In practical mathematics, answers are rarely needed to many decimal places. Rounding and significant figures are tools for expressing answers with an appropriate level of precision for the context.
KEY TAKEAWAY: Always match your precision to the context. Money → 2 decimal places. Building measurements → nearest mm or cm. Populations → nearest thousand.
Rule: Look at the digit immediately to the right of your rounding position.
- If it is \(\geq 5\), round up
- If it is \(< 5\), round down (truncate)
Examples:
| Original | Rounded to 2 d.p. | Rounded to 1 d.p. | Rounded to nearest whole |
|---|---|---|---|
| \(3.467\) | \(3.47\) | \(3.5\) | \(3\) |
| \(12.845\) | \(12.85\) | \(12.8\) | \(13\) |
| \(0.0354\) | \(0.04\) | \(0.0\) | \(0\) |
Significant figures (s.f.) count meaningful digits, starting from the first non-zero digit.
| Number | 1 s.f. | 2 s.f. | 3 s.f. |
|---|---|---|---|
| \(4567\) | \(5000\) | \(4600\) | \(4570\) |
| \(0.03842\) | \(0.04\) | \(0.038\) | \(0.0384\) |
| \(12.06\) | \(10\) | \(12\) | \(12.1\) |
How to count significant figures:
- Non-zero digits always count: \(342\) has 3 s.f.
- Zeros between non-zero digits count: \(3042\) has 4 s.f.
- Leading zeros do not count: \(0.0045\) has 2 s.f.
- Trailing zeros after the decimal point count: \(3.50\) has 3 s.f.
EXAM TIP: When a question says “correct to 2 significant figures”, find the first two non-zero digits and round from there.
Always round to 2 decimal places (nearest cent):
\$\(\$12.567 \to \$12.57\)\$
Context determines the rounding direction:
| Situation | Rule | Why |
|---|---|---|
| Buying paint (litres needed) | Round up | Don’t run short |
| Number of buses needed | Round up | Everyone must fit |
| Money to give as change | Round to nearest | Convention |
| Manufacturing tolerance | Round to nearest | Accuracy required |
COMMON MISTAKE: Students round \(2.1\text{ L}\) of paint to \(2\text{ L}\) because “it’s closest”. In context, you need \(3\text{ L}\) (a whole can) to have enough.
A builder needs \(23.4\text{ m}\) of timber. Timber is sold in \(3\text{ m}\) lengths. How many lengths are needed?
\(\$23.4 \div 3 = 7.8 \text{ lengths}\)\$
Round up to \(8\) lengths (can’t buy \(0.8\) of a length).
Total timber purchased: \(8 \times 3 = 24\text{ m}\)
Waste: \(24 - 23.4 = 0.6\text{ m}\)
APPLICATION: Rounding decisions in real life often have cost implications. Knowing when to round up — even if the number is close to the lower value — is a critical practical skill.
