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The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.
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 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 x {\displaystyle x} and y {\displaystyle y} .
Absolute value may also be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 70 and −70 is 70. For example, the absolute value of both 70 and −70 is 70.
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 Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane. This identification of the complex number x + i y {\displaystyle x+iy} as a vector in the Euclidean plane, makes the quantity x 2 + y 2 {\textstyle {\sqrt {x^{2}+y^{2}}}} (as first suggested ...
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The order of the natural numbers shown on the number line. A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
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