Measure of the magnitude of a real number , complex number , or vector .
Geometrically, the absolute value represents (absolute) displacement from the origin (or zero) and is therefore always nonnegative. If a real number a is positive or zero, its absolute value is itself; if a is negative, its absolute value is - a . A complex number z is typically represented by an ordered pair (a, b) in the complex plane. Thus, the absolute value (or modulus) of z is defined as the real number a 2 + b 2 , which corresponds to z 's distance from the origin of the complex plane. Vectors, like arrows, have both magnitude and direction, and their algebraic representation follows from placing their "tail" at the origin of a multidimensional space and extracting the corresponding coordinates, or components, of their "point." The absolute value (magnitude) of a vector is then given by the square root of the sum of the squares of its components. For example, a three-dimensional vector v , given by (a, b, c), has absolute value a 2 /n+/n b 2 /n+/n c 2 . Absolute value is symbolized by vertical bars, as in | x |, | z |, or | v |, and obeys certain fundamental properties, such as | a · b | = | a | · | b | and | a + b | ≤ | a | + | b |.