Every measurement is an approximation, and significant figures are the notation that says how good the approximation is. A length reported as 12.3 cm claims knowledge to the nearest tenth of a centimetre; reported as 12 cm, it claims only the nearest centimetre; reported as 12.30 cm, it claims the nearest hundredth. The digits are the same in spirit, but the claims are different, and treating them as interchangeable is how careful work quietly becomes wrong. This guide states the five rules for counting significant figures, the rules for rounding results of calculations, and the special cases — exact numbers, trailing zeros, and scientific notation — where intuition most often fails.

The five rules for counting significant figures

Rule 1: every non-zero digit is significant. 345 has three significant figures; 7.82 has three. Rule 2: zeros between non-zero digits — captive zeros — are significant. 1005 has four; 20.03 has four. Rule 3: leading zeros are never significant; they only place the decimal point. 0.00450 has its first significant digit at the 4, and 0.03 has one significant figure. Rule 4: trailing zeros after a decimal point are significant, because they show the measurement was read to that place. 12.30 has four significant figures; 0.00450 has three. Rule 5: trailing zeros in a whole number without a decimal point are ambiguous — 120,000 might carry two, three, or six significant figures, and only context or scientific notation resolves it.

Worked set, to fix the rules: 0.00056 has two significant figures (5 and 6; the zeros lead). 5.60 has three (the trailing zero counts). 100.0 has four (the decimal point makes the trailing zeros significant). 40,000 has one significant figure if written plainly — the zeros are placeholders — but 4.0 × 10⁴ has two and 4.00 × 10⁴ has three. When ambiguity matters, write the number in scientific notation; the mantissa’s digits are, by definition, exactly the significant figures.

Multiplication and division: fewest figures wins

When numbers are multiplied or divided, the result is rounded to the number of significant figures of the least precise input. 12.3 × 4.56 = 56.088, but 12.3 has three significant figures and 4.56 has three, so the answer is 56.1. With inputs of differing precision the limit bites: 12.345 × 2.1 = 25.9245, but 2.1 has two significant figures, so the answer is 26. The extra digits of the precise input are wasted — they describe a precision the other input does not support — and reporting them implies knowledge you do not have.

The reasoning is about relative uncertainty. A measurement of 2.1 is uncertain by about 1 part in 21, or roughly 5%; a result built from it cannot honestly be more certain than that, whatever the other input’s quality. In practice, count the figures of each input, take the smallest count, and round the computed result to that count — after computing with full precision, not before.

Addition and subtraction: fewest decimal places wins

Addition and subtraction follow a different rule, and confusing the two rules is the classic significant-figures error. Here the result is rounded to the fewest decimal places of any input, not the fewest figures. 12.34 + 1.2 = 13.54, but 1.2 is known only to the tenths place, so the result is 13.5. 100.05 − 0.003 = 100.047, rounded to 100.05, because the first input is known only to hundredths. The absolute uncertainty, not the relative uncertainty, governs addition.

The extreme case shows why the rule matters: 1,000,000 + 0.5 = 1,000,000.5 by arithmetic, but if the million is known only to the nearest whole unit, the half-unit is below its resolution, and the sum rounds to 1,000,000. Reporting 1,000,000.5 would claim the million was known to tenths — a precision it never had. Large numbers swallow small ones in addition; in multiplication, they do not.

Rounding: half-up, half-even, and when to round

To round to a given number of figures, look at the first digit to be dropped. If it is less than 5, drop and leave the last kept digit unchanged: 3.14159 to three figures is 3.14. If it is greater than 5, round the last kept digit up: 3.14659 to three figures is 3.15. If it is exactly 5 followed by nothing, the school rule rounds up (2.5 → 3), while the round-half-to-even convention used in statistics and computing rounds to the even neighbour (2.5 → 2, 3.5 → 4) to avoid upward bias over many roundings. Know which convention your context requires; for school and most laboratory work, half-up is expected.

One rule overrides all the detail: round once, at the end. Carry at least two guard digits through every intermediate step. A five-step calculation rounded to two figures at each step can accumulate several percent of error, and the final rounding cannot recover it. The calculator on the hub page rounds only the displayed result for exactly this reason.

Scientific notation, exact numbers, and logarithms

Scientific notation removes the trailing-zero ambiguity entirely: every digit in the mantissa is significant, and the exponent carries no precision claim. 1.20 × 10⁵ has three significant figures; 1.2 × 10⁵ has two. Converting a number to scientific notation therefore requires deciding its precision first — the notation displays the decision, it does not make it. When this page converts your input, it preserves the digits you typed, including trailing zeros, for exactly this reason.

Exact numbers never limit precision. Counting 25 items, using the defined conversion 1 inch = 2.54 cm exactly, or multiplying by 2 in a doubling formula contributes no uncertainty, so such numbers are treated as having infinite significant figures. Logarithms have their own rule, worth knowing for pH and decibel work: the number of decimal places in a logarithm equals the number of significant figures in its argument — log(2.5 × 10²) = 2.40, with two decimal places matching the two figures of 2.5. The characteristic (the part before the point) carries no precision information.

Frequently asked questions

How many significant figures are in 0.00450?

Three: 4, 5, and the trailing 0. Leading zeros never count; the trailing zero after a decimal point does count, because it shows the value was measured to the thousandths place.

How many significant figures are in 1005?

Four. Zeros between non-zero digits are always significant.

How many significant figures are in 120,000?

Ambiguous as written. The trailing zeros may be placeholders or significant. Write it as 1.2 × 10^5 for two figures, 1.20 × 10^5 for three, or 120,000. for six (the decimal point marks the zeros significant).

Do I round before or after calculating?

After. Carry at least two extra digits through intermediate steps and round only the final result to the correct number of significant figures. Rounding early compounds errors.

How do significant figures work in addition?

Round the result to the fewest decimal places of any input. 12.34 + 1.2 = 13.5, because 1.2 is known only to one decimal place. This differs from multiplication, where the fewest significant figures governs.

Are counted numbers significant figures?

Counted quantities and defined conversions are exact and never limit precision. Dividing a measured 87.5 g by a counted 5 samples gives 17.5 g, limited by the measurement’s three figures, not by the count.