Absolute Value

Q: What is the absolute value of zero?

$|0| = 0$. Zero is the only number whose absolute value is zero — every other number sits some positive distance away from zero.

The absolute value of a number is its distance from zero on the number line. Distance is never negative — you can't walk $-3$ blocks — so absolute value is never negative either. We write it with two vertical bars: $|-7| = 7$ and $|7| = 7$ , because both numbers sit $7$ units from zero.

That's the whole definition, but there is one trap worth learning up front: a negative sign written outside the bars survives. $|-6| = 6$ , but $-|6| = -6$ . The bars only strip the sign of what's inside them.

Distance from zero

Picture $-7$ on the number line. To get from $-7$ back to zero you travel $7$ units — direction doesn't matter, only the length of the trip. So $|-7| = 7$ .

Positive numbers are already a distance from zero: $|4| = 4$ . And zero is zero units from itself, so $|0| = 0$ . In short: the bars leave positive numbers and zero alone, and flip negative numbers to positive.

Opposites share an absolute value

Every number and its opposite sit the same distance from zero, just on opposite sides. $|-9| = 9$ and $|9| = 9$ . This is why an equation like $|x| = 5$ has two answers later in algebra — both $5$ and $-5$ are five units from zero.

On the number line below, $-9$ and $9$ sit the same $9$ units from zero — mirror images across the middle — so they share the absolute value $9$ .

A negative outside the bars

Work from the inside out. In $-|6|$ , first evaluate the bars: $|6| = 6$ . Then apply the negative sign that was waiting outside: $-|6| = -6$ .

Compare that with $|-6|$ , where the negative sign is inside the bars and gets stripped: $|-6| = 6$ . Same digits, different placement, opposite answers. On a test, check where the sign sits before you write anything.

Worked examples

Example 1: a negative inside the bars

Evaluate $|-9|$ .

-9

9

units from zero

Distance is never negative

|-9| = 9

Answer: $9$

Example 2: a positive inside the bars

Evaluate $|4|$ .

4

4

units from zero

The bars leave a positive number alone

|4| = 4

Answer: $4$

Example 3: a negative outside the bars

Evaluate $-|6|$ .

Evaluate the bars first

|6| = 6

Apply the negative sign waiting outside

-|6| = -6

Answer: $-6$

Try one yourself

Common questions

Can an absolute value ever be negative?

No. Absolute value measures distance from zero, and distance can't be negative. The smallest possible absolute value is $|0| = 0$ . But an expression like $-|6|$ can be negative, because the negative sign sits outside the bars.

What is the absolute value of zero?

$|0| = 0$ . Zero is the only number whose absolute value is zero — every other number sits some positive distance away from zero.

Why do

|-5|

and

|5|

give the same answer?

Because $-5$ and $5$ are opposites: they sit on opposite sides of zero but at the same distance, $5$ units. Absolute value only cares about the distance, not the side.

How is

-|6|

different from

|-6|

In $|-6|$ the negative sign is inside the bars, so it gets stripped: the answer is $6$ . In $-|6|$ the negative sign is outside the bars, so it stays: the bars give $6$ , then the sign makes it $-6$ .

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