What Absolute Value Means

Sometimes the direction of a number is beside the point and only its size matters. Walking three steps left is as far as walking three steps right. Absolute value is the tool that keeps the size and discards the direction.

Concept Distance from zero

The absolute value of a number is its distance from zero on the number line.

Both

-3

and

3

sit three units from zero, so both have an absolute value of 3. Distance is never negative, so an absolute value is never negative.

Concept The notation and the rule

Two vertical bars around a number mean "take its absolute value". The rule has two cases:

If the number is positive or zero, leave it unchanged:

|8| = 8

If the number is negative, drop the minus sign:

|-7| = 7

In short, keep the numerical value and discard the sign. Note also that $|0| = 0$ , since zero is no distance from itself.

Concept Bars behave like brackets

When an absolute value appears inside a longer expression, resolve it first, exactly as you would a bracket. Only then continue with the rest.

2 + |-3|

, work out

|-3| = 3

before adding.

|6| - 5

, work out

|6| = 6

before subtracting.

Example Two straightforward cases

|8| = 8

— the number is already positive, so nothing changes.

|-7| = 7

— the number is negative, so the sign is removed.

|-12| = 12

— the same rule, whatever the size.

Example Inside a calculation

Work out $2 + |-3|$ and $|6| - 5$ .

|-3| = 3

, so

2 + 3 = 5

|6| = 6

, so

6 - 5 = 1

⟹ 5 and 1

The bars come off before anything else happens. Adding first would give the wrong answer in both cases.

Example A temperature change

The temperature rises from $-5^\circ$ to $3^\circ$ . By how much did it change?

The change is

3 - (-5) = 8

Taking the absolute value:

|8| = 8

⟹ a change of 8 degrees

Had you subtracted the other way round you would have got $-8$ , but the size of the change is 8 either way. That is precisely what absolute value protects against.

Concept Where it is used

Distances — the gap between two points is positive whichever order you take them in.

Temperature — the size of a rise or fall.

Money — the amount of a debt or a profit.

Geometry — side lengths and heights, which cannot be negative.

Note Mistakes to avoid

Giving a negative answer — an absolute value is never negative.

Changing the sign of a positive number;

|8|

stays 8.

Working through the expression before removing the bars.

Assuming

|0|

is undefined; it is simply 0.

Treating absolute value as a sign change rather than a distance.

Summary

Absolute value is distance from zero, so it is never negative.
A positive number keeps its value; a negative number loses its sign.
|−3| and |3| are both 3, and |0| = 0.
Resolve the bars first, just as you would a bracket.
It is used wherever size matters but direction does not.