Absolute Value Calculator

The absolute value of a number is its distance from zero on the number line, regardless of direction. Absolute value is always non-negative — it strips away the sign and returns only the magnitude.

Definition: |x| = x if x ≥ 0 |x| = −x if x < 0

This means |7| = 7, |−7| = 7, and |0| = 0.

Key properties:

• Non-negativity: |x| ≥ 0 for all real x
• Definiteness: |x| = 0 only when x = 0
• Symmetry: |x| = |−x|
• Triangle inequality: |x + y| ≤ |x| + |y|
• Multiplicative: |x × y| = |x| × |y|
• Quotient: |x ÷ y| = |x| ÷ |y| (y ≠ 0)

Distance on the number line: Distance between a and b = |a − b|

This is the most practical application of absolute value — computing distance without caring about direction.

Worked examples:

• |−15| = 15
• |42| = 42
• |−3.7| = 3.7
• |0| = 0
• |7 − 12| = |−5| = 5 (distance between 7 and 12)
• |−3 + (−4)| = |−7| = 7 (triangle inequality: ≤ |−3| + |−4| = 3 + 4 = 7 ✓)

Absolute value equations: |x| = 5 → x = 5 or x = −5 (two solutions) |x − 3| = 7 → x − 3 = 7 or x − 3 = −7 → x = 10 or x = −4

Absolute value inequalities: |x| < 5 → −5 < x < 5 (between −5 and 5) |x| > 5 → x < −5 or x > 5 (outside −5 and 5)

Real-world applications:

• Finance: Profit/loss — |Revenue − Cost| gives the magnitude of deviation from break-even.
• Engineering: Tolerance — a part must be within |size − target| < 0.01mm.
• Statistics: Mean absolute deviation (MAD) = average of |xi − mean| — a measure of data spread.
• Navigation: Absolute error = |measured − actual|.
• Computer science: Used in sorting algorithms and distance metrics (Manhattan distance = Σ|Δx| + Σ|Δy|).

A couple of finer points. The function f(x) = |x| is continuous everywhere but has no derivative at x = 0 — the graph makes a sharp corner (a cusp) right at the origin. And the idea carries over to complex numbers: there the absolute value becomes the modulus, |a + bi| = √(a² + b²), which is the point’s distance from the origin in the complex plane.