TheIBAxolotl
Rounding to decimal places and significant figures
Decimal places count digits after the point. Significant figures count from the first non-zero digit. So \(0.004\,072\) is \(0.0041\) to 2 s.f. but \(0.004\) to 3 d.p., and \(38\,460\) is \(38\,000\) to 2 s.f. Zeros used only to hold place value (leading zeros, or trailing zeros in a rounded whole number) are not significant.
Bounds
Every rounded measurement hides a range of possible true values. If a length is 7.4 cm to 1 decimal place, the true value \(L\) satisfies
\[7.35 \le L < 7.45\]
The bounds are half a unit of the rounding either side. For "to the nearest 10", the half-unit is 5; for "to the nearest 0.01", it is 0.005.
Bounds in calculations
To find the largest possible result, choose the bounds that push the result up:
| Operation | Upper bound | Lower bound |
|---|---|---|
| \(a + b\) | \(a_U + b_U\) | \(a_L + b_L\) |
| \(a - b\) | \(a_U - b_L\) | \(a_L - b_U\) |
| \(a \times b\) | \(a_U \times b_U\) | \(a_L \times b_L\) |
| \(a \div b\) | \(a_U \div b_L\) | \(a_L \div b_U\) |
Worked example. A garden is 12 m by 8 m, each to the nearest metre. The length lies in \([11.5, 12.5)\) and the width in \([7.5, 8.5)\). The area lies between \(11.5 \times 7.5 = 86.25\ \text{m}^2\) and \(12.5 \times 8.5 = 106.25\ \text{m}^2\). The "rounded" area of \(96\ \text{m}^2\) could be wrong by about 10 m² either way: small rounding errors grow when multiplied.
Percentage error
To judge how good an approximation is, compare it with the exact value:
\[\varepsilon = \left|\frac{v_A - v_E}{v_E}\right| \times 100\%\]
where \(v_A\) is the approximate value and \(v_E\) the exact one. If a crowd was estimated at 250 people and the true count was 236, the percentage error is \(\frac{14}{236} \times 100\% \approx 5.93\%\). Always divide by the exact value.
Estimation
Before trusting a calculator, round each number to 1 significant figure and do a quick mental estimate. \(\dfrac{48.7 \times 0.198}{3.09} \approx \dfrac{50 \times 0.2}{3} \approx 3.3\); the calculator gives about 3.12, which is consistent. If the estimate and the answer disagree by a factor of 10, look for a misplaced decimal or a missing bracket.
Common mistakes
- Writing the upper bound as 7.44 or 7.449...; use 7.45 with a strict inequality.
- Dividing by the approximate value in the percentage error.
- Using upper ÷ upper for the largest quotient (it should be upper ÷ lower).
- Rounding intermediate answers, which adds avoidable error.
- Forgetting the absolute value, giving a negative percentage error.
Exam tips
- Answers must be exact or correct to 3 significant figures unless the question says otherwise; a 2 s.f. answer can lose an accuracy mark.
- State bounds as an inequality (\(7.35 \le L < 7.45\)) when asked for the range.
- In percentage error questions, write the formula with the numbers in it before evaluating.