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The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is ...
Absolute value (algebra) In algebra, an absolute value (also called a valuation, magnitude, or norm, [1] although "norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any ...
They do, however, share an attribute with the reals, which is called absolute value or magnitude. Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its absolute value. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3.
The same formula applies to octonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since and are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.
For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude can be defined as quantity or distance.
The absolute difference of two real numbers and is given by , the absolute value of their difference. It describes the distance on the real line between the points corresponding to and . It is a special case of the L p distance for all and is the standard metric used for both the set of rational numbers and their completion, the set of real ...
The converse, though, does not necessarily hold: for example, taking f as =, where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function.
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