Rounding and significant figures

Why significant figures count information rather than decimal places, why the count starts at the first non-zero digit, and why place value must survive the rounding.

Number · Rounding & bounds

Rounding and significant figures

Why significant figures count information rather than decimal places, why the count starts at the first non-zero digit, and why place value must survive the rounding.

Why it works

Rounding answers one question: how much detail is worth keeping? Decimal places measure detail from the decimal point; significant figures measure it from where the number actually starts — the first non-zero digit.

Why leading zeros don't count. In

0.00456

the zeros aren't information, they're scaffolding that holds the digits in place — the number in standard form is

4.56 \times 10^{-3}

, and the zeros vanish into the power. So the first significant figure is the 4, and

0.00456

to 2 s.f. is

0.0046

. Zeros between or after non-zero digits DO count: in

30.5

all three digits are significant.

Place value must survive. Rounding

4562

to 2 s.f. keeps two digits of information but the number must stay the same size:

4562 \to 4600, \text{ never } 46.

The 5 and 62 are replaced by zeros, not deleted —

46

is a hundred times too small. Quick check: a rounded number should be close to the original.

The cut-off rule is a distance argument.

4562

lies between

4500

and

4600

; it's nearer

4600

because the next digit (6) is 5 or more. "5 rounds up" isn't a decree — halfway needs a convention, and up is the one exams use. Rounding

18.965

to 1 d.p.: the digits after the 9 say "up", tipping

18.9 \to 19.0

— and the trailing zero must be written, because "to 1 d.p." promises one decimal place of information.

d.p. and s.f. are different promises.

0.030472

to 3 s.f. is

0.0305

(count 3, 0, 4 from the first non-zero, look at the 7, round up). To 3 d.p. it would be

0.030

. Read which one the question asks for — marks die on this more than on the arithmetic.