...very number field contains infinitely many elements. The field of rational numbers is contained in every number field.
...fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.
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...is representable in the form $a+b\sqrt{d}$, where $a$ and $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real numb
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...attice of all points with integral rational coordinates on the plane. Such numbers were first considered in 1832 by [[Gauss, Carl Friedrich|C.F. Gauss]] in hi
...osed into a non-trivial product) of $\Gamma$ (the Gaussian primes) are the numbers of the form
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where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,
is reduced to an integral of rational functions if at least one of the numbers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integ
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...field]] of constants or a finite extension of the field $\mathbb{Q}$ of [[rational number]]s (an [[algebraic number field]]).
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...hic to the projective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no
...the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere $\mathbf C\cup\{\infty\}$.
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...positive rational number $-r''$: $-r'>-r''$. The absolute value $|r|$ of a rational number $r$ is defined in the usual way: $|r|=r$ if $r\geq0$ and $|r|=-r$ if
...s, they are uniquely determined by $r'$ and $r''$ themselves. The rational numbers form an [[Ordered field|ordered field]], denoted by $\mathbf Q$.
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where the minimum is over all pairs $h_0,h_1$ of integral rational numbers such that
...ber $\xi$ can be approximated by rational numbers. For all real irrational numbers one has
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...mple, the set of real numbers is uncountable, whereas that of the rational numbers is countable.
The uncountability of the set of real numbers is sometimes proved by the Cantor diagonalization principle (cf. [[Cantor t
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An element of an extension of the field of rational numbers (cf.
obtained by completing the field of rational numbers with respect to a
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A number that is not a rational number (i.e. an integer or a fraction). Geometrically, an irrational number
...tween any two numbers there is an irrational number. The set of irrational numbers is uncountable, is a set of the second category and has type $G_\delta$ (cf
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...The $p$-adic [[valuation]] (or order) $\nu_p({\cdot})$ on the field of [[rational number]]s is defined by $\nu(a/b) = r$ where $a,b$ are integers and $a/b =
...$p$-adic norm, and the $p$-adic valuation extends to the field of $p$-adic numbers.
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where $a$ and $b$ are real numbers and $m$, $n$ and $p$ are rational numbers, cannot be expressed in terms of elementary functions for any $m$, $n$ and
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...f the same [[cardinality]]. For example, the set of integers, the set of [[rational number]]s or the set of [[algebraic number]]s.
...nite: that is, a set of the same cardinality as some subset of the natural numbers.
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''of numbers''
...e basic arithmetic operations. Multiplication consists in assigning to two numbers $a,b$ (called the factors) a third number $c$ (called the product). Multipl
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An [[Abelian number field]] is an Abelian extension of the field of rational numbers.
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.../\mathbf{Z}_p$, where $\mathbf{Q}_p$ is the additive group of the field of rational [[P-adic number|$p$-adic number]]s and $\mathbf{Z}_p$ is the additive group
...numbers, and also maximal $p$-subgroups of the additive group of rational numbers modulo 1. The ring of endomorphisms of a group of type $p^\infty$ is isomor
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$#C+1 = 19 : ~/encyclopedia/old_files/data/E035/E.0305730 Entire rational function,
are real or complex numbers, and $ z $
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In the case $P=1$,$Q=1$ the [[Fibonacci numbers]] and [[Lucas numbers]] are the Lucas sequences of the first and second kind respectively.
...Lucas sequence $\sum_{n=0}^\infty X_n z^{-n}$ satisfying \eqref{eq:1} is a rational function with denominator $z^2 - Pz + Q$. Indeed,
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...tions, in particular with approximations of irrational numbers by rational numbers. Approximations of curves, surfaces, spaces and mappings are studied in geo
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