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...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.

...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

...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

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

...field]] of constants or a finite extension of the field $\mathbb{Q}$ of [[rational number]]s (an [[algebraic number field]]).

...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 $\mathbf C\cup\{\infty\}$.

...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 $\mathbf Q$.

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

...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

An element of an extension of the field of rational numbers (cf. obtained by completing the field of rational numbers with respect to a

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

...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.

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

...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.

''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

An [[Abelian number field]] is an Abelian extension of the field of rational numbers.

.../\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

$#C+1 = 19 : ~/encyclopedia/old_files/data/E035/E.0305730 Entire rational function, are real or complex numbers, and $ z $

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,

...tions, in particular with approximations of irrational numbers by rational numbers. Approximations of curves, surfaces, spaces and mappings are studied in geo

...]] $\mathbf{Q}(e^{2\pi i/p})$ 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]]]).

which is an identity in [[formal power series]] over the rational numbers. Over the field of $p$-adic numbers we define

...for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with coefficients in $F$.

$#C+1 = 101 : ~/encyclopedia/old_files/data/R077/R.0707590 Rational function A rational function is a function $ w = R ( z) $,

...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

...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

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 $H$.

...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

...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

''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

...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

...et series]] with exponents that are independent over the field of rational numbers; etc.

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

''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.

...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

...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/

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 $,

...^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 $G$

...a field|characteristic]] 0 is [[Isomorphism|isomorphic]] to the field of [[rational number]]s. A prime field of [[Characteristic of a field|characteristic]] $p

...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

...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

...ield of [[algebraic number]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree.

''(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

...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

...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]]].

...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}

...e construction described above gives the completion of the set of rational numbers by Dedekind sections.

...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 $

...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

is an integer, while each one of the numbers $ b _ {j} $, the numbers

...ree $n$ is an extension of degree $n$ of the field $\mathbf Q$ of rational numbers. Alternatively, a number field $K$ is an algebraic number field (of degree

...c numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then ...numbers, linearly independent over the field of rational numbers, then the numbers $e^{\beta_1},\dots,e^{\beta_n}$ are algebraically independent.

...R$ of real numbers is a Euclidean field. The field $\mathbf Q$ of rational numbers is not a Euclidean field. ...D valign="top"> G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wile

...ndamental sequences of rational numbers one arrives at the concept of real numbers (cf. [[Real number|Real number]]); by identifying isomorphic groups with ea

* The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]] * The field of real numbers is not quadratically closed as it does not contain a square root of $-1$.

...of the concept of irrationality (cf. [[Irrational number]]). Thus, the two numbers $\alpha$ and $1$ are linearly independent if and only if $\alpha$ is irrati

...ing $\mathbf Z$ is the minimal ring which extends the semi-ring of natural numbers $\mathbf N=\{1,2,\dots\}$, cf. [[Natural number|Natural number]]. Cf. [[Num ...s an algebraic field extension, where $\mathbf Q$ is the field of rational numbers, the [[field of fractions]] of $\mathbf Z$, then the integers of $k$ are th

...field]] $\mathbf Q(e^{2\pi i/p})$ is divisible by $p$. All other odd prime numbers are called regular. ...one of the numerators of the first $(p-3)/2$ [[Bernoulli numbers|Bernoulli numbers]] $B_2,B_4,\dots,B_{p-3}$ is divisible by $p$ (cf. [[#References|[1]]]).

An Abelian [[Extension of a field|extension]] of the field of rational numbers $\mathbf{Q}$, i.e. a [[Galois extension]] $K$ of $\mathbf{Q}$ such that the

...an infinite number of solutions in integers $q \ge 1$ for almost-all real numbers $\alpha$ if the series ...s. For example, for almost-all $\alpha$ there exists an infinite number of rational approximations $p/q$ satisfying the inequality

variables with integer rational coefficients not all of which are divisible by $ m $. are different prime numbers, is equivalent to the solvability of the congruences

...sponds a unique real logarithm (logarithms of negative numbers are complex numbers). The main properties of the logarithm are: These make it possible to reduce multiplication and division of numbers to the addition and subtraction of their logarithms, and the raising to pow

over an algebraically closed field is a [[Rational curve|rational curve]], i.e. it is birationally isomorphic to the projective line $ P ^ the rational mapping $ \phi _ {| K _ {X} | } : X \rightarrow P ^ {g-1} $

...\alpha_n$ be arbitrary real numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that

==Algebraic independence of numbers.== Complex numbers $ \alpha _{1} \dots \alpha _{n} $

...own. A classical example of such a sequence is the sequence of [[Fibonacci numbers]] $1,1,2,3,5,8$ defined by $a_{n+2}=a_{n+1}+a_n$ with $a_0=0$, $a_1=1$. ...orm a recursive sequence. Such a series represents an everywhere-defined [[rational function]]: its denominator is the reciprocal polynomial $X^p F(1/X)$.

...om]] for the real line can be formulated in terms of Dedekind cuts of real numbers.

for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, nam ...laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a pr

...]; 3) $a_{n+2}=a_{n+1}+a_n$, the sequence of [[Fibonacci numbers|Fibonacci numbers]]. ...only if the formal power series $\alpha(x)=\alpha_0+\alpha_1x+\dotsb$ is a rational function of the form $\alpha(x)=p(x)/q(x)$, with $p(x)=1-p_1x-\dotsb-p_mx^m

...he same kind, and which is accepted as a unit" . Rigorous theories of real numbers were constructed at the end of the 19th century by K. Weierstrass, G. Canto Real numbers form a non-empty totality of elements which contains more than one element

...Cantor's construction of the set of real numbers from the set of rational numbers.

