Absolute value and opposites

In the previous article we saw that negative is to the left of zero and positive is to the right. Here we'll look at two ideas that follow naturally: the opposite of a number and its absolute value.

The opposite of a number

The opposite of a number is the number sitting on the number line at the same distance from zero, but on the other side.

The opposite of 3 is −3. The opposite of −5 is 5. The opposite of 0 is 0 itself.

In other words: the opposite has the same distance from zero but the opposite sign.

On the number line the pair `3` and `−3` look like mirror images — zero is the "mirror":

Absolute value

The absolute value of a number is its distance from zero on the number line. Distance can never be negative, so the absolute value is never less than 0.

We write it between vertical bars: `|x|`.

`|3| = 3` — there are 3 units from 0 to 3.

`|−3| = 3` — there are also 3 units from 0 to −3.

`|0| = 0`.

A simple rule:

• If the number is positive or 0, the absolute value is the same: `|7| = 7`.
• If the number is negative, the absolute value is the number without the minus: `|−7| = 7`.

How the two ideas connect

• A number and its opposite have the same absolute value: `|−4| = |4| = 4`.
• A number plus its opposite is always zero: `4 + (−4) = 0`.

Where you see this in real life

• How far a temperature is from freezing. If it's −5 °C outside, the "swing from 0" is 5 degrees. That's `|−5|`.
• Loss vs. gain. "Losing £50" has the same size as "gaining £50" — just the opposite sign.
• Above vs. below sea level. A helicopter 120 m up (+120) and a submarine 120 m down (−120) are the same distance from sea level.

Common traps

• `|−3|` is not `−3`. An absolute value is always positive or zero.
• The opposite of 0 is still 0. Zero has no other side of the mirror.
• `−|3|` means "minus the absolute value of 3", so `−3`. That's different from `|−3|` = 3. Watch whether the minus sits inside or outside the bars.

What you'll try

• Absolute value practice — the generator gives an integer and asks `|x|`.
• Comparing negative numbers.