Rounding

Video lesson4 worked examples

Theory

You need to round answers to a specified number of significant figures or decimal places.

You should consider the effects of rounding appropriately. For example, there is a precision limitation if you round an angle prematurely after calculating it using trigonometry, especially as distances increase.

The Golden Rule: significant figures are counted from the first non-zero digit; decimal places are counted after the point. Whichever you are asked for, look at the next digit to decide whether to round up, and keep any place-holding zeros.

⚠️ Common Examiner Traps

Confusing the two: 2 significant figures is not the same as 2 decimal places. $0.057134$ to 2 s.f. is $0.057$ ; to 2 d.p. it is $0.06$ .
Leading zeros are not significant: in $0.057$ the first significant figure is the 5, not the zeros.
Dropping place-value zeros: $45\,872$ to 2 s.f. is $46\,000$ , not $46$ — the zeros hold the size of the number.
Rounding too early: in a multi-step problem, keep full accuracy until the final line. Rounding a trig angle mid-calculation shifts the final answer, and the error grows with distance.

Worked examples

Example 1

Rounding to Significant Figures

Round 45,872 to 2 significant figures.

Step 1: Identify the second significant figure (5).

Step 2: Look at the next digit (8). Because it is 5 or more, round the 5 up to 6. Replace the remaining digits before the decimal with zeros.

Answer: 46,000.

Example 2

Rounding to Decimal Places

Round 3.14159 to 3 decimal places.

Step 1: Identify the third decimal place (1).

Step 2: Look at the next digit (5). Because it is 5 or more, round up.

Answer: 3.142.

Example 3

Significant Figures with Leading Zeros

Round 0.008451 to 2 significant figures.

Step 1: Significant figures start at the first non-zero digit. The leading zeros do not count, so the first significant figure is 8 and the second is 4.

Step 2: Look at the next digit (5). Because it is 5 or more, round the 4 up to 5. The leading zeros must be kept — they fix the size of the number.

Answer: 0.0085.

Example 4

Counting Significant Figures

State how many significant figures each number has: (a) $25.00$ and (b) $0.00680$ .

Step 1 (a): Count from the first non-zero digit. In $25.00$ every digit counts — the trailing zeros after a decimal point are significant, as they show the precision.

(a) 4 significant figures.

Step 2 (b): In $0.00680$ the leading zeros do not count, so start at the 6. The 6, the 8 and the trailing 0 all count.

(b) 3 significant figures.

Ready to practise?

22 Rounding questions with answers and video solutions