Limits of accuracy and bounds

What a rounded measurement really says

A measurement given as $46$ cm to the nearest cm is not the number $46$. It is a statement about an interval: the true length $L$ satisfies

The half-unit $0.5$ is half of the rounding unit. To the nearest $10$ the half-unit is $5$; to $1$ decimal place it is $0.05$; to $2$ significant figures on a value near $46$ it is again $0.5$.

Why the two ends are not symmetric

The lower end is included ($\le$) and the upper end is excluded ($<$), because $46.5$ rounds up to $47$. Written work that puts $\le$ at both ends is marked wrong even when the numbers are right.

Combining bounds

Let $a$ and $b$ be measurements with bounds $a_{\min},a_{\max},b_{\min},b_{\max}$.

• $(a+b)_{\max}=a_{\max}+b_{\max}$ and $(a+b)_{\min}=a_{\min}+b_{\min}$;
• $(ab)_{\max}=a_{\max}b_{\max}$ and $(ab)_{\min}=a_{\min}b_{\min}$;
• $(a-b)_{\max}=a_{\max}-b_{\min}$ — the subtracted quantity flips;
• $\left(\dfrac{a}{b}\right)_{\max}=\dfrac{a_{\max}}{b_{\min}}$ — the divisor flips.

The rule behind all four: to make the answer as large as possible, make every quantity that increases the answer as large as possible, and every quantity that decreases it as small as possible.