The absolute value of a number is its distance from $0$. With subtraction you can find the distance between any two numbers.
Key idea
The distance between $a$ and $b$ on a number line is $|a - b|$. Subtracting in either order works, because $|a - b| = |b - a|$.
Why it works
$a - b$ tells how far you move to get from $b$ to $a$, and its sign tells the direction. A distance has no direction, so we drop the sign with the absolute value. When the numbers are on opposite sides of $0$, you can also add their distances from $0$.
Worked examples
Worked example: Opposite sides of zero
How far apart are $-4$ and $7$?
- $|-4 - 7| = |-11| = 11$.
- Check: $-4$ is 4 from $0$ and $7$ is 7 from $0$, and $4 + 7 = 11$ ✓.
Answer: 11
Worked example: Both negative
How far apart are $-12$ and $-3$?
- $|-12 - (-3)| = |-12 + 3| = |-9| = 9$.
- Both are on the same side of $0$, so do not add their distances from $0$: $12 - 3 = 9$ ✓.
Answer: 9
Worked example: Above and below sea level
A bird flies at 15 m above sea level and a fish swims at 6.5 m below ($-6.5$ m). How far apart are they?
- $|15 - (-6.5)| = |15 + 6.5| = 21.5$.
Answer: 21.5 m
Common mistake
Getting a negative distance, like $-4 - 7 = -11$, and stopping there. Distances are never negative: take the absolute value. Another slip is adding the distances from $0$ when both numbers are on the same side.