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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 ((), ()) = = ((), ()).
The following proof follows the one of Theorem 10.1 in Schikhof (2007). Let | | be an absolute value on the rationals. We start the proof by showing that it is entirely determined by the values it takes on prime numbers.
The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute
Absolute zero, the lowest limit of the thermodynamic temperature scale; Absolute magnitude, a measure of the luminosity of a celestial object; Relative change and difference, used to compare two quantities taking into account the "sizes" of the things being compared; Absolute (disambiguation) Number (disambiguation)
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. This is written in symbols as | −3 | = 3 and | 3 | = 3. In general, any arbitrary real value can be specified by its ...
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A subderivative value 0 occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval [ − 1 , 1 ] {\displaystyle [-1,1]} , which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin ...