Trust your intuition

We are going to use the relative error $\epsilon_{r} = \large \frac{\text{| Aproximation - exact value|}}{\text{| exact value |}}$ which is better aproximation (for the error) than the absolute error $\epsilon_{a} = \text{| Aproximation - exact value|}$

One mile = 1.609344 km (it's not exactly equal)

$\epsilon_{r} (1) = \dfrac { | 1.6 - 1.609344 |}{1.609344} = 0.005...$ (it's not a exact equality)

One km = 0.6213... miles

$\epsilon_{r} (2) = \dfrac { | 0.6 - 0.6213... |}{| 0.6213.. |} = 0.034..$ (it's not a exact equality)

Therefore, $\epsilon_{r} (1) < \epsilon_{r} (2)$ ...

Question: Why the relative error is better than absolute error for measuring the error? This is equivalent to the question: why one missile launched to 1000 kms from the mark getting one absolute error of 3 metres has less error(or better aproximation) than measuring one wall of 3 metres and getting one absolute error of 1 cm?