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MEASUREMENTS — APPROXIMATION AND ERRORS

These notes explain how to approximate measurements and how to describe the errors that result. They are written for learners aged about 14 (Kenya). Read the rules, study examples and try the short exercises.

1. Key ideas

Exact value — the true value (often unknown).
Approximation — a close value we write instead of the exact value (e.g., rounding).
Rounding — changing a number to the nearest chosen place (10, 1, 0.1, ...).
Truncation — cutting off extra digits (not rounding).
Significant figures (s.f.) — digits that carry meaning about precision.

2. Rounding rules (simple)

To round to a certain place:

Look one place to the right of the place you want to keep.
If that digit is 5 or more, increase the kept digit by 1; if less than 5, leave it.

Examples:

Round 347 to the nearest 10 → 350 (because 7 ≥ 5).
Round 4.276 to 2 decimal places (0.01) → 4.28 (6 → round up).
Round 0.0349 to 2 significant figures → 0.035 (first two s.f. are 3 and 4; 4 rounds up because next digit 9 ≥ 5).

3. Significant figures (s.f.) — quick rules

Non-zero digits are always significant (123 → 3 s.f.).
Zeros between non-zero digits are significant (1003 → 4 s.f.).
Leading zeros are not significant (0.0045 → 2 s.f.).
Trailing zeros in a decimal number are significant (2.300 → 4 s.f.).

4. Errors — absolute and relative

Suppose the exact value is E and the measured/approximate value is A.

Absolute error = |E − A| (how much the approximation differs from the exact value).
Relative error = |E − A| / |E| (the absolute error compared with the exact value).
Percentage error = (Relative error) × 100%.

Worked example:
A length is exactly 2.45 m (E) but you read 2.44 m (A).

      Absolute error = |2.45 − 2.44| = 0.01 m

      Relative error = 0.01 / 2.45 ≈ 0.00408

      Percentage error ≈ 0.408% (≈ 0.41%)

5. Upper and lower bounds (important in measurement)

When a measurement is given to a certain precision, it actually means the true value lies in an interval:

If you measure 4.2 m to the nearest 0.1 m, the half of last place is 0.05.

      Lower bound = 4.2 − 0.05 = 4.15

      Upper bound = 4.2 + 0.05 = 4.25

      So the true value x satisfies: 4.15 ≤ x < 4.25

6. Reporting and avoiding false precision

Report measurements and answers with the correct number of significant figures or decimal places.
Do not report more digits than the measurement tool can justify (e.g., a ruler reading to 1 mm should not give extra decimal places).
When doing calculations, keep a few extra digits then round the final result correctly.

7. Small visual: number line showing bounds

      4.15 |───●────| 4.2 |────●───| 4.25

      (lower)     (measured)   (upper)

8. Short practice (try these)

Round 367.89 to 2 s.f.
A rod measured as 1.234 m to 3 decimal places. Give the upper and lower bounds.
Measured value 5.0 m, actual 4.96 m. Find absolute and percentage error.

Answers:

      1) 370 (2 s.f.: 3 and 6 → 6 causes 3 to round to 4? Wait: correct method: first two s.f. are 3 and 6; look next digit 7 ≥5 so 36 → 37 then 3 becomes 3? Better: 367.89 → to 2 s.f. = 370).

      2) Measured 1.234 to 3 decimal places means ±0.0005: lower = 1.2335, upper = 1.2345 (so 1.2335 ≤ true < 1.2345).

      3) Absolute error = |4.96 − 5.0| = 0.04 m. Percentage error = (0.04 / 4.96) × 100% ≈ 0.806% (≈ 0.81%).

9. Extra notes for tests (KCSE style)

When a question asks for bounds, include the inequality with ≤ for the lower bound and < for the upper bound.
When multiplying or dividing measured values, use the rules for significant figures carefully — often report the answer to the least number of s.f. used in the data.

Good practice: always write the unit (m, cm, g, etc.) and the uncertainty together, e.g., 4.2 m ± 0.05 m.

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