absolutevalue

The absolute value of a real number measures its distance from zero on the real number line. It is denoted |x| and is defined by |x| = x if x ≥ 0, and |x| = -x if x < 0. For example, |−5| = 5 and |3| = 3.

Key properties include non-negativity (|x| ≥ 0, with equality only if x = 0); multiplicativity (|xy| = |x| |y|); and the quotient rule |x/y| = |x| / |y| when y ≠ 0. It also satisfies the triangle inequality |x + y| ≤ |x| + |y| and its reverse form ||x| − |y|| ≤ |x − y|. The absolute value also represents distance on the real line: the distance between a and b is |a − b|.

The graph of the absolute value function is a V-shaped curve with its vertex at the origin. It is continuous for all real x, and differentiable for x ≠ 0, with the derivative equal to 1 for x > 0 and −1 for x < 0 (the derivative is not defined at x = 0).

Extensions and related concepts: For a complex number z = a + ib, the modulus |z| = sqrt(a^2 + b^2) gives its distance from zero in the complex plane. The absolute value is a simple example of a norm; in higher dimensions, analogous p-norms generalize this idea, with p = 1 corresponding to the real absolute value on a line. The absolute value is also used to define metrics, inequalities, and various analytic tools.