Significant figures — counting, rounding, and how many to keep

Significant figures are where careful physics quietly turns into lost marks. The rules are short and a mark scheme does not negotiate them: an answer with too many digits claims a precision the measurement never had, and one with too few throws away information you actually measured.

Counting them

Decimal numbers. Find the first non-zero digit and count everything to the right of it. Trailing zeros are significant.
Whole numbers with no decimal point. Trailing zeros are not significant.

NUMBER S.F. WHY

002.00 3 count from the 2: 2, 0, 0

0.02050 4 count from the 2: 2, 0, 5, 0

2.04 3 all three digits count

1.005 4 count from the 1: 1, 0, 0, 5

0.002455 4 count from the 2: 2, 4, 5, 5

2450 3 trailing zero not significant

10205 5 zeros between digits count

1000 1 all trailing zeros dropped

A whole number like 1000 is genuinely ambiguous, and that is a problem with the notation rather than with you. Scientific notation removes it: 1 × 10³ is one significant figure, 1.00 × 10³ is three. When the precision matters, write it that way.

Rounding

542.48 rounded to:

2 s.f. → 540 ( better written 5.4 × 10² )

3 s.f. → 542

4 s.f. → 542.5

Exact halves. Rounding 3.25000 and 3.35000 to 2 s.f. depends on which convention you use:

Half away from zero — the IB and standard school convention: 3.3 and 3.4
Half to even — used by some calculators and software: 3.2 and 3.4

Know that both exist, so that a calculator disagreeing with you does not throw you. Use the first.

A habit worth building. Round only at the very end. Carry the full calculator value through every intermediate step — rounding early accumulates error, and a mark scheme allows for the correctly carried value, not the prematurely rounded one.

How many to keep in a calculation

Two rules. They are different from each other, and merging them is the most common mistake in this topic.

OPERATION THE ANSWER KEEPS

Addition / subtraction the fewest decimal places of any input

Multiplication / division the fewest significant figures of any input

Example 1 — addition

3.21 + 4.2 = 7.41

4.2 has 1 decimal place → 7.4

Example 2 — subtraction

This is the one worth studying, because the rule bites hardest here.

8.24 − 8.19 = 0.05 both have 2 decimal places → 0.05

Both measurements were quoted to three significant figures, and the answer has one. Subtracting two numbers that are close together destroys precision, and no rule can put it back — the digits that agreed cancelled, and only the digits that disagreed survived.

This is the same fact the uncertainties page states from the other side: subtracting two measurements does not subtract their uncertainties, it makes the result less certain. If an experiment asks you to find a small difference between two large readings, expect the answer to be imprecise, and design around it where you can.

Example 3 — division

(27 × 578) / 12.33 = 1265.7…

27 has only 2 significant figures → 1.3 × 10³

Not 1265.7, and not 1266. One two-figure measurement in the chain limits everything downstream of it, however precisely the other quantities were measured.

Where marks are actually lost

Applying the significant-figures rule to an addition, or the decimal-places rule to a multiplication. They are not interchangeable.
Rounding at every intermediate step instead of only at the end.
Writing 2450 when you meant three significant figures, and 2.45 × 10³ was available.
Copying out the calculator's whole display as the answer.
Quoting more significant figures than the least precise measurement supports.

Notes and practice

Reading through a topic can make it feel clear; working through the questions is what tells you whether it really is.

Lecture notes — Significant figure .pdf