...ional variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theorems for linear algebraic groups. of complex numbers were in fact proved by E. Picard and, in contemporary terminology, establis

...ample, any real number is an accumulation point of the set of all rational numbers in the ordinary topology. In a [[discrete space]], no set has an accumulati

geometry, which reduces the problem of the existence of rational non-trivial absolute valuations $\nu$ on $K$ the set of $K_\nu$-rational

...ctively defined. Thus, in the case of rational approximations to algebraic numbers, the bound for the denominators of "good" approximations, which is establ ...olves the application of effective methods of the theory of transcendental numbers (cf. [[Linear form in logarithms|Linear form in logarithms]]). The best res

are finite or infinite sequences of complex numbers or functions. For continued fractions one uses the notation ...w.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564026.png" /> are rational functions. Famous examples of explicit continued fractions are those for hy

...hematical novelette [[#References|[a7]]]. Only in later years have surreal numbers become the subject of more traditional mathematical papers and books [[#Ref ...nes. In fact, the construction never terminates, and therefore the surreal numbers do not form a set but a proper class (cf. [[Types, theory of|Types, theory

...{ n } ( z , \tau )$, and as weights the [[Christoffel numbers|Christoffel numbers]] ...ces|[a2]]]. The underlying ideas have been generalized from polynomials to rational functions. See [[#References|[a1]]].

...[6]</TD> <TD valign="top"> A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)</TD></TR></table> ...theorem to the problem of simultaneous approximation of several algebraic numbers. This was extended by H.P. Schlickewei [[#References|[a2]]] to include $p$-

...to a [[Field|field]], for example, the field of rational, real or complex numbers. ...into factors of the first and second degree, and over the field of complex numbers into factors of the first degree (cf. [[Algebra, fundamental theorem of|Alg

...additive number theory. Let $X_1,\ldots,X_k$ be arbitrary sets of natural numbers, let $N$ be a natural number and let $J_k(N)$ be the number of solutions of where $n_1\in X_1,\ldots,n_k\in X_k$. It is with the investigation of the numbers $J_k(N)$ that additive number theory is concerned; for example, if it can b

has rational coefficients, is called a rational trigonometric sum; if $ P = q $,

...] in connection with the calculation of the sum of equal powers of natural numbers: The values of the first Bernoulli numbers are:

The Apéry numbers $a _ { n }$, $b _ { n }$ are defined by the finite sums ...irrationality proof of $\zeta ( 3 )$, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrat

...s of primes: $\theta(x)$ is the sum of the natural logarithms of the prime numbers up to $x$, while $\psi(x)=\sum_{n\leq x}\Lambda(n)$ (cf. [[Chebyshev functi Arithmetic functions appear and are employed in studies on the properties of numbers. However, the theory of arithmetic functions is also of independent interes

A conjecture on the finiteness of the set of rational points on an [[Algebraic curve|algebraic curve]] of genus $ g > 1 $. ...'s conjecture is taken to be the assertion of the finiteness of the set of rational points $ X ( L) $

...\lim_{r_n\to\a} a^{r_n}$, where $r_n$ is an arbitrary sequence of rational numbers tending to $\a$. Powers with a complex base (see

...p^\nu})$ is the field obtained by adjoining to the field $\mathbb{Q}$ of [[rational number]]s a primitive root of unity $\zeta_{p^\nu}$ of order $p^\nu$ and $k

group of the field of rational numbers are sequences of the form The multiplicative group of the field of rational numbers is

If $k$ is the field of rational numbers, the problem becomes one of constructing a normal algebraic number field wi ...bf C(X)$, the field of rational functions in one variable over the complex numbers, which is well under control. Under certain conditions on the group one can

...ortant examples of divisible groups are the additive group of all rational numbers and the group of all complex roots of unity of degrees $p^k$, $k=1,2,\ldots ...ive module|Injective module]]). Let $\mathbf Q_p$ be the field of $p$-adic numbers and $\mathbf Z_p$ its ring of integers. Then the quasi-cyclic group for the

...equations|Diophantine equations]] arises from the theory of transcendental numbers, from the solution by A.O. Gel'fond of Hilbert's seventh problem and from s the empty set. For instance, for the field of rational numbers, given a finite set $ S = \{ p _ {1} \dots p _ {s} \} $

is the greatest common divisor of the numbers $ n $ It follows immediately from Euler's criterion that the numbers $ 1 \dots p - 1 $

==Liouville's theorem on approximation of algebraic numbers== ...r]] of degree $n \ge 2$ and $p$ and $q$ are any positive integral rational numbers, then

...ree operations (addition, multiplication, passing to the limit by rational numbers), performed not more than a countable number of times, starting from an arg

...set of Lagrange constants in the problem of rational approximation to real numbers. The Lagrange spectrum is strictly contained in the Markov spectrum (see [[

of a recurrence is a rational function $ { {r ( X ) } / {s ( X ) } } $ If so, the distinct complex numbers $ \alpha _ {i} $

rational numbers $\Q$, the field of real numbers $\R$, the field of complex numbers $\C$, finite fields (see

...ho defined the space of real numbers as the completion of that of rational numbers, see [[Real number|real number]]

...n|Simply-connected domain]]) is called rational if its homotopy groups are rational vector spaces (cf. also ...e of homotopy categories between the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algeb

...many solutions: $x_k=k\pi$, $k=0,\pm1,\pm2,\ldots,$ in the domain of real numbers. If an equation has as solution all numbers of a domain $M$, then it is called an identity on $M$.

...the cases where $F$ is an algebraic number field (finite over the rational numbers; cf. also [[Algebraic number|Algebraic number]]; [[Number field|Number fiel Schur indices over the real numbers are computed by means of the Frobenius–Schur count of involutions. Let $L

of integral $ \ell $-adic numbers which are continuously acted upon from the left by the fundamental group of is the field of rational $ \ell $-adic numbers, then the $ \mathbf Q _ {\ell} $-spaces $ H _ {\ell} ^ {i} ( \overline

...ficients and roots of an equation that are numbers of a certain kind (e.g. rational, real or complex). The case of the coefficients and roots being elements of ...icity). In particular, this statement also applies to the field of complex numbers.

are different prime numbers. In any finite non-nilpotent group there are subgroups that are Shmidt grou of the rational numbers). A group is locally finite if every finite subset generates a finite subgr

...uence, and the theorem on the existence of exact bounds of bounded sets of numbers. From the point of view of traditional mathematics, Specker's result shows

...ge's theorem]] implies that the Pythagoras number of the field of rational numbers is $4$. A finite field has Pythagoras number $1$ (in characteristic $2$) o

...ral numbers $(a,b)$ yields the [[continued fraction]] development of the [[rational number]] $a/b$.

