Absolute Value Calculator

What does absolute value mean?

Absolute value measures a number's distance from zero on the number line, without regard to direction. |−5| = 5 and |5| = 5 because both are 5 units from zero. The absolute value is always non-negative and equals the number itself if the number is already positive or zero.

What notation is used for absolute value?

Vertical bars: |x|. In programming, you typically use the abs() function (Python, C, Java) or Math.abs() (JavaScript). In complex number contexts, |z| denotes the modulus (distance from the origin in the complex plane), which is a generalization of absolute value.

How do you solve absolute value equations?

To solve |x| = a (for a ≥ 0), split into two cases: x = a or x = −a. For |x − 3| = 5: x − 3 = 5 → x = 8, or x − 3 = −5 → x = −2. Always check both solutions in the original equation. If a < 0, there is no solution since absolute values are never negative.

What is the triangle inequality for absolute values?

The triangle inequality states |a + b| ≤ |a| + |b| for all real numbers a and b. Equality holds when a and b have the same sign (or one is zero). This extends to complex numbers and is fundamental in analysis, metric spaces, and vector geometry.