres” and write 100 m, you mean one significant figure. If you measure it carefully as 100.0 m and then drop the decimal, the same three characters now understate what you know. Nothing in “100” distinguishes the two.

Three notations for the same value, and what each one declares
Written asValueDeclaresVerdict
12001200nothing — 2, 3 or 4 significant figuresAmbiguous. Read as 2 significant figures by the common classroom convention.
1200.12004 significant figuresUnambiguous, but the trailing point is easy to miss (and easy to lose in a spreadsheet).
1.2 × 10³12002 significant figuresUnambiguous.
1.20 × 10³12003 significant figuresUnambiguous.
1.200 × 10³12004 significant figuresUnambiguous.

This calculator refuses to pretend the ambiguity is not there. It reports the common classroom interpretation — trailing zeros in a bare integer are not counted, so 100 reads as one significant figure, 1000 as one and 1200 as two — but it labels that reading as a convention, keeps the value flagged as ambiguous, lists every unambiguous way to write it, and lets you declare which precision you actually meant before the receipt becomes final.

Significant Figures in Calculations

There is no single rule. Multiplication and division count digits; addition and subtraction count decimal places. Using the wrong one is the most common significant-figure error there is.

Multiplication & Division

The result carries as many significant figures as the measured operand with the fewest significant figures. For 4.56 × 1.4 the exact product is 6.384, but 1.4 has only two significant figures, so the answer is reported as 6.4 and 1.4 is the limiting operand. Powers and roots follow from this: raising a measurement to a whole-number power is repeated multiplication by itself, so 2.0³ = 8.0 keeps two significant figures.

Addition & Subtraction

The result is rounded to the leftmost least-significant decimal place among the measured operands — the place where the least precise number ran out. For 18.0 + 1.013 the exact sum is 19.013, but 18.0 is only known to the tenths, so the answer is 19.0. Note what did not happen: 1.013 has four significant figures and 18.0 has three, yet the answer has three — not because of any digit count, but because the tenths place is where the precision stopped.

This is also why subtraction can destroy precision. 100.0 − 99.9 = 0.1: two four-significant-figure measurements produce a one-significant-figure answer, because almost every significant digit cancelled. The calculator flags this rather than quietly reporting a confident-looking number.

Mixed Operations

When the two rules meet in one expression, apply each at its own step and never round in between. For (12.11 + 0.3) × 4.20 the bracket is limited by the tenths place, so it reports as 12.4 — but 12.41 is what enters the multiplication. The product 52.122 is then limited by the three significant figures of the bracket, giving 52.1. Round the bracket first and you inject an error the multiplication then amplifies; the step panel on this page shows the reported intermediate as a view while the unrounded value carries forward.

Exact Numbers vs Measured Numbers

A measured number came from an instrument and carries a limited precision. An exact number did not, and has unlimited significant figures. Three kinds show up constantly:

Why it matters: 4.56 × 2 gives 9 if the 2 is a measurement (one significant figure) and 9.12 if the 2 is a count. Same arithmetic, different reported precision. In Calculate mode each operand has a Measured/Exact control, so this is a decision you make explicitly instead of one the tool makes for you.

Rounding to Significant Figures

The algorithm is the same whatever the magnitude:

  1. Find the first significant digit — the leftmost non-zero digit.
  2. Count the target number of significant figures from there. That last digit is the one you will keep.
  3. Look at the very next digit. Above 5, round up; below 5, leave it; exactly 5 with nothing after it is a tie, resolved by your rounding policy.
  4. Replace every discarded digit before the decimal point with a zero placeholder, and simply drop the discarded digits after it.
  5. Check that the written answer still declares the precision you claimed. If it cannot, switch to scientific notation.

Worked at three targets: 0.0045678 becomes 0.005 at 1 significant figure, 0.0046 at 2 and 0.00457 at 3. Going the other way, 12345 becomes 1.2 × 10⁴ at 2 significant figures — written as 12000 it would read as two, three, four or five, which is exactly the ambiguity the previous section describes.