Examples of constructive metric spaces. a) The space of natural numbers $ H $. is the set of natural numbers (the natural numbers are treated as words of the form $ 0 , 01 , 011 ,\dots $),

of real numbers, which satisfies the following conditions: the field of real numbers, then $ | x | = \max \{ x, - x \} $,

...ms of Diophantine approximations concerned rational approximations to real numbers, but the development of the theory gave rise to problems in which certain r ...egers (linear homogeneous Diophantine approximations), i.e. the problem of rational approximations to <img align="absmiddle" border="0" src="https://www.encycl

with coefficients in the field of rational numbers $ \mathbf Q $, is the group of rational $ i $-

...the [[Completion of a uniform space|completion]] of the field of rational numbers with respect to its additive uniform structure.

...the methods of algebraic geometry. Estimates from below of the number of (rational) points [[#References|[1]]], [[#References|[4]]] are also important. of rational points with values in $ K $

...onstant vector whose components are linearly independent over the rational numbers. Examples of conditionally-periodic functions are given by partial sums of

is a rational integer and $\alpha$ are rational numbers, with the usual addition and multiplication. Then $A$

is a finite extension of the field of rational numbers $ \mathbf Q $; ...D valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD

...ss with rational coefficients, then the corresponding Chern number will be rational. The Chern number $ x [ M ^ {2n} ] $ The Chern numbers are quasi-complex bordism invariants, and hence the characteristic class $

...natural numbers $a_1,\ldots,a_n$. The greatest common divisor of a set of numbers not all of which are zero always exists. The greatest common divisor of $a_ ...on divisor of $a_1,\ldots,a_n$ is divisible by any common divisor of these numbers.

...l numbers, that satisfies the following condition: There exists a faithful rational representation $\rho : G \rightarrow \mathrm{GL}_n$ defined over $\mathbb{Q

is the field of rational numbers. adic numbers is called the cyclotomic $ \Gamma $-

...1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined by ...g lattice points and Fourier theory (cf. [[Geometry of numbers|Geometry of numbers]]; [[Fourier transform|Fourier transform]]). There are many generalizations

...thod of undetermined coefficients is its use in the expansion of a regular rational function in a complex or real domain in elementary fractions. Let $ P ( z The regular rational function $ P ( z) / Q ( z) $

such that, for any numbers $ r _ {1} \dots r _ {n} \in \mathbf Z $ as a topological group if and only if the numbers $ 1, \omega _ {1} \dots \omega _ {n} $

...ing mentioned above concerning the vectors $X_\alpha$, $h_\alpha'$ and the numbers $N_{\alpha\beta}$, carry over verbatim to the case of an arbitrary finite-d

...oefficients, the solutions of which are sought for in integers or rational numbers. It is usually assumed that the number of unknowns in Diophantine equations of rational numbers, of the field of $ p $-adic numbers, etc.

the numbers $ a _ {n} $ that have rational ordinates for $ p \in P $,

...y class $B$ of birationally-equivalent surfaces, except for the classes of rational and ruled surfaces, there is a (moreover, unique) minimal model. ...$B$ are exhausted by the projective plane $P^2$ and the series of minimal rational ruled surfaces $F_n=P(\mathcal O_{P^1}+\mathcal O_{P^1}(n))$ for all intege

...t box principle]] a more general theorem can be demonstrated: For any real numbers $\alpha_1,\ldots,\alpha_n$ and any natural number $Q$ there exist integers over the field of rational numbers $ \mathbf Q $

be the set of all rational numbers. Then a set $ X $( be the set of numbers of the form $ n + m \xi $,

...a distributive quasi-group one may quote the set $\mathbf{Q}$ of rational numbers with the operation $x \cdot y = (x+y)/2$. Any idempotent [[medial quasi-gro ...p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, with $p_1,\ldots,p_k$ distinct prime numbers and $\alpha_1,\ldots,\alpha_k$ non-negative integers, is isomorphic to the

...r and denominator. The construction generalises the construction of the [[rational number]]s from the ring of integers.

over the field of complex numbers with the usual operations (see ...nsional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see

holds, where $P$ and $Q$ are positive odd coprime numbers, and the supplementary formulas ...acter. This real character plays an important role in the decomposition of rational primes in a [[Quadratic field|quadratic field]] (see [[#References|[a1]]]).

...df}}{=} \overline{\mathbf{Z}^{+}} $. If the set $ \mathbf{Q} $ of rational numbers is ordered also by the relation $ \leq $, then $ \eta \stackrel{\text{df}}{ ...A totally ordered set is of type $ \lambda $ (the type of the set of real numbers $ \mathbf{R} $ ordered by $ \leq $) if and only if it is continuous and con

tuples of complex numbers $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $; such that the numbers $ c _ { 0 }, \dots, c _ { {n-1} } $

...arrow G : \gamma ( 1 ) = 1 \}$; here, $S ^ { 1 }$ is the circle of complex numbers $z = e ^ { i \theta }$. One can take spaces of polynomial, rational, real-analytic, smooth, or $L _ { 1 / 2 } ^ { 2 }$ loops, in decreasing ord

and at least one of the numbers $ e _ {i} $ of rational numbers there exists a ramified prime ideal. This is not true for arbitrary algebra

...ditionally, proofs are deemed to be non-elementary if they involve complex numbers. ...partitions, of additive representations, of the approximation by rational numbers, and of continued fractions. Quite often, the solution of such problems lea

are finite or infinite sequences of complex numbers. Instead of the expression (1) one also uses the notation determine two sequences $\{P_n\}_{n=0}^\o$ and $\{Q_n\}_{n=0}^\o$ of complex numbers. As a rule, it is assumed that the sequences (2) and (3) are such that $Q_n

An important result on the arithmetic of the [[Bernoulli numbers|Bernoulli numbers]] $B _ { n }$, first published in 1840 by Th. Clausen [[#References|[a1]]] ...theorem is the complete determination of the denominators of the Bernoulli numbers: If $B _ { 2 n } = N _ { 2 n } / D _ { 2 n }$, with $\operatorname { gcd }

be a [[Rational mapping|rational mapping]] defined by a [[Linear system|linear system]] $ | m K _ {V} | $, Suppose that the ground field is the field of the complex numbers $ \mathbf C $.

