Math Core

If you walk 5 blocks east or 5 blocks west, you've walked the same distance: 5 blocks. Direction changes where you end up, but not how far you went. Absolute value is the math tool that measures "how far" without caring about direction.

Distance from zero

The number $-5$ is 5 units from $0$ . The number $5$ is also 5 units from $0$ . Both have the same size, even though they point in opposite directions.

Both -5 and 5 are 5 units from 0.

Definition

Absolute value

The absolute value of a number is its distance from $0$ on the number line. The absolute value of $a$ is written $|a|$ .

So $|5| = 5$ and $|-5| = 5$ .

Because absolute value is a distance, it is never negative. A distance can be $0$ (the distance from $0$ to itself), but it can't be less than $0$ .

Finding absolute value

The absolute value of a positive number is the number itself: $|9| = 9$ .
The absolute value of a negative number is its opposite: $|-9| = 9$ .
The absolute value of zero is zero: $|0| = 0$ .

An absolute value is always positive or zero.

Worked example: Finding absolute values

Find each absolute value.

$|-14|$
$|6|$
$\left|-\dfrac{2}{3}\right|$
$|-0.8|$

Solutions.

$-14$ is 14 units from $0$ , so $|-14| = 14$ .
$6$ is 6 units from $0$ , so $|6| = 6$ .
$-\dfrac{2}{3}$ is $\dfrac{2}{3}$ of a unit from $0$ , so $\left|-\dfrac{2}{3}\right| = \dfrac{2}{3}$ .
$|-0.8| = 0.8$ .

Common mistake

Absolute value bars are not the same as "the opposite." $|4|$ is $4$ , not $-4$ . And the minus sign outside the bars stays: $-|-4|$ means "the opposite of $|-4|$ ," which is $-4$ . Find the absolute value first, then apply any sign outside the bars.

Absolute value in real life

In real situations, absolute value tells you the size of an amount, ignoring whether it is a gain or a loss, above or below.

Worked example: Elevation

A shipwreck lies at an elevation of $-60$ feet. A lighthouse top is at $45$ feet. Which is farther from sea level?

Sea level is $0$ . Compare distances from $0$ :

Shipwreck: $|-60| = 60$ feet.
Lighthouse: $|45| = 45$ feet.

The shipwreck is farther from sea level, by 15 feet.

Worked example: Debt

Sam's account balance is $-30$ dollars. That means he owes $30. Priya's balance is $-12$ dollars. Who owes more money?

The size of each debt is the absolute value of the balance: $|-30| = 30$ and $|-12| = 12$ . Sam owes $30 and Priya owes $12, so Sam owes more.

Notice that $-30$ is the lower balance, but $30$ is the bigger debt. A balance less than $-20$ dollars means a debt of more than $20.

Tip

Two different numbers can have the same absolute value: they are opposites. If $|x| = 7$ , then $x$ could be $7$ or $-7$ .