Rounding Error/Examples/73.854 mm

Example of Rounding Error
Let $x$ be a length expressed as $73.854 \, \mathrm {mm}$ to $5$ significant figures.

The maximum rounding error in $x$ is $0.0005 \, \mathrm {mm}$.

Proof
By definition of rounding, we have:
 * $73.8535 \le x \le 73.8545$

Whether the inequalities are strict or not depends on the treatment of the half.

Rounding up, we have:
 * $73.8535 \le x < 73.8545$

because $73.8545$ will round up to $73.855$.

Rounding down, we have:
 * $73.8535 < x \le 73.8545$

because $73.8535$ will round down to $73.853$.

Rounding to even, we have:
 * $73.8535 \le x \le 73.8545$

because both $73.8535$ and $73.8545$ will round to even to $73.854$.

Hence we have:
 * $\size {73.854 - 73.8535} \le 0.0005 \ge \size {73.854 - 73.8545}$