...example, the integers form an essential submodule of the group of rational numbers. Each module is an essential submodule of its injective envelope (see [[Inj

...''general'' complex quintic threefold, there are only finitely many smooth rational curves of fixed degree d ≥ 1.''' ...math>, there is an incidence scheme <math>\Phi_d</math> that parameterizes rational curves of degree <math>d</math> embedded in hypersurfaces of degree <math>m

of positive real numbers. By definition one puts $ \mathop{\rm mod} _ {K} ( 0) = 0 $. ...ld is isomorphic to either the field of real numbers, the field of complex numbers or the skew-field of quaternions.

are polynomials with integer rational coefficients. In other words, the polynomials $ a( x) $ with rational coefficients are called congruent modulo the double modulus $ ( p, f( x))

is the ring of rational equivalence classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cy In the case of a variety over the field of complex numbers, there is a homomorphism $\CH(X) \to \mathrm{H}(X,\mathbb Z)$ into the sing

...$ is a uniform subgroup of $G_A$ if and only if: 1) $G$ has no non-trivial rational characters defined over $\mathbf Q$; and 2) all unipotent elements of $G_{\

...e solution of the problem of trisecting an angle $\phi$ reduces to finding rational roots of a cubic equation $4x^3-3x-\cos\phi=0$, where $x=\cos(\phi/3)$, whi

...$M$ is $h_M(n) = \dim_K M_n$ and there exists a polynomial $P_M(t)$ with rational coefficients such that, for sufficiently large $n$, $P_M(n) = h_M(n) = \di

are pairwise distinct complex numbers, $ \alpha , \alpha ^ \prime $( Riemann, under certain additional assumptions on the numbers $ \alpha , \alpha ^ \prime \dots \gamma ^ \prime $,

...$K(L)$, called the field of rational functions on $L$. It consists of the rational functions $p(x, y)/q(x, y)$, where $q$ is not divisible by $f$, considered ...erse to each other; here the fields $K(L)$ and $K(M)$ are isomorphic. Such rational mappings are called birational, or Cremona, transformations. All Cremona tr

.... In certain cases, for example for rational $\alpha$ and $\beta$, and for numbers $\alpha$ and $\beta$ one of which can be represented by a [[Continued fract ...is the problem whether for an arbitrary increasing sequence $M$ of natural numbers $(m_k)$, $k=1,2,\ldots$, one has

algebraic equation over the field of rational numbers, and the problem splitting field has degree $2^s$ over the field of rational numbers for

...$E$-functions). The development of methods of the theory of transcendental numbers has proved to have a strong influence on new studies in [[Diophantine equat ...n="top">[a1]</TD> <TD valign="top"> K. Mahler, "Lectures on transcendental numbers" , ''Lect. notes in math.'' , '''546''' , Springer (1976) {{MR|0491533}} {{

...]] is term-generated, whereas the $\Sigma_F$-algebra of [[Real number|real numbers]] is not.

[[Least common multiple|least common multiple]] of the numbers $b$ and $d$ is taken as the common denominator. Multiplication and division ...uchy sequence|Cauchy sequences]]) leads to $\R$, the ordered field of real numbers (cf.

...functions (growth estimates, inequalities for derivatives, polynomials and rational functions, least deviation from zero, etc.). ==The approximation of functions of a complex variable by polynomials and rational functions.==

...primes $p$, where $\mathbf Q$ and $\mathbf Z_p$ are the field of rational numbers and the ring of $p$-adic integers, respectively. The class of the element $

plane, and such that there exists two numbers $ p _ {1} , p _ {2} $ is real and rational, $ f ( z) $

...fficients whose roots cannot be expressed in terms of radicals of rational numbers. For a modern formulation of Abel's theorem for equations over an arbitrary are complex numbers, converges at $ z = z _ {0} $,

...ok like a complete curve. For instance, if $f \in \mathbf{Q} ^ { * }$ is a rational number, the identity for the differential $d f / f$, when $f$ is a non-zero rational function on a smooth complex projective curve $C$.

...a constructive (computable) real number. First one introduces the natural numbers as words of the form $ 0 , 01 , 011 \dots $ Analogously one defines rational numbers as words of a certain type over the alphabet $ \{ 0 , 1 , - , / \} $.

that is, the representation of all possible rational functions of $ w _ {1} $, the rational statement of the Jacobi inversion problem is as follows: Suppose one is giv

of polynomials, and is called a rational function of $ x _ {1}, \dots, x _ {n} $. an algebraic function can be expressed as square and cube roots of rational functions in the variables $ x _ {1}, \dots, x _ {n} $;

An extension of degree 2 of the field of rational numbers $\Q$ (cf. [[Extension of a field|Extension of a field]]). Any quadratic fie ...the ring of integers of the field $\Q\bigl(\sqrt d\bigr)$ over the ring of rational integers $\Z$, one can take

...ng normal numbers was indicated. For other methods for constructing normal numbers and for connections between the concepts of normality and randomness see {{ |valign="top"|{{Ref|Pi}}|| S. Pillai, "On normal numbers" ''Proc. Indian Acad. Sci. Sect. A'' , '''12''' (1940) pp. 179–184 {{MR|0

...itial objects and formation rules present no difficulties). Also, rational numbers, algebraic polynomials, algorithms, and calculi of various well-defined typ ...valign="top">[4]</TD> <TD valign="top"> N.A. Shanin, "Constructive real numbers and constructive function spaces" , ''Transl. Math. Monogr.'' , '''21''' ,

where the numbers $a_n$ all belong to a fixed [[algebraic number field]] (cf. also [[Algebrai ...t|_\nu$ of $K$ (cf. also [[Norm on a field]]). When the $x_i$ are rational numbers, $H(x_0,\ldots,x_n)$ is simply the maximum of the absolute values of the $x

...mate for those values of $\alpha$ that could not be well approximated by a rational number with a small denominator. Vinogradov's estimate used the sieve of Er ...ghan [[#References|[a6]]] found a much simpler approach to sums over prime numbers.

