Absolute Value Definition
|x| = x if x ≥ 0
|x| = −x if x < 0
Distance formula: |a − b| = distance between a and b on number line
|x| = −x if x < 0
Distance formula: |a − b| = distance between a and b on number line
Properties of Absolute Value
| Property | Formula | Example |
|---|---|---|
| Non-negativity | |x| ≥ 0 | |−5| = 5 ≥ 0 |
| Identity | |x| = 0 ↔ x = 0 | |0| = 0 |
| Multiplication | |x × y| = |x| × |y| | |−3 × 4| = 12 |
| Triangle inequality | |x + y| ≤ |x| + |y| | |−3 + 4| ≤ 7 |
| Even function | |−x| = |x| | |−7| = |7| = 7 |
| Square root | |x| = √(x²) | |−5| = √25 = 5 |
Real-World Uses
- Distance: Distance between two points = |x₂ − x₁|
- Error/deviation: |measured − expected| = absolute error
- Finance: |profit − loss| to find the magnitude regardless of direction
- Temperature change: |T₂ − T₁| for the magnitude of temperature shift
- Physics: Speed = |velocity| (magnitude without direction)
Frequently Asked Questions
Can absolute value ever be negative?
No — by definition, |x| ≥ 0 for all real numbers x. The absolute value strips the sign, returning the magnitude (distance from zero). The only way |x| = 0 is if x = 0 itself.
What is the absolute value of a complex number?
For a complex number z = a + bi, the absolute value (or modulus) is |z| = √(a² + b²). This is the distance from the origin in the complex plane. This calculator handles real numbers only.
How do you solve absolute value equations?
|x − 3| = 5 means x − 3 = 5 OR x − 3 = −5, giving x = 8 or x = −2. Always consider both positive and negative cases when solving absolute value equations.