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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 ...
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 mapping |x| from D to the ...
Norm (mathematics) In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space ...
In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient ...
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.
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 ...
Ostrowski's theorem. In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p -adic absolute value. [1]
The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to ...