What Is Absolute Value?
Have you ever described how far away something is? When you say "the supermarket is 3 km away," you never say "negative 3 km" — distance is always positive. That's exactly the idea behind absolute value.
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. We write it using vertical bars: , read as "the absolute value of ".
For example:
- (5 is already 5 steps from zero)
- (−5 is also 5 steps from zero, just in the opposite direction)
- (zero is zero steps from zero)
Think of a number line like a road. You are standing at zero. Whether you walk 5 steps to the right or 5 steps to the left, you are still 5 steps away from where you started. Absolute value measures those steps — never negative!
The Formal Definition
Absolute value has two different rules depending on the value of — one rule for non-negative numbers and one for negative numbers:
In plain English:
- If is positive or zero, the absolute value is just itself.
- If is negative, the absolute value is (which flips the negative to a positive).
When is negative, is actually positive. For example, if , then . So . The negative sign in the formula is doing the job of "removing" the original negative sign.
A common misconception: students think and are the same thing. They are not!
- (absolute value of a negative number → positive)
- (the negative sign is outside the bars → result is negative)
The expression always gives a non-positive result.
12 more sections in this topic
Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions.