Rounding, estimation and limits of accuracy · ⁨Arrondis, estimations et limites de précision⁩

Close enough?

• A builder needs 3.47 m of wood but the shop only measures to the nearest 0.1 m. They'll buy 3.5 m.
• An engineer needs 3.470 m measured to the millimetre. Precision matters — and knowing how much precision you need is a mathematical skill.

Rounding to decimal places 小数位

• Count digits after · ⁨après⁩ the decimal point.
• $3.14159$ to 2 d.p. $= 3.14$ (look at the third digit: $1 < 5$, round down).
• $3.146$ to 2 d.p. $= 3.15$ (look at the third digit: $6 \geq 5$, round up).

Rounding to significant figures 有效数字

• Count from the first non-zero digit (left to right).
• $5764$ to 1 s.f. $= 6000$ (the first s.f. is $5$; the next digit $7 \geq 5$, round up).
• $0.003\,47$ to 2 s.f. $= 0.0035$ (the first two s.f. are $3$ and · ⁨ et ⁩$4$; the next digit $7 \geq 5$, round up).

Rounding is about finding the nearest tick — which value is $3.47$ closest to?

Estimation 估算

• Round every number in the calculation to 1 significant figure, then calculate.
• $\dfrac{41.3}{9.79 \times 0.765} \approx \dfrac{40}{10 \times 0.8} = \dfrac{40}{8} = 5$.
• Estimation is a quick sanity check — if your calculator says $500$, something went wrong.

Limits of accuracy · ⁨ Limites de précision⁩ 精度范围 (Core and Extended)

• A rounded measurement has a range of possible true values.
• $635$ m to the nearest metre: $634.5 \leq h < 635.5$.
• Extended: for combined measurements, make the result as big (or small) as possible. For a product · ⁨produit⁩ (area = length × width): max = UB × UB, min = LB × LB. For a quotient (speed = distance ÷ time): max = UB distance ÷ LB time, min = LB distance ÷ UB time — biggest top over smallest bottom.

Measurement bounds are also Core

• A reading of 8.6 cm to nearest 0.1 cm has half-step 0.05 cm: $8.55\le l<8.65$. Include the lower boundary; the upper boundary rounds to 8.7.
• Extended calculated bounds: for positive $d\in[99.5,100.5)$ km and $t\in[1.95,2.05)$ h, speed has lower bound $99.5/2.05\approx48.5366$ and upper bound $100.5/1.95\approx51.5385\text{ km/h}$. Core stops at bounds of the measured inputs.

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