Mathematics
What Are Significant Figures?
Four rules decide which digits count, and 0.00420 has three of them. The one genuinely ambiguous case is a whole number ending in zeros, such as 3200.
Significant figures
Significant figures are the digits in a number that carry information about its size, counted from the first non-zero digit onwards.
- Also called
- s.f., sig figs
- Where students meet it
- Grade 7 to 9; rounding to a given number of significant figures sits in the key stage 3 programme of study in England, and Cambridge IGCSE Mathematics papers ask for three significant figures whenever an answer is not exact.
The short answer
Significant figures are the digits that tell you the size of a number, counted from the first non-zero digit onwards. 0.00420 has three: the 4, the 2 and the final 0. Leading zeros never count, zeros between other digits always do, and trailing zeros after a decimal point do.
An example
48 361 → 50 000 (1 s.f.) → 48 000 (2 s.f.) → 48 400 (3 s.f.)
One number rounded three ways. At 3 s.f. the digits kept are 4, 8 and 3, and the next digit is 6, so the 3 rounds up to 4. The zeros that follow are not significant figures at all — they are place holders, and dropping them would turn 48 400 into 484 and change the number by a factor of a hundred.
Four rules, and there are no others
The whole system fits in four lines. Learn them once and every counting question after that is mechanical.
Notice that only zeros are ever in doubt. Every other digit is significant wherever it stands, so the whole skill reduces to deciding what a particular zero is doing in a particular number.
- Every non-zero digit is significant. 427 has three.
- Zeros between non-zero digits are significant. 4007 has four.
- Leading zeros are never significant. 0.00420 begins counting at the 4, so it has three, not six.
- Trailing zeros after a decimal point are significant. 3.60 has three, and the final zero is a claim that the value is accurate to hundredths.
3200: two figures, three or four?
This is the case where the rules stop deciding for you, and the honest answer is that you cannot tell from the number alone. 3200 might be a population rounded from 3162, in which case it has two significant figures, or an exact count of 3200 items, in which case it has four.
Standard form removes the doubt, which is one of the reasons it exists: 3.2 × 10³ is unmistakably two significant figures and 3.200 × 10³ is unmistakably four. If a question asks how many significant figures a plain whole number ending in zeros has, it is asking about the intended precision, and that has to come from the context rather than from the digits.
Why boards ask for figures rather than places
Decimal places count positions after the point; significant figures count meaningful digits wherever they sit. On a number like 0.00420 the difference is stark. Rounded to two decimal places it becomes 0.00, which has destroyed the answer. Rounded to two significant figures it is 0.0042, which has kept it.
That is why examiners prefer significant figures for answers that could come out at any scale, and why the standard instruction on Cambridge IGCSE Mathematics papers is to give non-exact answers to three significant figures unless the question asks for something else. One practical rule goes with it: round at the end only. Rounding partway through a calculation and then continuing puts an error into every line that follows.
- Significant figures Cambridge IGCSE Mathematics papers ask for when an answer is not exact
- 3Significant figures Cambridge IGCSE Mathematics papers ask for when an answer is not exact[2]
Common questions
How many significant figures does 0.00420 have?
Three. The zeros in front of the 4 are only holding the decimal point in position and never count. The 4 and the 2 are significant because they are non-zero, and the final 0 is significant because it comes after the decimal point and after a non-zero digit, where it makes a claim about precision.
Are zeros ever significant?
Often. A zero between two non-zero digits always counts, as in 4007. A zero after the decimal point and after a non-zero digit counts, as in 3.60. A zero before the first non-zero digit never counts. A zero at the end of a whole number is the one case that cannot be settled without context.
What is the difference between significant figures and decimal places?
Decimal places count digits after the point, so 0.0042 has four. Significant figures count meaningful digits from the first non-zero one, so 0.0042 has two. On small numbers the two instructions give wildly different answers, which is why the question always specifies which it wants.
How many significant figures should I give in an exam?
Read the question first; if it names an accuracy, use that. Cambridge IGCSE Mathematics papers instruct candidates to give non-exact answers to three significant figures otherwise. Whatever the target, keep full accuracy in your working and round only the final line, since rounding early moves the answer.
Sources
- National curriculum in England: mathematics programmes of study — Department for Education
- Cambridge IGCSE Mathematics (0580) — Cambridge Assessment International Education
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