...ently, that integral polynomials cannot be small and non-zero at algebraic numbers (cf. also [[Liouville theorems|Liouville theorems]]). A Łojasiewicz inequa

is a non-zero rational number, and the Birch–Tate conjecture is about a relationship between $ is a rational integer, and the Birch–Tate conjecture states that if $ F $

...n-example.'' The [[quotient group]] $\R/\Q$ (real numbers modulo rational numbers, additive) may be thought of as a quotient measurable space, $\R$ being en

field tower is the extension of the field of rational numbers obtained

is the group of rational numbers and $ \mathbf Z $

is not rational (cf. [[Rational surface|Rational surface]]); ...<TR><TD valign="top">[10]</TD> <TD valign="top"> Y. Miyaoka, "On the Chern numbers of surfaces of general type" ''Invent. Math.'' , '''42''' (1977) pp. 225–

exist, then for any real numbers $ \lambda _ {1} $ ...educed by means of substitution and integration by parts to integration of rational functions:

each rational mapping of a non-singular variety into an Abelian The theory of Abelian varieties over the field of complex numbers $\C$

...fact, which was already known to P. Fermat. The only prime divisors of the numbers $ x ^ {2} + 1 $ These relations for the Legendre symbol show that the prime numbers $ p $

the étale or (over the field of complex numbers) the topological Betti numbers (cf. [[Betti number|Betti number]]) $ b _ {0} = b _ {4} = 1 $, only for smooth rational curves. It also follows that $ ( D) ^ {2} $

...he arithmetic of corresponding mathematical objects: the field of rational numbers, algebraic fields, algebraic varieties over finite fields, etc. The simples

...0, its genus 1) or has a unique singular double point (in which case it is rational). Cubic curves are the curves of lowest degree for which there exist moduli is the field of complex numbers, $ X ( \mathbf C ) $

...constant has not been studied; it is not even known (2022) whether it is a rational number or not. * {{Ref|HaWr}} G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|042

where the numbers $a_n$ belong to a fixed [[algebraic number field]] (cf. also [[Algebraic nu ...s $\nu$ of $K$ (cf. also [[Norm on a field]]). When the $x_i$ are rational numbers, $H(x_0,\ldots,x_n)$ is simply the maximum of the absolute values of the $x

...t sum]]s of cyclic groups are Abelian. Also the additive group of rational numbers $\mathbf{Q}^+$ is Abelian; it is moreover a [[locally cyclic group]], i.e. ...uely into a direct sum of primary groups that correspond to distinct prime numbers.

..., composite numbers, squares) were distinguished; the structure of perfect numbers (cf. [[Perfect number|Perfect number]]) was studied; and the solution in in ...mbers. Somewhat later, Eratosthenes discovered a method of obtaining prime numbers, which is still called the sieve of Eratosthenes (cf. [[Eratosthenes, sieve

is the field of complex numbers $ \mathbf C $, is a rational variety, the algebraic cycles $ Z $

of rational numbers by adjoining a primitive $ n $-th root of unity $ \zeta _ {n} $, is the field of rational $ p $-adic numbers. Since $ K _ {n} = K _ {2n} $

...rly, these numbers are in fact perfect or abundant (i.e. "non-deficient") numbers. ...$b = 5$, $c = 7$, $abcd$ is abundant for any prime number $d > c$. Of the numbers $\leq 1000$, $52$ are abundant.

is the group of rational characters of the torus $ S $ , be the groups of rational characters of the tori $ S $

...character is the completion of the real number system by two "improper" numbers $ + \infty $ by the transfinite numbers $ \omega , \omega + 1 \dots 2 \omega , 2 \omega + 1 ,\dots $(

denote, respectively, the minimum and maximum of the numbers $ x _ {0} $, classes of rational functions $ \Pi _ {m , n } $ (of the form $ p _ {m} / q _ {n} $,

...of rational integers. Let $\tilde {\bf Q }$ be the field of all algebraic numbers (cf. also [[Algebraic number|Algebraic number]]) and let $\widetilde{\bf Z} ...Z}$. In 1984, D.C. Cantor and P. Roquette [[#References|[a1]]] considered rational functions $f _ { 1 } , \dots , f _ { m } \in {\bf Q} ( X _ { 1 } , \dots ,

...="top">[a5]</TD> <TD valign="top"> D. Zagier, "On the number of Markoff numbers below a given bound" ''Math. Comp.'' , '''39''' (1982) pp. 709–723</TD ...TD> <TD valign="top"> A. Baragar, "Asymptotic growth of Markoff–Hurwitz numbers" ''Compositio Math.'' , '''94''' (1994) pp. 1–18</TD></TR><TR><TD vali

where the first terms are rational irreducible fractions with positive denumerators, with lowest common multip 3) Vinogradov's estimates for trigonometric sums with prime numbers. Let $ \epsilon \leq 0.001 $.

and complex numbers $ w _ {1} \dots w _ {n} $. be positive definite, and in this case there is a rational $ ( 2 \times 2 ) $-

...K. Bari, 1921); for example, the [[Cantor set|Cantor set]] with a constant rational ratio $ \theta $ is an integer, that is, whether a set of numbers is a $ U $-

...athbb{Q}/\mathbb{Z}$, where $\mathbb{Q}$ is the additive group of rational numbers and $\mathbb{Z}$ is the additive group of integers. This fact is of importa ...hic Brauer–Severi varieties of dimension $n-1$, possessing a point that is rational over $K$ (cf. [[Brauer–Severi variety|Brauer–Severi variety]]).

