Absolute Value
“Absolute value drops the sign and keeps the 'distance to 0'.”
The formula
|a| = a (a ≥ 0), |a| = −a (a < 0)How to read it: the absolute value is how far a number sits from 0 — its distance
- |a|
- — the absolute value of a — its distance from 0
- a ≥ 0
- — if a is 0 or positive, leave it as is
- −a (a < 0)
- — if a is negative, strip the sign to make it positive
The hook
−3 and 3 have opposite signs, but they're the same distance from 0 — and that distance is the absolute value.
In plain words
How far a number is from 0. Throw away the direction (the sign) and keep only the distance.
The intuition
Think of it as the length you'd measure with a ruler out to that number on the line. A distance can never be negative, whether you go left or right, so absolute value is always 0 or more.
How it's built
Just two rules — if it's 0 or positive, leave it; if it's negative, drop the leading − to make it positive. So |−3|=3, |3|=3, |0|=0.
Example
Take |−4|. The number −4 sits 4 steps to the left of 0. The distance is 4, so |−4| = 4.
Common mistake
Don't blindly memorize |−a| = a — because a itself might be negative. Think of absolute value as 'measuring distance', not 'erasing a minus sign'.
Where it's used
Distance between two points, the size of an error, speed (the absolute value of velocity) — anywhere the direction doesn't matter but the size does.
Where it came from
The bar notation | | was introduced by the 19th-century mathematician Weierstrass, and the name comes from a Latin word meaning 'magnitude'.
Prerequisites
Quick check
What is |−7|?
- −7
- 0
- 7✓
- 14
Practice
What is |−9|?
Answer: 9
- −9 sits 9 steps from 0
- → |−9| = 9
Takeaway: For a negative, just drop the sign.
What is |6| + |−4|?
Answer: 10
- |6| = 6, |−4| = 4
- 6 + 4 = 10
Takeaway: Take each absolute value first, then compute.
What is |−3| − |−1|?
Answer: 2
- |−3| = 3, |−1| = 1
- 3 − 1 = 2
Takeaway: Absolute value is always ≥ 0 — make it positive first, then compute.
Find all values of a with |a| = 5.
Answer: undefined
- Two points are distance 5 from 0
- 5 to the right, −5 to the left
- a = 5 or a = −5
Takeaway: Two numbers share an absolute value — same size, opposite signs.