The same algorithm at a range of targets — every row below is produced by the calculator above:

Worked rounding results at a range of targets
NumberTarget sig figsRoundedWhat decided the last digit
45.5147445.51First discarded digit 4 → round down.
2.00922.0First discarded digit 0 → round down.
0.0365930.0366First discarded digit 9 → round up.
250032.50 × 10³Zeros added — the input declares less precision than this. Plain decimal form would be ambiguous, so scientific notation is reported.
1234531.23 × 10⁴First discarded digit 4 → round down. Plain decimal form would be ambiguous, so scientific notation is reported.
999.531.00 × 10³First discarded digit 5 (exact tie) → round up. Plain decimal form would be ambiguous, so scientific notation is reported.
0.004567830.00457First discarded digit 7 → round up.
1.231.20Zeros added — the input declares less precision than this.

Step 5 is the one most calculators skip. Rounding 999.5 to three significant figures carries through every nine and gives 1000, which plainly written declares one significant figure, not three. The honest answer is 1.00 × 10³, and that is what this page reports.

Half away from zero vs half to even

The two policies differ only on an exact tie. Half away from zero — standard classroom rounding — always pushes a tie away from zero, so 12.345 becomes 12.35 at four significant figures and −2.5 becomes −3 at one. Half to even, also called banker's rounding, sends a tie to the nearest even digit, so 12.345 becomes 12.34 and 2.5 becomes 2; it is the IEEE 754 default because always rounding ties up introduces a small but systematic upward bias across many values. Both are available here, and the receipt always names which one decided the digit.

Significant Figures and Scientific Notation

Scientific notation writes a value as a mantissa between 1 and 10 times a power of ten, and that split is exactly what makes it unambiguous: the exponent carries all the magnitude, so every digit left in the mantissa is there to declare precision. 1.2 × 10³, 1.20 × 10³ and 1.200 × 10³ are the same value with two, three and four significant figures — a distinction plain 1200 simply cannot express.

Two consequences worth remembering. First, converting to scientific notation never changes how many significant figures a number has; it only makes the count visible. Second, an exponent is not a measurement: the 3 in 10³ is exact and contributes nothing to the count.

To convert between the two forms in either direction, use the Scientific Notation Calculator; to round to decimal places rather than significant figures, use the Rounding Calculator.

Logarithms, Powers and Exact Constants

Calculate mode implements four further conventions, and each one is named in the receipt when it fires.

π and e are treated as exact and never reduce a result's precision. Where no convention is safe — a measurement raised to a measured power, for instance — the calculator reports the unrounded value and issues a precision notice instead of asserting a significant-figure count it cannot justify. Logarithms, antilogarithms and non-integer powers are evaluated in double precision (about 15 digits) and labelled as such; everything else is exact decimal arithmetic to 40 guard digits.

Quick Answers: How Many Significant Figures?

Every row below is produced by running the calculator above, so the table cannot disagree with the tool.

How many significant figures common numbers have
NumberSignificant figuresUnambiguous formWhy
0.069004 SF6.900 × 10⁻²The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
2.0254 SF2.025 × 10⁰1 zero sits between significant digits, so it is significant.
2.003 SF2.00 × 10⁰The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
5.02 SF5.0 × 10⁰The 1 trailing zero is significant because the notation declares it — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
5.003 SF5.00 × 10⁰The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
100.04 SF1.000 × 10²The 3 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
0.011 SF1 × 10⁻²The 2 zeros before the first non-zero digit only place the decimal point, so they are not significant.
20.03 SF2.00 × 10¹The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
1.003 SF1.00 × 10⁰The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
0.502 SF5.0 × 10⁻¹The 1 trailing zero is significant because the notation declares it — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
0.010304 SF1.030 × 10⁻²The 1 trailing zero is significant because the notation declares it — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
0.0045604 SF4.560 × 10⁻³The 1 trailing zero is significant because the notation declares it — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
10024 SF1.002 × 10³2 zeros sit between significant digits, so they are significant.
1.200e34 SF1.200 × 10³The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
1200.4 SF1.200 × 10³The 2 trailing zeros are significant because the notation declares them — a written decimal point (or the mantissa of scientific notation) is a claim about measured precision.
100ambiguous (1–3); common classroom interpretation 1 SF1 × 10²Trailing zeros with no decimal point declare nothing.
1000ambiguous (1–4); common classroom interpretation 1 SF1 × 10³Trailing zeros with no decimal point declare nothing.
1200ambiguous (2–4); common classroom interpretation 2 SF1.2 × 10³Trailing zeros with no decimal point declare nothing.

Worked Examples

18 examples covering counting, rounding, all four operations, ambiguity, scientific notation, exact numbers and mixed expressions. Each answer is generated by the engine on this page.

  1. 1. Counting with leading and trailing zeros

    0.004560 → 4 significant figures

    The three leading zeros place the decimal point; the final zero is a deliberate claim of precision.

  2. 2. Captive zeros inside a number

    1002 → 4 significant figures

    A zero trapped between significant digits cannot be a placeholder, so it always counts.

  3. 3. Every kind of zero at once

    0.01030 → 4 significant figures

    Leading zeros out, captive zero in, trailing zero in — the answer is four.

  4. 4. An ambiguous integer

    1200 → 2 significant figures

    Without a decimal point the notation simply does not say how precise 1200 is.

  5. 5. Scientific notation settles it

    1.200e3 → 4 significant figures

    The same value, written so the precision is impossible to misread.

  6. 6. Rounding a large integer

    12345 → 3 SF → 1.23 × 10⁴

    The plain answer would end in zeros that cannot declare precision, so it is reported in scientific notation.

  7. 7. A carry that changes every digit

    999.5 → 3 SF → 1.00 × 10³

    Rounding up a run of nines lifts the exponent — and makes the plain form ambiguous.

  8. 8. Rounding a small decimal

    0.0045678 → 3 SF → 0.00457

    Leading zeros never count towards the target, they only locate the decimal point.

  9. 9. Asking for more precision than you have

    1.2 → 3 SF → 1.20

    The zeros can be written, but they assert precision the measurement does not support.

  10. 10. Multiplication

    4.56 * 1.4 → 6.4

    Raw result 6.384 · Multiplication & division — fewest significant figures

    Limited by 1.4

    The operand with the fewest significant figures decides the answer.

  11. 11. Division

    12.5 / 3.0 → 4.2

    Raw result 4.16666666666666666666666666666666666666666 · Multiplication & division — fewest significant figures

    Limited by 3.0

    Division uses the same rule as multiplication.

  12. 12. Addition

    18.0 + 1.013 → 19.0

    Raw result 19.013 · Addition & subtraction — least precise decimal place

    Limited by 18.0

    Place value, not significant-figure count, limits a sum.

  13. 13. Subtraction and cancellation

    100.0 - 99.9 → 0.1

    Raw result 0.1 · Addition & subtraction — least precise decimal place

    Limited by 100.0 and 99.9

    Four significant figures in, one out — subtraction can destroy precision.

  14. 14. A measured “2”

    4.56 * 2 → 9

    Raw result 9.12 · Multiplication & division — fewest significant figures

    Limited by 2

    Read as a measurement, a bare 2 has a single significant figure.

  15. 15. The same “2”, marked exact

    4.56 * 2 → 9.12

    Raw result 9.12 · Multiplication & division — fewest significant figures

    Limited by 4.56

    A counted or defined 2 imposes no limit, so the measurement keeps its precision.

  16. 16. Mixed operations with guard digits

    (12.11 + 0.3) * 4.20 → 52.1

    Raw result 52.122 · Multiplication & division — fewest significant figures

    Limited by 0.3 and 4.20

    The bracket is reported to three significant figures but 12.41 — unrounded — is what gets multiplied.

