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  2. Absolute value - Wikipedia

    en.wikipedia.org/wiki/Absolute_value

    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.

  3. Absolute value (algebra) - Wikipedia

    en.wikipedia.org/wiki/Absolute_value_(algebra)

    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 ((), ()) = = ((), ()).

  4. p-adic valuation - Wikipedia

    en.wikipedia.org/wiki/P-adic_valuation

    The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers R {\displaystyle \mathbb {R} } , the completion of the rational numbers with respect to the p {\displaystyle p} -adic absolute value ...

  5. Integer - Wikipedia

    en.wikipedia.org/wiki/Integer

    For example, 21, 4, 0, and −2048 are integers, while 9.75, ⁠5 + 1 / 2 ⁠, 5/4, and √ 2 are not. [8] The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers.

  6. Algebraic number theory - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number_theory

    Consider, for example, the integers. In addition to the usual absolute value function |·| : Q → R, there are p-adic absolute value functions |·| p : Q → R, defined for each prime number p, which measure divisibility by p. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence).

  7. p-adic number - Wikipedia

    en.wikipedia.org/wiki/P-adic_number

    The 3-adic integers, with selected corresponding characters on their Pontryagin dual group. In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p ...

  8. Valuation (algebra) - Wikipedia

    en.wikipedia.org/wiki/Valuation_(algebra)

    When the ordered abelian group Γ is the additive group of the integers, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field K. If K is complete with respect to this metric, then it is called a complete valued field .

  9. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology ; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order ...

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