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# absolute value

1. the magnitude of a quantity, irrespective of sign; the distance of a quantity from zero. The absolute value of a number is symbolized by two vertical lines, as |3| or |-3| is 3.
2. the square root of the sum of the squares of the real and imaginary parts of a given complex number, as |a + bi| is equal to the square root of a2+b2. Also called modulus.
[1905-10]

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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 a2 + b2, 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 a2/n+/nb2/n+/nc2. 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|.

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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 √(a2 + b2) , 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 √(a2 + b2 + c2) . 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|.

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Universalium. 2010.

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