  17. 17. An exact constant

    pi * 2.50^2 → 19.6

    Raw result 19.634954084936206875 · Multiplication & division — fewest significant figures

    Limited by 2.50

    π carries unlimited precision, so the radius alone sets the answer.

  18. 18. Scientific notation in a calculation

    6.022e23 * 2.0 → 1.2 × 10²⁴

    Raw result 1204400000000000000000000 · Multiplication & division — fewest significant figures

    Limited by 2.0

    Very large and very small measurements obey exactly the same rules.

Frequently asked questions

How many significant figures are in 0.01030?

Four. The two leading zeros only place the decimal point and are not significant. The 1 and the 3 are non-zero, so they count. The zero between them is a captive zero and always counts, and the final zero is significant because it sits after the decimal point, where it can only be there to declare precision. The significant digits are 1, 0, 3 and 0.

Are trailing zeros significant?

Only when the notation declares them. After a decimal point they always are: 4.500 has four significant figures and 0.50 has two. In an integer written without a decimal point they are ambiguous — 1200 could carry two, three or four. Writing 1200., or switching to 1.20 × 10³, removes the ambiguity.

How many significant figures are in 100?

It is genuinely ambiguous: one, two or three. Nothing in the notation says whether the zeros are measured or are just placeholders. The common classroom reading is one significant figure, and this calculator reports that reading as a labelled convention while still flagging the value as ambiguous. Write 1.00 × 10² for three, 1.0 × 10² for two, or 100. for three with an explicit decimal point.

How many significant figures are in 1000?

One, two, three or four — the notation does not say. By the common classroom convention it is read as one significant figure. To be unambiguous write 1 × 10³, 1.0 × 10³, 1.00 × 10³ or 1.000 × 10³, or add an explicit decimal point (1000.) to declare all four.

Does zero count as a significant figure?

It depends entirely on where the zero sits. A zero between significant digits always counts (1002 has four). A zero before the first non-zero digit never counts (0.0045 has two). A zero after the last non-zero digit counts only if a decimal point or a scientific-notation mantissa declares it (0.0400 has three; 400 is ambiguous).

Are exact numbers limited by significant figures?

No. Counted quantities (12 students), defined conversion factors (1 inch = 2.54 cm exactly, 1 foot = 12 inches) and mathematical constants such as π and e carry unlimited significant figures and never limit a result. In Calculate mode, mark an operand Exact and it drops out of the precision decision: 4.56 × 2 gives 9 if the 2 is a measurement, and 9.12 if it is a count.

Do I use sig figs or decimal places for addition?

Decimal places. Addition and subtraction are limited by the leftmost least-significant place among the measured operands, not by significant-figure counts. In 18.0 + 1.013 the sum is 19.013, but 18.0 is only known to the tenths, so the answer is 19.0 — even though 1.013 has more significant figures than 18.0.

When should I round intermediate calculations?

Never. Round once, at the end. Rounding an intermediate result injects an error that the rest of the calculation then amplifies. This calculator keeps every intermediate value at full precision (forty guard digits) and rounds only the reported answer; the step panel shows an intermediate to the correct number of significant figures as a view, while the unrounded value is what actually carries forward.

How are logarithms handled?

Only the part after the decimal point in a logarithm carries precision information, so a logarithm is written with as many decimal places as its argument has significant figures: log(1.0 × 10³) is reported as 3.00, because the argument has two significant figures. Antilogarithms invert the rule — 10^x keeps as many significant figures as x has decimal places. Both are computed in double precision and labelled as such.

Is scientific notation better for showing significant figures?

Yes, and it is the only notation that is never ambiguous. In a × 10ⁿ every digit written in the mantissa is significant, so 1.2 × 10³, 1.20 × 10³ and 1.200 × 10³ state two, three and four significant figures for the same value of 1200. Whenever a rounded answer cannot be written unambiguously in plain decimal form, this calculator reports it in scientific notation instead.

What does 45.5147 rounded to 4 significant figures give?

