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Absolute Value?

A number's distance from zero on the number line — regardless of direction. It strips away the sign to reveal magnitude.

Simple Definition

The absolute value of a number is its distance from zero on the real number line. It is always non-negative. For any real number \(x\), the absolute value is denoted \(|x|\).

Think of it as a "value without direction." Whether you walk 3 steps to the right (+3) or 3 steps to the left (-3), you have still traveled a distance of 3. That distance is the absolute value: \(|3| = 3\) and \(|-3| = 3\).

Absolute values are used everywhere: in measuring magnitudes (like speed or length), in real-world distances, and in solving equations and inequalities where both positive and negative solutions are possible.

Input
x
→
Absolute Value
|x|
→
Output
Distance from 0
Intuition
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The distance model

Imagine a number line. The absolute value of a number is simply how far that number is from 0. It doesn't matter if it's on the positive side (right) or negative side (left) — only the length of the jump matters. That's why \(| -5 | = 5\): you're five units away from zero.

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The direction-free magnitude

Think of absolute value as a "magnitude operator". It tells you how much of something you have, ignoring whether it's positive or negative. In real life, speeds, distances, and absolute temperatures are all magnitudes — they have no direction.

Formal Definition

The absolute value of a real number \(x\) is defined piecewise:

Piecewise definition
$$ |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases} $$

In words: if the number is positive or zero, leave it as is. If the number is negative, make it positive by multiplying by -1. The result is always \(\ge 0\).

Key Properties
Non-negativity\(|x| \ge 0\) for all real \(x\), and \(|x| = 0\) iff \(x = 0\)
Symmetric\(|x| = |-x|\) — mirror symmetry around zero
Multiplicative\(|x \cdot y| = |x| \cdot |y|\) — absolute value of a product
Triangle inequality\(|x + y| \le |x| + |y|\) — the absolute value of a sum
Reverse triangle inequality\(||x| - |y|| \le |x - y|\)
Relation to square root\(\sqrt{x^2} = |x|\) — the square root of \(x^2\) is the absolute value
Quick Examples
Example 1 — Positive number
\(|7| = 7\)
7 is already positive and 7 units from zero.
Example 2 — Negative number
\(|-4| = 4\)
-4 is 4 units from zero, so its absolute value is 4.
Example 3 — Product property
\(|-3 \cdot 2| = |-6| = 6\)
This equals \(| -3 | \cdot |2| = 3 \cdot 2 = 6\) — property confirmed.
Example 4 — Equation
\(|x - 2| = 5\)
\(x - 2 = 5\) or \(x - 2 = -5\)
\(\Rightarrow x = 7\) or \(x = -3\). Two solutions.
Example 5 — Inequality
\(|x| < 3\)
Interpreted as distance from zero less than 3:
\(-3 < x < 3\). All numbers strictly between -3 and 3.
Common Mistakes
✗ Thinking \(|-x| = -x\) (only for negative x)
The formula \(|-x| = -x\) is only true when \(x \le 0\). If \(x\) is positive, \(|-x| = x\). Always remember: \(|-x| = |x|\) for all \(x\).
✗ Claiming \(|x + y| = |x| + |y|\)
This is false in general. The correct relationship is the triangle inequality: \(|x+y| \le |x| + |y|\). Equality holds only when \(x\) and \(y\) have the same sign or one is zero.
✗ Solving \(|f(x)| = a\) as if it's a single equation
For \(a > 0\), \(|f(x)| = a\) is two equations: \(f(x) = a\) and \(f(x) = -a\). You must solve both. For \(a = 0\), it's just one: \(f(x) = 0\).
Related Concepts
Function→ Exponent→ Quadratic Equation→ Complex Number→ Derivative→

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