...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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...]] $\mathbf{Q}(e^{2\pi i/p})$ is not divisible by $p$. All other odd prime numbers are called irregular (see [[Irregular prime number|Irregular prime number]]
...ernoulli numbers]] $B_1,\ldots,B_{(p-3/2)}$, when these numbers (which are rational) are written as irreducible fractions (see [[#References|[a1]]]).
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which is an identity in [[formal power series]] over the rational numbers.
Over the field of $p$-adic numbers we define
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...for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with coefficients in $F$.
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$#C+1 = 101 : ~/encyclopedia/old_files/data/R077/R.0707590 Rational function
A rational function is a function $ w = R ( z) $,
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...ber field]] with a non-Abelian [[Galois group]] over the field of rational numbers $\QQ$,
of algebraic numbers, and the term "non-Abelian" is understood to refer to the Galois group ov
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...ts; moreover, any factorization of $\phi(x)$ into irreducible factors with rational coefficients leads to a factorization of $f(x)$ into irreducible factors wi
...Thus, $g(c_i)$ divides $f(c_i)$. Choosing arbitrary divisors $d_i$ of the numbers $f(c_i)$, one obtains
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The measure of algebraic independence of the numbers $\alpha_1,\dots,\alpha_m$ is the function
where the minimum is taken over all polynomials of degree at most $n$, with rational integer coefficients not all of which are zero, and of height at most $H$.
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...of degree $n$. All rational numbers, and only such numbers, are algebraic numbers of the first degree. The number $i$ is an algebraic number of the second de
...n by zero) are algebraic numbers; this means that the set of all algebraic numbers is a [[Field|field]]. A root of a polynomial with algebraic coefficients is
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...for any $x \in X \subset \mathbf{R}$ (or $x \in X \subset \mathbf{C}$) the numbers $x+T$ and $x-T$ also belong to $X$ and such that the following equality hol
The numbers $\pm nT$, where $n$ is a natural number, are also periods of $f$. For a fun
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''of algebraic numbers''
...\alpha_1,\ldots,\alpha_n$, $\beta_1,\ldots,\beta_n$ are [[Rational number|rational]] or [[algebraic number]]s and $\log\alpha_1,\ldots,\log\alpha_n$, with fix
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...cteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to
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...et series]] with exponents that are independent over the field of rational numbers; etc.
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When $m$ is rational, this is an [[algebraic curve]]. In particular, when $m=1$ it is a circle,
...s case the pole is a multiple point (see Fig.). When $m=p/q$ is a positive rational number, the curve consists of $p$ intersecting branches. When $m$ is a nega
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''Mahler's 3/2 problem'' concerns the existence of "Z-numbers". A ''Z-number'' is a real number $x$ such that the [[Fractional part of
...natural numbers $n$. Kurt Mahler conjectured in 1968 that there are no Z-numbers.
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...nd the rational numbers $x$ in the prime decomposition of which only prime numbers from the set $S$ appear.
...s, every element of this set is of the form $|.| v$, where $v$ is either a rational prime number or the symbol $\infty$. One now modifies the definition of the
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...cyimages/s/s085/s085000/s0850009.png" /> and if there exists a sequence of
rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofm
...pediaofmath.org/legacyimages/s/s085/s085000/s08500020.png" /> are
rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
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A method for isolating the algebraic part in indefinite integrals of rational functions. Let $ P( x) $
are real numbers, $ ( p _ {j} ^ {2} /4)- q _ {j} < 0 $,
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...^G$ of $K$ consisting of all elements fixed under $G$ is itself a field of rational functions in $n$ (other) variables with coefficients in $\mathbf Q$. This q
...l, the answer to Noether's problem is negative. The first example of a non-rational field $K^G$ was constructed in [[#References|[2]]], and in this example $G$
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...a field|characteristic]] 0 is [[Isomorphism|isomorphic]] to the field of [[rational number]]s. A prime field of [[Characteristic of a field|characteristic]] $p
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...without remainder) by $b$; this is noted as $b\mid a$. Division of complex numbers is defined by the formula
while division of the complex numbers in their trigonometric form is given by the formula
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...is in fact $A$-rational. Recall that a formal power series $\alpha$ is $R$-rational, $R$ a commutative ring, if there exist two polynomials $P , Q \in R [ X ]$
For a rational function $R \in K ( X )$, there are several representations of the form $R
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...ield of [[algebraic number]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree.
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''(in the geometry of numbers)''
...tional polyhedron, i.e. is defined by a system of linear inequalities with rational coefficients, then the "non-zero volume condition" in the flatness theore
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...thmetic condition (usually one looks for solutions in integers or rational numbers). The study of such equations forms the topic of the theory of [[Diophantin
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...and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere $S^n$:
is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[1]]].
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...o element other than the identity is (aperiodic). The additive group of [[rational number]]s $\mathbb{Q}^+$ is an aperiodic example, and the group $\mathbb{Q}
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...e construction described above gives the completion of the set of rational numbers by Dedekind sections.
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...characteristic number]] defined for closed oriented manifolds and assuming rational values. Let $ x \in H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) $
the rational number $ x [ M ] = \langle x ( \tau M ) , [ M] \rangle $
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...algebraic numbers (cf. [[Algebraic number|Algebraic number]]) by rational numbers: Find a quantity $\nu=\nu(n)$ such that for each algebraic number $\alpha$
has a finite number of solutions in rational integers $p$ and $q$, $q>0$, for any $\epsilon>0$, and an infinite number o
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is an integer, while each one of the numbers $ b _ {j} $,
the numbers
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