Every measurement is really an approximation. If a length is recorded as $21$ cm to the nearest cm, the true value could be anything from $20.5$ cm up to (but not including) $21.5$ cm — any of those values round back to $21$. These two edges are the lower bound (LB) and upper bound (UB), written as an error interval:

The upper bound uses a strict $<$ because $21.5$ itself would round up to $22$, not $21$.

Use the half-unit rule: halve the rounding unit and add/subtract it from the given value.

• Nearest cm (unit $=1$): half-unit $=0.5$ → $21\pm0.5$.
• Nearest 5 m (unit $=5$): half-unit $=2.5$ → $105\pm2.5 \Rightarrow 102.5\le L<107.5$.
• 2 d.p. (unit $=0.01$): half-unit $=0.005$ → $5.36\pm0.005$.
• 2 s.f., e.g. $x=240$: the last significant digit (4) is in the tens column, so the unit is $10$, half-unit $5$ → $235\le x<245$.

Truncation just chops digits off without rounding — $5.87$ truncated to 1 d.p. is $5.8$, not $5.9$. The truncated value is always the lower bound; the upper bound is one unit of the last digit's place value higher.

Addition and multiplication are intuitive — to get the biggest result, use the biggest inputs. Subtraction and division are not intuitive, because making the result as large as possible means pairing a big value with a small one.

Worked check (subtraction): $b=\sqrt{c^2-a^2}$, with $c=25$ cm, $a=10$ cm (both nearest cm). Bounds: $24.5\le c<25.5$, $9.5\le a<10.5$. For $LB(b)$ we need $LB(c^2)-UB(a^2) = 24.5^2-10.5^2 = 600.25-110.25=490$, so $LB(b)=\sqrt{490}\approx22.14$ cm.

Calculate both the final LB and UB, then find the most precise rounding that gives the same value for both — any digit they agree on, you're certain of.

Worked exam-style question: $P = F/A$ where $F=28.5$ (3 s.f.) and $A=0.195$ (3 s.f.). Find $P$ to a suitable degree of accuracy, with justification.

1. Bounds: $28.45\le F<28.55$ and $0.1945\le A<0.1955$.
2. $LB(P)=LB(F)\div UB(A) = 28.45\div0.1955\approx145.52$.
3. $UB(P)=UB(F)\div LB(A)=28.55\div0.1945\approx146.79$.
4. Round both to increasing accuracy: at 1 s.f. both give $100$; at 2 s.f. both give $150$; at 3 s.f. they disagree ($146$ vs $147$).
5. Answer: $P\approx150$ (2 s.f.), justified because both the LB and UB round to $150$ at that accuracy — rounding further is not supported by the data.

• Using $\le$ on both sides of the error interval — the upper bound is always a strict $<$ for continuous data.
• Flipping the subtraction/division rule and pairing LB with LB, UB with UB.
• Treating $\pi$ or other exact constants as having bounds — only measured quantities do.
• Confusing truncation with rounding — truncated numbers don't have a symmetric half-unit either side.