Absolute Value

The absolute value of an integer is its distance from zero, with the direction thrown away. We write it with two bars: |−7| = 7 and |7| = 7. Both numbers sit 7 steps from zero.

The question it answers is simply: how far, regardless of which way? That is why it shows up wherever the gap matters but the sign does not — measuring error, comparing speeds, judging how off a guess was.

Absolute value also gives the distance between any two integers: subtract them and take the absolute value.

How far is −3 from 5? |5 − (−3)| = |8| = 8 units. Do it the other way and |−3 − 5| = |−8| = 8 — same answer, because distance never cares about order.

This is the everyday version of the idea: it turns "which is bigger?" into a clean "how far apart?".