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.