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Mathematics · Chapter 2: Real Numbers
Rounding, Approximation and Significant Figures
Rounding to decimal places and significant figures, estimating answers, and upper and lower bounds.
Key termRememberExampleWatch out
01Concept
Exact numbers are not always useful. If someone asks the height of K2, you say "about 8,600 metres", not "8,611 metres". Rounding gives a simpler number that is close enough for the purpose.
Rounding to decimal places
To round to a number of decimal places (d.p.), look at the next digit after the last one you keep: 5 or more, round up; 4 or less, leave it.
- 3.14159 to 2 d.p. = 3.14 (next digit 1)
- 2.678 to 1 d.p. = 2.7 (next digit 7)
- 0.995 to 2 d.p. = 1.00 (keep the zeros to show 2 d.p.)
Significant figures
Significant figures (s.f.) count from the first non-zero digit.
| Rule | Example | Significant figures |
|---|---|---|
| Non-zero digits are significant | 2.47 | 3 |
| Zeros between non-zero digits are significant | 4007 | 4 |
| Leading zeros are not significant | 0.0052 | 2 |
| Trailing zeros after a decimal point are significant | 3.20 | 3 |
- 8,611 to 2 s.f. = 8,600 (zeros keep the place value)
- 0.004362 to 2 s.f. = 0.0044
- 54.96 to 3 s.f. = 55.0
Estimating
Round every number to 1 significant figure and work out a rough answer. This checks that a calculator answer is sensible.
Example: 48.7 × 21.3 ÷ 9.8 ≈ 50 × 20 ÷ 10 = 100. The calculator gives 105.8…, so it is reasonable.
Upper and lower bounds (O Level)
If a length is 12 cm to the nearest cm, the real length could be anything from 11.5 cm up to (but not including) 12.5 cm.
- Lower bound = 11.5 cm, upper bound = 12.5 cm
- We write 11.5 ≤ l < 12.5
Half the unit of rounding is added and subtracted. A mass of 3.4 kg to 1 d.p. has bounds 3.35 kg and 3.45 kg.
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Real World Examples
Heights of mountains
K2 is 8,611 m high. Textbooks often write "about 8,600 m" (2 s.f.), which is easier to remember.
Prices
A shopkeeper rounds Rs 497 to "about Rs 500". Petrol prices per litre are given to 2 d.p., because there are 100 paisa in a rupee.
Measurements
Weather reports give temperatures to the nearest degree, and cricket scorers round run rates to 2 d.p.
04Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Round 7.4862 to (a) 2 d.p. (b) 3 s.f.Show answer
Answer
(a) 7.49 (b) 7.49
Q2.How many significant figures in 0.03050?Show answer
Answer
4 (3, 0, 5, 0: the leading zeros do not count).
Q3.Round 263,481 to 2 s.f.Show answer
Answer
260,000
Q4.Estimate 61.2 × 3.87 ÷ 0.198.Show answer
Answer
≈ 60 × 4 ÷ 0.2 = 240 ÷ 0.2 = 1,200.
Q5.A rope is 25 m long to the nearest metre. Write its bounds.Show answer
Answer
Lower 24.5 m, upper 25.5 m: 24.5 ≤ l < 25.5.
Q6.A rectangle measures 8 cm by 5 cm, each to the nearest cm. Find the largest possible area.Show answer
Answer
Upper bounds: 8.5 × 5.5 = 46.75 cm².
Long questions
Q7.A car travels 120 km (to the nearest 10 km) in 1.5 hours (to the nearest 0.1 h). Find the greatest possible average speed.Show answer
Answer
- Distance bounds: 115 ≤ d < 125 km.
- Time bounds: 1.45 ≤ t < 1.55 h.
- Greatest speed = greatest distance ÷ least time
- = 125 ÷ 1.45 = 86.2 km/h (3 s.f.)