What you'll learn
This topic covers rounding numbers, estimating answers and understanding the limits of accuracy. In this guide you will learn how to round to decimal places and significant figures, how to estimate calculations quickly, how to find error intervals, and how upper and lower bounds work. These skills help you check answers and judge how precise a value really is.
Key terms and definitions
Decimal place (d.p.) — the position of a digit after the decimal point.
Significant figure (s.f.) — a digit that contributes to the precision of a number, starting from the first non-zero digit.
Estimate — an approximate answer found by rounding values first.
Error interval — the range a rounded value could actually lie in.
Upper/lower bound — the largest/smallest value a rounded measurement could be.
Core concepts
Rounding to decimal places
To round to a number of decimal places, look at the next digit: if it is 5 or more, round up; if less than 5, round down. For example, 3.847 to 2 d.p. is 3.85 (the third digit, 7, rounds the 4 up to 5).
Rounding to significant figures
Significant figures start at the first non-zero digit. To round to a given number of s.f., keep that many significant digits and round the next one. For example, 0.004982 to 2 s.f. is 0.0050; 38 740 to 2 s.f. is 39 000. Keep place-value zeros so the number stays the right size.
Estimating calculations
To estimate, round each number to 1 significant figure and calculate. For example, (31.2 × 4.9) ÷ 0.21 ≈ (30 × 5) ÷ 0.2 = 150 ÷ 0.2 = 750. Estimation gives a quick check that a detailed answer is sensible. Use the ≈ symbol.
Limits of accuracy and error intervals
A rounded measurement could be anywhere in a range. A length given as 8 cm to the nearest cm could be from 7.5 cm up to (but not including) 8.5 cm. The error interval is written 7.5 ≤ x < 8.5. The halfway value rounds up, so the lower bound is included and the upper bound is not.
Upper and lower bounds in calculations
When calculating with rounded values, use bounds for the most/least the answer could be. For a sum or product, use upper bounds for the maximum; for a difference or quotient, the maximum uses the upper bound of the first and lower bound of the second. This matters in accuracy questions.
Worked examples
Example 1: Significant figures
Round 0.03608 to 2 significant figures.
First two significant digits are 3 and 6; the next is 0, so round down: 0.036.
Example 2: Estimation
Estimate 19.6 × 4.1.
≈ 20 × 4 = 80 (actual is 80.36, so the estimate checks out).
Example 3: Error interval
A mass is 12 kg to the nearest kg. Write the error interval.
11.5 ≤ m < 12.5.
Common mistakes and how to avoid them
Counting leading zeros as significant. Significant figures start at the first non-zero digit.
Losing place value. Keep zeros so the rounded number is the right size (39 000, not 39).
Wrong inequality for bounds. The lower bound is included (≤), the upper is not (<).
Rounding too early in a calculation. Round only at the end unless estimating.
Using the wrong bound for subtraction or division.
Exam technique for Rounding and Estimation
Identify d.p. or s.f. that is required.
Look at the next digit to decide round up or down.
Round to 1 s.f. for quick estimates and use ≈.
Write error intervals as a ≤ x < b.
Choose the right bounds for the largest or smallest result.
Quick revision summary
To round to decimal places, look at the next digit and round up if it is 5 or more. Significant figures start at the first non-zero digit; keep place-value zeros so the number stays the right size (38 740 → 39 000 to 2 s.f.). To estimate, round each value to 1 significant figure and calculate, using ≈ as a quick check. A rounded measurement has an error interval: 8 cm to the nearest cm gives 7.5 ≤ x < 8.5 — the lower bound is included, the upper is not. When calculating with rounded values, use upper and lower bounds; for a difference or quotient the maximum uses the upper bound of the first value and the lower bound of the second. The common errors are counting leading zeros as significant, losing place value, using the wrong inequality, rounding too early, and choosing the wrong bound. Identify d.p. or s.f., look at the next digit, estimate with 1 s.f., and write error intervals carefully.