Bounds and Error Intervals

Measure a pencil and someone says it is 5 cm "to the nearest cm". But it is almost certainly not exactly 5 cm — no real ruler is that perfect. It could be 4.7 cm, or 5.3 cm, and either would still round to 5. In fact anything from 4.5 cm up to (but not quite) 5.5 cm rounds to 5. Every rounded measurement secretly hides a whole range of possible true values.

The two ends of that hidden range are the lower bound and the upper bound. They matter enormously the moment things have to fit together or hold weight — a bolt into a hole, a beam across a gap, a wheel onto an axle. An engineer who ignores the range gets nasty surprises; one who knows the bounds builds things that actually work.

A rounded value has a lower bound half a unit below it, and an upper bound half a unit above it. For the pencil (nearest cm, so the unit is 1 cm, half of which is 0.5 cm):

4.5 \le x < 5.5

The band between the two bounds is called the error interval. Notice the upper end uses <, not \le: a length of exactly 5.5 would round up to 6, so it sits just outside. The upper bound is really a "just below" limit — we write it as 5.5 by convention, meaning "anything up to, but not reaching, 5.5".

Worked example 1 — reading off the bounds

A road sign says a town is 80 km away, to the nearest 10 km. Find the bounds.

The rounding unit is 10, so half a unit is 5. Step 5 either side of 80:

75 \le x < 85

So the town could really be anywhere from 75 km up to (just under) 85 km away. The same recipe works for any accuracy: to the nearest whole number the half-unit is 0.5; to the nearest 0.1 it is 0.05; to the nearest 100 it is 50.

A value rounded to a unit u lies within \tfrac{u}{2} of the rounded value:

lower bound = \text{value} - \tfrac{u}{2};
upper bound = \text{value} + \tfrac{u}{2};
write the error interval as \text{lower} \le x < \text{upper} — the upper bound uses < because that value would round up.

Seeing the interval

Here is x = 80 (to the nearest 10) on a number line. The shaded band is the error interval 75 \le x < 85 — an open circle at 85 marks that the upper bound is not included.

Worked example 2 — combining bounds (biggest area)

A rectangle measures 6 cm by 4 cm, each to the nearest cm. What is the largest its area could possibly be?

First get the bounds of each side (half-unit 0.5):

5.5 \le \text{length} < 6.5 \qquad 3.5 \le \text{width} < 4.5

A bigger side makes a bigger area, so the maximum area uses both upper bounds:

\text{max area} = 6.5 \times 4.5 = 29.25 \text{ cm}^2

And the minimum area uses both lower bounds: 5.5 \times 3.5 = 19.25 cm². The "true" area could be anywhere in between — a surprisingly wide spread for a rectangle we casually called "6 by 4"!

Worked example 3 — will it fit?

A shelf gap is 30 cm wide, to the nearest cm. A box is 29 cm wide, to the nearest cm. Are we certain the box fits?

To be safe we must ask: could the widest possible box still beat the narrowest possible gap? Use the worst case for each:

\text{gap, lower bound} = 29.5 \text{ cm} \qquad \text{box, upper bound} = 29.5 \text{ cm}

The narrowest the gap could be is 29.5 cm; the widest the box could be is 29.5 cm. They are equal — so we cannot guarantee it fits! This is exactly why engineers reason with bounds instead of the rounded numbers: "29 into 30" looks fine, but the worst case is a dead heat.

Two subtle traps catch people out again and again:

The upper bound is the halfway point above — even though it would round up. A length of 5 cm to the nearest cm has upper bound 5.5, even though 5.5 itself rounds up to 6. We still write 5.5 by convention, meaning "everything less than 5.5". Do not chicken out and write 5.49 — the bound is 5.5 with a < sign.
Dividing flips which bound to use. For a product like a \times b, the maximum uses the maximum of both. But for a quotient a \div b, the maximum uses the maximum a and the minimum b — because dividing by a smaller number gives a bigger result. Reach for the biggest bottom when you want the smallest answer, and vice versa. This flip is one of the most-missed marks in the whole topic.

Bounds are the reason every serious engineering drawing carries tolerances. A part machined to "10 mm \pm\, 0.05 mm" is really an instruction about bounds: the real size must land between 9.95 mm and 10.05 mm, or it might not fit its neighbour. It is why a bridge is built with safety margins, why a piston is made a hair narrower than its cylinder, and why every scientific graph sprouts little error bars — no measurement in the real world is ever exact.

And it is not just tidiness: sloppy bounds have caused genuine disasters. Parts machined at the edge of their tolerance, gaps that were "close enough", loads assumed exact when they weren't — knowing the range of possible true values, not just the rounded headline number, is what keeps real structures standing up.