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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 , which corresponds to zs 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 . Absolute value is symbolized by vertical bars, as in
This entry comes from Encyclopædia Britannica Concise. For the full entry on absolute value, visit Britannica.com.