45.51. The first four significant digits are 4, 5, 5 and 1; the first discarded digit is 4, which is below 5, so the last kept digit stays as it is. The answer's last significant digit sits in the hundredths place.

Why does rounding 999.5 to 3 significant figures give 1.00 × 10³?

Rounding up the tie carries through every 9, so 999.5 becomes 1000 — but written plainly, 1000 reads as one significant figure, not three. Scientific notation is the only way to say the answer is precise to three significant figures, so it is reported as 1.00 × 10³.

What is the difference between half up and half to even rounding?

They differ only on an exact tie, where the discarded part is exactly half. Half away from zero (standard classroom rounding) always rounds a tie away from zero, so 12.345 becomes 12.35 at four significant figures. Half to even (banker's rounding) sends a tie to the nearest even digit, so the same number becomes 12.34; it avoids the small upward bias that always-round-up introduces across many values.

Methodology

Every number you type is read lexically before it is read numerically: the calculator records the raw token, the normalized token, which characters are significant digits, the place value of the last significant digit, the notation used and whether the notation actually declares a precision. Only then is the value built, as an exact decimal (a BigInt coefficient and a power of ten) rather than a floating-point double, so 0.1 + 0.2 is exactly 0.3 in every receipt. Counting follows the standard rules — non-zero digits always count, captive zeros count, leading zeros never count, trailing zeros count only when a decimal point or a scientific-notation mantissa declares them. Trailing zeros in an integer written without a decimal point are reported as a structured ambiguity with an explicit minimum and maximum, alongside a clearly labelled common classroom interpretation that is never presented as the declared precision. Multiplication and division take the fewest significant figures among the measured operands; addition and subtraction are rounded to the leftmost least-significant decimal place among the measured operands; operands marked exact impose no limit at all, and pi and e are exact constants. Intermediate values are carried unrounded to forty guard digits and rounded exactly once, on the reported answer, so display precision can never feed back into the arithmetic. Powers with whole-number exponents keep the base's significant figures, roots keep the radicand's, logarithms take as many decimal places as the argument has significant figures, and antilogarithms take as many significant figures as the exponent has decimal places; a measurement raised to a measured power has no accepted convention, so the calculator reports the unrounded value and declines to assert a precision. Expressions are parsed by a dedicated recursive-descent parser — never eval — and every failure returns a specific error code. The result is a serializable receipt containing the raw result, the final result, the rule applied, the limiting operand or place, the rounding decision and its policy, warnings and steps; the same receipt drives the page, the tests and an independent second implementation that re-derives every counting and rounding decision by different arithmetic.

Assumptions

Limitations

No independent third-party mathematical review yet. The author is responsible for the methodology; correctness is enforced by golden vectors, invariant tests and an independent second implementation that re-derives counting and rounding by different arithmetic and must agree on every published case.

How this calculator is tested

Engine: significant-figures v1.0.0 · Receipt: v1.0.0 · Last updated:

Every build runs 103 golden vectors against this engine — 34 counting cases, 20 rounding cases, 22 arithmetic-propagation cases, 4 ambiguity cases and 23 structured-error cases — plus 6 families of invariant (property) tests covering leading-zero invariance, sign invariance, scientific-notation equivalence, exact-number monotonicity, guard-digit invariance and displayed-result/receipt consistency.

A second, independent implementation re-derives every counting and rounding decision by different arithmetic — a right-to-left digit scanner and exact rational scaling, rather than the engine's left-to-right scan and digit-string carry — and the two are compared on every published case: 307 of 307 cross-checks agree, with 0 disagreements. A single disagreement fails the build.

The 18 rows of the quick-answer table, the 10 example chips and the 18 worked examples on this page are generated by running the same engine, so a published answer cannot drift from the calculator that produced it. Every calculation runs locally in your browser: nothing you type is transmitted.

Your recent calculations

Calculations you save are listed here, stored only in this browser — never sent to a server. At most 20 are kept.

Sources

References for the reporting conventions, the rounding rules and the measurement context (these document the method — they are not input datasets):

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