Rounding, estimation and limits of accuracy
Rounding and estimation
- decimal places: count after the point ($3.14159 \to 3.14$ to 2 d.p.).
- significant figures: count from the first non-zero digit ($5764 \to 6000$ to 1 s.f.).
- To estimate, round each number to 1 s.f.: $\dfrac{41.3}{9.79 \times 0.765} \approx \dfrac{40}{8} = 5$.
Practice
Estimate 41.3 / (9.79 × 0.765) by rounding each to 1 s.f.
≈ 40 / (10 × 0.8) = 40 / 8 = 5.
Practice
What is 5764 rounded to 1 significant figure?
The first significant figure is 5; the next digit 7 rounds it up to 6000.
Limits of accuracy
- A rounded value has a lower and upper bound, half a unit each side.
- $635$ m to the nearest metre → $634.5 \leq h < 635.5$.
- (Extended) For an area, the largest uses both upper bounds; for a speed, largest distance ÷ smallest time.
Practice
A height is 635 m to the nearest metre. What is the lower bound (m)?
The bounds are half a metre each side: 634.5 ≤ h < 635.5.
You've got it
Key idea
- d.p. count after the point; s.f. from the first non-zero digit
- estimate by rounding each value to 1 s.f.
- bounds: nearest unit → $\pm$ half a unit ($635 \to 634.5 \leq h < 635.5$)