...\}$ be a fixed set of rational prime numbers and let $a \neq 0$ be a fixed rational integer. The Diophantine equation (cf. also [[Diophantine equations|Diophan ...nd complex as well as $p$-adic) of linear forms in logarithms of algebraic numbers (cf. [[Linear form in logarithms|Linear form in logarithms]]) and reduction

...of real functions on a topological space $X$, indexed by countable ordinal numbers, which are defined inductively iterating the operation of taking pointwise ...ich takes the values $0$ on the irrational numbers and $1$ on the rational numbers, is a classical example of function which does not belong to the first Bair

...and solvable. In particular, the elementary theory of the field of complex numbers is solvable. Two formulas in the same signature as that of a theory $ T $ ...ng the other solvable elementary theories are those of addition of natural numbers and integers, of Abelian groups, of $ p $-

be an infinite triangular table of arbitrary but fixed complex numbers: also consisting of arbitrary fixed complex numbers.

...elds $ K $ of finite degree over the field $ \mathbf Q $ of rational numbers (cf. [[Algebraic number|Algebraic number]]). ...\dots + x _{n} \omega _{n} $ where all the $ x _{i} $ run through the rational integers (i.e. $ \mathbf Z $ ). Moreover, such a representation is unique

For simplicity, below the objects are considered over the complex numbers. The book [[#References|[a1]]] is a general reference with full coverage of ...th century geometers, especially G. Castelnuovo and F. Enriques, used this rational mapping to study the embedding of $ S $

in his proof that the set of real numbers in the segment $ [ 0, 1 ] $ ...infinite sequences of zeros and ones (hence, for instance, the set of real numbers is non-denumerable). But the argument does not depend on the indices rangin

...h [[Algebra|algebra]], which includes the study of operations performed on numbers. The properties of integers form the subject of number theory (cf. [[Elemen ...ncient Egyptians to perform addition and subtraction operations on natural numbers in a relatively simple manner. Multiplication was carried out by doubling,

is a fixed vector whose components are linearly independent over the rational numbers. When $ G= \mathop{\rm SL} ( 2, \mathbf R ) $

...same holds for the relative algebraic closure of the field $Q$ of rational numbers in $Q_p$. There are also $p$-adically closed fields whose value group is no ...67) and Roquette (1971) proved a $p$-adic analogue which characterizes the rational functions $f\in K(X_1,...,X_n)$ over a $p$-adically closed field $K$ which

...a finite group and finitely many copies of the additive group of rational numbers. Local perfect rings are characterized by the fact that any linearly indepe

...ic exponential sums that generalize quadratic Gauss sums and the number of rational points on certain elliptic curves (cf. also [[Elliptic curve|Elliptic curve is a rational function on the projective line $ \mathbf P $,

The numbers $ v,\ b,\ r _ {i} $, different numbers $ \lambda _ {1} \dots \lambda _ {m} $

They reduce to the [[Lucas numbers]] irreducible polynomials over the rational numbers if and only if $n=2^k$

...of algebraic functions developed in parallel with the theory of algebraic numbers. The fundamental analogy between the two, which was stressed by D. Hilbert while their solutions are replaced by rational points with values in the field $ K $

and the numbers $ f ( \Gamma ) = \sigma ( \Gamma ) / \Delta ( \Gamma ) ^ {- 1/3} $, of all real numbers are invariants of the equivalence relation $ \rho $;

...elliptic curves (cf. [[Elliptic curve|Elliptic curve]]) over the rational numbers and modular forms (cf. [[Modular form|Modular form]]). It has been complete Let $E$ be an [[Elliptic curve|elliptic curve]] over the rational numbers, and let $L(E,s)$ denote its Hasse–Weil $L$-series. The curve $E$ is said

if that limit in fact exists. The concept of derived numbers along the web $ N $ ...the differentiation of the increment of a function in one real variable by rational dyadic intervals of the form $ ( j / 2 ^ {i} , ( j + 1) / 2 ^ {i} ] $.

The field of rational functions of an irreducible algebraic curve over $ k $ ...of degree zero is a principal divisor. This property is characteristic for rational smooth projective curves.

of complex numbers and for any integer $ p \geq 0 $ ...-dimensional uniruled variety, that is, a variety such that there exists a rational mapping of finite degree $ P ^ {1} \times Y \rightarrow X $,

having a solution that is rational over some field extension of $ F $ has a solution that is rational over $ F $.

...ts, etc. At these stage there was no such thing as an abstract number, and numbers were merely names. ...resentations of numbers begin to appear (cf. [[Numbers, representations of|Numbers, representations of]]).

...the representation of a given number as a sum of not more than three prime numbers (cf. [[Goldbach problem|Goldbach problem]]); and the representation of a gi located in intervals distributed over neighbourhoods of rational points, is extracted. Instead of the analytic properties of $ F(z) $

...]). Kummer [[#References|[5]]] proved the theorem for some irregular prime numbers and also established its validity for all $p<100$. In case 1 he showed that ...is is equivalent to the requirement that only one of the numerators of the numbers $B_{2n}$, where $2n = 2,4,\ldots,p-3$, is divisible by $p$); 2) $B_{2np} \n

...differs from numerical packages in that it manipulates symbols rather than numbers; thus, it does calculations in exact mode. For instance, integers are regar ...eric set of $7$ equations in $7$ variables of degree $4$ over the rational numbers in a reasonable time.

