Truncation means cutting off digits beyond a specified position without rounding — even when the dropped digit is 5 or above. On a GCSE paper it is a common approximation technique that always rounds towards zero, and it defines an error interval that is different from the one produced by ordinary rounding.
What does truncation mean in maths?
When you truncate a number you remove all digits beyond a specified position. Unlike rounding, you never look at the first removed digit to decide whether to adjust the last kept digit — you simply discard everything beyond the cut-off.
Examples:
- Truncate 7.836 to 1 decimal place → 7.8 (discard .036)
- Truncate 7.876 to 1 decimal place → 7.8 (discard .076; even though 7.876 rounds to 7.9, truncation keeps 7.8)
- Truncate 15.999 to the nearest integer → 15 (discard .999)
This "always chop, never look ahead" rule is the essential difference between truncation and rounding.
How do you truncate to a given number of decimal places?
- Identify the decimal place you want to keep last.
- Write all digits up to and including that place.
- Discard every digit after that place — do not round up.
Worked example: Truncate 3.14159 to 3 decimal places.
| Step | Action | Result |
|---|---|---|
| 1 | Decide to keep to 3 d.p. | — |
| 2 | Write 3.141 | 3.141 |
| 3 | Discard 59… | 3.141 |
Compare this with rounding: 3.14159 rounded to 3 d.p. is 3.142 (because the 4th decimal place is 5, which rounds up). Truncated it remains 3.141. The truncated value is always ≤ the rounded value for positive numbers.
How do you truncate to significant figures?
The process is identical, but you count significant figures rather than decimal places. Leading zeros are never significant.
Worked example: Truncate 0.074 892 to 2 significant figures.
The first significant figure is 7, the second is 4. Truncated to 2 s.f.: 0.074.
For large numbers, replace discarded digits before the decimal point with placeholder zeros rather than removing them:
- Truncate 58 743 to 3 s.f.: keep 5, 8, 7 → 58 700.
How does truncation differ from rounding?
| Feature | Rounding | Truncation |
|---|---|---|
| Decision rule | Look at next digit; ≥ 5 rounds up | Always discard — never look ahead |
| Direction | Rounds to nearest value | Always rounds towards zero |
| 4.97 to 1 d.p. | 5.0 | 4.9 |
| −4.97 to 1 d.p. | −5.0 | −4.9 |
The negative-number row is the key insight: truncation rounds towards zero in both directions. Truncating −4.97 to 1 d.p. gives −4.9 (closer to zero), not −5.0.
What error interval does truncation give?
When a number x has been truncated to a value t, the error interval has the truncated value as its lower bound, with the upper bound one unit in the last place above t, reached but not equalled.
Rule: if x is truncated to t, then t ≤ x < t + (1 unit in last place)
| Truncated value | Precision | Error interval |
|---|---|---|
| 6.3 | 1 d.p. | 6.3 ≤ x < 6.4 |
| 12 | nearest integer | 12 ≤ x < 13 |
| 0.050 | 3 d.p. | 0.050 ≤ x < 0.051 |
Note the contrast with rounding, where the lower bound is 0.5 units below the rounded value. With truncation, the true value can never be below t; it can only be at or above it.
How is truncation tested in GCSE exam questions?
The most common question types are:
- Truncate a given number to a stated precision. Read the precision carefully — "2 decimal places" is different from "2 significant figures".
- State the error interval for a truncated value. Remember to use ≤ at the lower bound and < at the upper bound.
- Distinguish truncation from rounding in a context. Some calculators truncate their displays; the question may ask you to explain the difference.
What mistakes should you avoid?
Mistake 1 — Rounding up instead of truncating. Truncating 8.996 to 2 d.p. gives 8.99, not 9.00.
Mistake 2 — Wrong inequality signs in the error interval. The lower bound uses ≤ (the true value can equal t); the upper bound uses < (the true value cannot reach t + 1 unit). Reversing the signs costs marks.
Mistake 3 — Forgetting that negative numbers truncate towards zero. Truncating −7.6 to the nearest integer gives −7 (not −8), because −7 is closer to zero.
Frequently asked questions
Does truncation always produce a smaller number?
For positive numbers, yes — truncation removes positive digits, so the result is ≤ the original. For negative numbers, the truncated value is closer to zero, making it larger (less negative) than the original. Truncating −8.9 to the nearest integer gives −8, which is greater than −8.9.
Why do some calculators truncate instead of rounding?
Calculators hold a fixed number of internal digits. When a result has more digits than the display allows, many devices simply drop the excess rather than round, because truncation requires only one pass through the digits. Some calculators do round; checking the device manual clarifies which behaviour it uses.
How do I tell whether a GCSE question wants truncation or rounding?
The question will use the word "truncate" or "truncated to" when it wants truncation. The words "round", "correct to", "give to" or "to the nearest" indicate ordinary rounding. The two operations have distinct GCSE meanings that cannot be interchanged.
What is the maximum error when truncating to 1 decimal place?
The truncated value t satisfies t ≤ x < t + 0.1. The error is x − t, which can be anywhere from 0 up to (but not including) 0.1. So the maximum possible error is just under 0.1 — it approaches 0.1 but never reaches it.
For step-by-step GCSE maths coaching that builds understanding behind every rule, visit aitutors.me.