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|

That is the most practical use of absolute value: distance without caring about direction. It is also the one-dimensional case of the distance formula, which is what √((x₂-x₁)²) collapses to when there is no y to worry about.

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 and loss, where |Revenue - Cost| gives the size of the gap from break-even without saying which side of it you are on
• Engineering: tolerance, where a part must satisfy |size - target| < 0.01 mm
• Statistics: mean absolute deviation, the average of |xᵢ - mean|, a spread measure that does not square the outliers the way standard deviation does
• Navigation: absolute error = |measured - actual|
• Computer science: distance metrics. Manhattan distance between two points is |Δx| + |Δy|, and in n dimensions it is the sum of the absolute differences along every axis

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 there, and a corner is not the same as a cusp: at a corner the two one-sided slopes are different and finite (here -1 and +1), while at a cusp both run off to infinity. Textbooks blur this constantly.
The idea also carries over to complex numbers, where absolute value becomes the modulus, |a + bi| = √(a² + b²), which is again the point’s distance from the origin.