...al fractions. Let $\{a_n\}_{n=1}^\infty$ be a sequence of distinct complex numbers, and let $\{g_n(z)\}$ be a sequence of rational functions of the form

They reduce to the [[Fibonacci numbers|Fibonacci numbers]] $F _ { n }$ for $x = 1$ and they satisfy several identities, which may be ...ey showed that the $U _ { n } ( x , y )$ are irreducible over the rational numbers if and only if $n$ is a prime number. They also generalized (a5) and proved

...he language of quadratic forms (cf. also [[Geometry of numbers|Geometry of numbers]]; [[Quadratic form|Quadratic form]]). A.K. Lenstra, H.W. Lenstra and L. Lo ...[[#References|[a5]]]. There is a polynomial-time algorithm which, given a rational point $ x \in \mathbf R ^ {n} $

of other complex numbers of modulus $1$, called local Artin root numbers (Deligne calls them simply "local constants" ). Given $\rho$, there is one ..., and to the existence of a global normal integral basis, while local root numbers are connected to Stiefel–Whitney classes, to Hasse symbols of trace forms

...lopediaofmath.org/legacyimages/c/c025/c025650/c0256502.png" /> of the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l ...nction]] either to the set of rational numbers or to the set of irrational numbers is continuous, but the Dirichlet function itself is discontinuous at all po

...group $\mathbf Q/\mathbf Z$, where $\mathbf Q$ denotes the set of rational numbers. This pairing establishes a [[Pontryagin duality|Pontryagin duality]] betwe

...ding the transcendence of the values $B ( a , b )$ of the beta-function at rational points $( a , b ) \in ( \mathbf{Q} \backslash \mathbf{Z} ) ^ { 2 }$. Next, ..., \ldots , \beta _ { n }$ are $\mathbf{Q}$-linearly independent algebraic numbers, but also yields a similar result for elliptic functions (and, more general

...ive periods $ 2 \omega _{1} ,\ 2 \omega _{3} $ may be represented as a rational function of $ \wp (z) $ and $ \wp ^ \prime (z) $ , i.e. $ \wp (z) $ ...he function $ \wp (z) $ . An absolute invariant of $ \wp (z) $ is any rational function of $ j = g _{2} ^{3} / g _{3} ^{2} $ or of $ J =g _{2} ^{3} /

...aic irrational $x$ the complexity is $b^n$ (which would follow if all such numbers were [[Normal number|normal]]) but all that is known in this case is that $

the resulting numbers, unlike $p$-values, are exactly those of a probability as being shared by all rational persons, whereas

...the order of the one-dimensional Galois cohomology group of the module of rational characters $\hat T$ of the torus $T$ and the order of its Shafarevich Tate |valign="top"|{{Ref|Ko2}}||valign="top"| R.E. Kottwitz, ''Tamagawa numbers'' Ann. of Math., '''127''' (1988) pp. 629 646 {{MR|0942522}} {{ZBL|0678.

...$\mathcal{M}$ there exist expressions $K$ and $L$ constructed from natural numbers and variables $a,z_1,\ldots,z_n$ by addition, multiplication and exponentia ...or the recognition of the solvability of Diophantine equations in rational numbers is equivalent to the problem of the existence of an algorithm for the recog

...metrized. For example, the unit circle $X ^ { 2 } + Y ^ { 2 } = 1$ has the rational parametrization ...ever, the non-singular cubic $Y ^ { 2 } = X ^ { 3 } - 1$ does not have any rational parametrization. To obtain the parametrization of the circle, one cuts it b

of rational numbers, or to the additive group of $ p $-

...nd ring]]). For example, if $K = \mathbf{Q}$ and $P$ consists of all prime numbers, then $O _ { K } = \mathbf{Z}$. ...a finite subset $S$ of $P$. Consider the field of totally $\text{p}$-adic numbers:

...theorems in set theory or about non-constructive entities such as the real numbers. Another weakness of NQTHM is that, like most other contemporary automated ...pplicative subset of Common Lisp. Unlike NQTHM, ACL2 supports the rational numbers, complex rationals, character objects, character strings, and symbol packag

are some specially-chosen natural numbers. In all these cases the Schwarz functions are rational automorphic functions; their group is a finite group of motions of a sphere

...Gokhberg, M.G. Krein, "Fundamental aspects of deficiency numbers, root numbers and indexes of linear operators" ''Uspekhi Mat. Nauk'' , '''12''' : 2 (1 ....encyclopediaofmath.org/legacyimages/i/i051/i051410/i05141085.png" /> is a rational matrix function. In that case <img align="absmiddle" border="0" src="https:

...number-theoretic functions, and the theory of algebraic and transcendental numbers. ==Distribution of prime numbers==

If the ground field is the field $\C$ of complex numbers, a semi-simple the space $E=X(T)\otimes\R$. For a rational

...rings of integers in finite algebraic extensions of the field of rational numbers.

be a given sequence of complex numbers. The Nehari extension problem is the problem to find (if possible) all $ ...see [[#References|[a8]]]), and when the data are Fourier coefficients of a rational matrix function, the formulas for the coefficients in the linear fractional

...a correspondence between two arbitrary sets (not necessarily consisting of numbers) was formulated by R. Dedekind in 1887 [[#References|[3]]]. ==Functions on numbers.==

..., \underline { \beta } ^ { ( n ) }$ are linearly independent. Then the $n$ numbers ...er whose expansion in an integral basis is given by an automaton is either rational or else transcendental.

...ients of integers that is essentially equivalent to the theory of rational numbers. With the latter as basis, a classification is given in Book X of quadratic ...s, namely: the seventh, eighth and ninth, containing the general theory of numbers of the Ancient geometers (1835, transl. from the Greek); M.E. Vashchenko-Za

of rational numbers; then there exists an infinite set of values $ t _{1} ^{0} \dots t _{k} ^ of rational functions in $ t _{1} \dots t _{n} $,

which are linearly independent over the field of real numbers and are such that $ f(z + w _ \nu ) = f(z) $ It is then possible to define the special Abelian functions as all rational functions in the $ p $

with integral rational coefficients $ a _ {i _ {1} } \dots a _ {i _ {n} } $ rational points of the [[Algebraic variety|algebraic variety]] defined by the system

is an algebra over the field of rational numbers and the module of differentials $ \Omega _ {R/S} ^ {1} $

...homogeneous solutions). Examples are the rings of integers and of rational numbers, algebraic number fields, and finite rings. For such a ring $ R $,

are complex numbers, $ p $ are non-negative rational numbers, $ F _ {n} ( x) \not\equiv 0 $,

...tion]]; [[Pole (of a function)|Pole (of a function)]]; [[Rational function|Rational function]]). is the least among the numbers for which $ a ^ {n} = 1 $.

is rational and diagonalizable, so $ \mathfrak g $ is the group of rational characters of $ T $ )

are rational numbers, $ n/2 \leq r _ {0} < \dots < r _ {k} \rightarrow + \infty $. ...n="top"> V.I. Arnol'd, "Remarks on the stationary phase method and Coxeter numbers" ''Russian Math. Surveys'' , '''28''' : 5 (1973) pp. 19–48 ''Uspekhi Mat.

