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The term absolute value has been used in this sense from at least 1806 in French [3] and 1857 in English. [4] The notation | x |, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. [5] Other names for absolute value include numerical value [1] and magnitude. [1]
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 ((), ()) = = ((), ()).
In general, the value of the norm is dependent on the spectrum of : For a vector with a Euclidean norm of one, the value of ‖ ‖ is bounded from below and above by the smallest and largest absolute eigenvalues of respectively, where the bounds are achieved if coincides with the corresponding (normalized) eigenvectors.
This template may be used to enclose text between two vertical bars (U+007C | VERTICAL LINE), such as to denote the absolute value. It adds padding (of width 0.1 em) on each side inside the bars. It adds padding (of width 0.1 em) on each side inside the bars.
When two sets, and , have the same cardinality, it is usually written as | | = | |; however, if referring to the cardinal number of an individual set , it is simply denoted | |, with a vertical bar on each side; [3] this is the same notation as absolute value, and the meaning depends on context.
The vertical bar is used as a mathematical symbol in numerous ways. If used as a pair of brackets, it suggests the notion of the word "size". These are: absolute value: | |, read "the absolute value of x" [2] cardinality: | |, read "the cardinality of the set S" or "the length of a string S"
The norm of a vector is represented with double bars on both sides of the vector. The norm of a vector v can be represented as: ‖ ‖ The norm is also sometimes represented with single bars, like | |, but this can be confused with absolute value (which is a type of norm).
Here, and in the rest of the article, the absolute value bars denote the standard (Euclidean) norm of a vector. The function f is called the integrand, the curve C {\displaystyle {\mathcal {C}}} is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length of the curve C {\displaystyle {\mathcal {C ...