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The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric ...
The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value. [6]
"The limit of a n as n approaches infinity equals L" or "The limit as n approaches infinity of a n equals L". The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value | a n − L | is the distance between a n and L. Not every sequence has a limit.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
The following is an example of an image sequence zooming to a selected c value. The magnification of the last image relative to the first one is about 10 10 to 1. Relating to an ordinary computer monitor , it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.
The absolute value bars can be dropped when is a rational number with an even numerator in its reduced form, and is drawn from the set of real numbers, or one of its subsets. The Euclidean norm from above falls into this class and is the 2 {\displaystyle 2} -norm, and the 1 {\displaystyle 1} -norm is the norm that corresponds to the rectilinear ...
The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring. A DVR is compact if and only if it is complete and its residue field R/M is a finite field. Examples of complete DVRs include the ring of p-adic integers and; the ring of formal power series over ...