Rounding, and the difference between the two kinds
Decimal places counts digits after the point. Significant figures counts digits from the first non-zero one, wherever the point happens to be. They give different answers for the same number, and questions specify which is wanted.
0.004 638 to 2 decimal places is 0.00 — everything has vanished. To 2 significant figures it is 0.0046, which actually says something. For small numbers, significant figures are almost always the sensible measure.
| Number | 2 d.p. | 2 s.f. | 3 s.f. |
|---|---|---|---|
| 3.14159 | 3.14 | 3.1 | 3.14 |
| 0.004638 | 0.00 | 0.0046 | 0.00464 |
| 58 621 | 58 621.00 | 59 000 | 58 600 |
| 0.9971 | 1.00 | 1.0 | 0.997 |
Zeros before the first significant figure do not count
In 0.00408 the first significant figure is the 4. The zeros before it are only holding the decimal point in place. But the zero between the 4 and the 8 does count, because it is between two significant digits — so 0.00408 to 2 s.f. is 0.0041.
Estimating before calculating
An estimate is not a rough answer offered instead of the real one. It is a check on the real one, and the syllabus asks for it explicitly.
The method: round every number to 1 significant figure, then do the arithmetic mentally. If the calculator says something far from the estimate, a key was pressed wrongly — and finding that out takes ten seconds rather than losing every mark that followed.
Estimate (48.7 × 6.13) ÷ 0.198, then say what it tells you if a calculator returns 15.1.
- Round each to 1 significant figure:
50,6and0.2.One significant figure is enough — the point is a check, not a second answer. - Numerator:
50 × 6 = 300.Easy mental arithmetic is the whole reason for rounding this far. - Divide:
300 ÷ 0.2 = 1500.Dividing by 0.2 is the same as multiplying by 5. - A calculator answer of 15.1 is a hundred times too small, so something was mistyped — most likely 0.198 entered as 19.8.The estimate does not give the right answer; it tells you the one you have is wrong.
About 1500. An answer of 15.1 is out by a factor of 100, so re-enter the calculation.
Limits of accuracy
A length recorded as 8 cm to the nearest centimetre was not exactly 8 cm. It was anything that rounds to 8 — from 7.5 cm up to (but not including) 8.5 cm.
Those two values are the lower and upper bounds, and the rule is always the same: half of the rounding unit either side. Rounded to the nearest 10, the bounds are ±5. Rounded to 1 decimal place, ±0.05.
A recorded value stands for a whole interval, not a point. The finer the instrument, the narrower that interval — which is what "more precise" actually means.
Bounds in calculations
When bounded values are combined, the bounds of the answer come from choosing the right combination — and it is not always the obvious one.
For a maximum: add two upper bounds, or multiply two upper bounds. But to make a subtraction as large as possible, take the largest first value and the smallest second. To make a division as large as possible, take the largest numerator and the smallest denominator.
| To find the | Use |
|---|---|
| maximum of a + b | upper a + upper b |
| minimum of a + b | lower a + lower b |
| maximum of a − b | upper a − lower b |
| minimum of a − b | lower a − upper b |
| maximum of a × b | upper a × upper b |
| maximum of a ÷ b | upper a ÷ lower b |
| minimum of a ÷ b | lower a ÷ upper b |
Why subtraction and division reverse
Taking away less leaves more, so the biggest difference comes from subtracting the smallest possible amount. Dividing by less gives more, so the biggest quotient comes from the smallest possible denominator. Both follow from thinking about what makes the answer large, rather than from memorising the table.
Before you leave this chapter
- Decimal places count after the point; significant figures count from the first non-zero digit.
- Leading zeros are not significant; zeros between significant digits are.
- To estimate, round everything to 1 s.f. and check the calculator against it.
- Bounds are half the rounding unit either side: 8 cm to the nearest cm means 7.5 ≤ x < 8.5.
- For a maximum difference or quotient, use the SMALLEST second value.
How many figures should an answer have?
An answer cannot be more accurate than the data it came from. If a measurement was given to 3 significant figures, quoting a result to eight is claiming a precision that does not exist — and examiners penalise it.
The convention: round the final answer to the same number of significant figures as the least accurate value used, or to 3 s.f. if the question does not indicate otherwise. Angles are usually given to 1 decimal place, and money to 2.
| Given data | Sensible answer |
|---|---|
| 12.4 and 3.7 (3 s.f. and 2 s.f.) | 2 significant figures |
| no accuracy stated | 3 significant figures |
| money | 2 decimal places |
| an angle | 1 decimal place |
| a number of people or objects | a whole number |
Round once, at the end
Carry the full calculator value through every intermediate step and round only when you write the final answer. Rounding at each stage compounds the error, and a chain of three or four steps can shift the last figure. If a question asks for working, write the intermediate values to more figures than the final answer needs.