The Rules
Five rules for counting, two for arithmetic, and one honest admission about where the notation gives out.
Counting significant figures
1. Every non-zero digit is significant.
123 has 3. 45.6 has 3. 9 has 1.
Nothing about a non-zero digit is in doubt — it was measured or it was not written.
2. Zeros between non-zero digits are significant.
1002 has 4. 10.05 has 4. 40.08 has 4.
A zero with digits on both sides cannot be a placeholder; it is reporting that the measurement was zero at that place.
3. Leading zeros are never significant.
0.00450 has 3. 0.5 has 1. 0.0001 has 1.
They exist only to place the decimal point. 0.00450 and 4.50 × 10⁻³ are the same number, and the second makes it obvious that three digits were measured.
4. Trailing zeros after a decimal point are significant.
2.50 has 3. 1.000 has 4. 10.0 has 3.
Nobody writes 2.50 by accident. The zero is a claim about precision: the hundredths place was measured and found to be zero.
5. Trailing zeros in a whole number with no decimal point are ambiguous.
100 could be 1, 2 or 3.
This is the one case where the rules genuinely run out. The convention is to read them as not significant, but a convention is not a fact — write 1.00 × 10² if you mean three.
Arithmetic
Multiplication and division keep significant figures.
The answer carries as many significant figures as the least precise input.
2.5 × 3.42 = 8.55 → 8.6 (2 s.f., set by 2.5)
Addition and subtraction keep decimal places.
The answer carries as many decimal places as the input with the fewest — because adding lines up the decimal point, and a column nobody measured cannot be filled in.
2.5 + 3.42 = 5.92 → 5.9 (1 d.p., set by 2.5)
Exact numbers do not limit anything.
Counted quantities and defined conversions — 12 in a dozen, exactly 2.54 cm in an inch — carry unlimited significant figures and never set the precision of a result.
Loss of significance
Subtracting two close numbers destroys precision, and the decimal-places rule is what exposes it. 100.0 − 99.9 = 0.1: both inputs carry four significant figures, the answer carries one. Nothing went wrong — the measurement simply never contained the information the answer would need.