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Number · Rounding & bounds

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Upper and lower bounds

Why a rounded value hides half a unit of uncertainty each way, why the upper bound is a strict "less than", how truncation differs, and how to pick the right bounds when calculating — the ÷ and − traps included.

Number · Rounding & bounds

Upper and lower bounds

Why a rounded value hides half a unit of uncertainty each way, why the upper bound is a strict "less than", how truncation differs, and how to pick the right bounds when calculating — the ÷ and − traps included.

Why it works

A measurement of "45 cm to the nearest cm" doesn't mean 45.000… — it means the true length landed somewhere that rounds to 45. Everything from 44.544.5 up to (but not including) 45.545.5 does that, so the error interval is

44.5≤L<45.544.5 \le L < 45.5

— half the rounding unit each way. Nearest 10 ⇒ ±5\pm 5; nearest 0.1 ⇒ ±0.05\pm 0.05. The unit being halved is the rounding step, not 1.

Why the top is << and not ≤\le. 45.545.5 itself rounds up to 46, so it isn't in the interval — but everything below it is, no matter how close. That's why the upper bound is stated as 45.545.5 with a strict inequality: there's no "last number before 45.5" (44.9, 45.49, 45.499… never finish). Writing the UB as 45.49 or "45.4 recurring" is the classic mark-loser: use the clean half-unit value and let the << carry the meaning.

Truncation cuts, rounding balances. A value truncated to 8.4 had its tail chopped, not rounded — anything from 8.48.4 up to 8.58.5 truncates to 8.4, so the interval is 8.4≤y<8.58.4 \le y < 8.5: the stated value is the bottom, and the whole step (not half) sits above it.

Calculating with bounds: push everything the same way. To make a result as BIG as possible, make everything that increases it big and everything that decreases it small:
  • Multiply / add: UB × UB, UB + UB for the maximum.
  • Divide: maximum of ab\frac{a}{b} is UB(a)LB(b)\frac{\text{UB}(a)}{\text{LB}(b)}
— a big top over a small bottom. Using UB/UB is the single most common bounds error at grade 8–9.
  • Subtract: maximum of a−ba - b is UB(a)−LB(b)\text{UB}(a) - \text{LB}(b); the
minimum is LB(a)−UB(b)\text{LB}(a) - \text{UB}(b) — the bounds cross over.

Reason it out each time ("what makes this biggest?") rather than memorising the table; the reasoning is what the E-marks want.

"To a suitable degree of accuracy." Compute the UB and LB of the answer; quote only the figures they AGREE on. If UB=3.11\text{UB} = 3.11 and LB=2.89\text{LB} = 2.89, both round to 3, so the answer is 3 (1 s.f.) — any more precision would be pretending to know digits the measurement can't support.