The absolute value of a number is its distance from zero on the number line, always non-negative.
The absolute value of a real number x, denoted |x|, is defined as x if x ≥ 0 and -x if x < 0. For example, |5| = 5 and |-5| = 5. Absolute value represents magnitude without regard to sign. Properties include |ab| = |a||b|, |a/b| = |a|/|b|, and the triangle inequality: |a + b| ≤ |a| + |b|. Absolute value is used in distance measurement, error analysis, and complex numbers.
The concept of absolute value was introduced by Jean-Robert Argand in 1806 for complex numbers. Karl Weierstrass used it in real analysis in the 19th century. The notation |x| was popularized by Karl Weierstrass in 1841. Earlier mathematicians like Euclid and al-Khwarizmi used the concept of magnitude without formal notation.