...cyclopediaofmath.org/legacyimages/m/m064/m064430/m06443076.png" /> are odd numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.o ...yclopediaofmath.org/legacyimages/m/m064/m064430/m064430159.png" /> and the rational points on the real axis are often referred to as cusps.

if $G$ is the field of rational numbers or the field of residues modulo a prime number, and

...ecause of poles of the coefficients $c_j ( \lambda )$. These solutions are rational functions of $\lambda$ with possible poles at the poles of $c _ { 1 } ( \la ...special cases are given first. Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Note that neither of the special cas

...nt coefficients, every $ \ell $-adic cohomology theory for various prime numbers $ \ell $ modulo a suitable (rational, algebraic, numerical, etc.) equivalence relation, or $ C ( X) = K ( X) $

...or $H _ { \phi }$ has finite rank if and only if $\mathcal{P} - \phi$ is a rational function. Moreover, $\operatorname{rank} H _ { \phi } = \operatorname { deg ...or a function $\phi$ on $\bf T$ in $\operatorname{BMO}$ one can define the numbers $\rho _ { n } ( \phi )$ by

over the field of rational numbers. An isogeny $ f: A \rightarrow A _ {1} $

These were concerned with index numbers, a critique of Elderton and [[Pearson, Karl|Pearson]] (see below), reviews he wrote a long, prize-winning essay on Index Numbers.

are numbers. the analogous assertion is true for these classes with any rational number $ r > 0 $(

elliptic curve with distinguished point $Q_0$, then any rational mapping curve over an arbitrary field $k$. If the set of $k$-rational points

...tion does one associate the set of all real numbers, or the set of natural numbers, as a single object of study? Modern physical conceptions do not give a goo ...ind an extremum for functions of a very restricted class (polynomials with rational coefficients in several variables), and, what is essential, the theorem ind

...the foundations of mathematical analysis, which unifies the theory of real numbers (cf. [[Real number|Real number]]), the theory of limits, the theory of [[Se of numbers, is associated by some rule a number $ y $,

which admits no rational parametrization, but which may be uniformized by means of elliptic function are rational functions in the Weierstrass $ \wp $-function and its derivative, with co

and all sufficiently large numbers $ n $, ...ace. A non-singular projective algebraic surface over the field of complex numbers is a compact four-dimensional oriented real manifold; in particular, the in

The standard example of $(K,\infty,A)$ is given by a rational function field $K=\F_q(T),$, $\infty$ being the usual place at infinity, $A ...d $\Q$ of rationals. This also works for general base rings $A$ with class numbers $>1$; here the situation resembles the theory of complex multiplication of

...cal]], if $K$ contains a transcendental element over the field of rational numbers.

Over the field of complex numbers $ \mathbf C $ The field of rational functions $ k(X) $

...$\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.

...eger|Remainder of an integer]]). In order to express the congruence of the numbers $ a $ with just one of the numbers $ 0 \dots m-1 $;

...\operatorname { cos } ( n \operatorname { arccos } x )$. Over the complex numbers, if $u = e ^ { i \alpha }$ so that $x = u + 1 / u = 2 \operatorname { cos } ...a )$, and linear polynomials $a x + b$ (here, $a$ and $b$ may be rational numbers). The first proof of this conjecture was given by M. Fried [[#References|[a

study of linear algebraic groups over the field of complex numbers was complex numbers $\mathbb C$ can be regarded as an analytic subgroup (cf.

...tant $c_3$ (cf. also [[Distribution of prime numbers|Distribution of prime numbers]]). where $m,m_1,\ldots,m_s \in \mathbb{Z}$ and $p_1,\ldots,p_s$ are fixed prime numbers. Till recently, the proofs of this theorem suffered from a common drawback

...l axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axi ::''Is Hilbert's second problem about the real numbers or the natural numbers?''

...th for rotation numbers which can be very rapidly approximated by rational numbers, even if the transformation $ S $ is rational or irrational. Thus, in the absence of equilibrium positions there exist on

is some set of complex numbers, and $ \alpha $ may be one of various spaces of sequences of complex numbers. In connection with the class $ H ^ \infty $

...ative algebra with a unit over $ k $ . In general, the group of $ k $ -rational points of an [[Algebraic group|algebraic group]], defined over $ k $ , is

i) In the graded case: Is the Poincaré series rational? The Poincaré series of a graph is a rational function of the form $ P _ {G} ( z) = q( z) p( z) ^ {-1} $

...t of all open intervals (it is sufficient to take only open intervals with rational end points). The remaining open sets are unions of such intervals. ...balls for which the radius and the coordinates of the centre are rational numbers. A topology is often defined by some (natural) standard procedure on a set

the field of rational functions in one variable over an algebraically classification of rational varieties over not algebraically closed

...s,i_n } x_1^{i_1}\cdots x_n^{i_n}$ over $A$ such that there exist positive numbers $r_1,\ldots,r_n$ and $C$ such that $\Vert c_{ i_1,\ldots,i_n } \Vert \le C

...of a topological space|weight]] all coincide. The non-coincidence of these numbers is an indication of the non-metrizability of the corresponding space. ...ce is metrizable by a complete metric: an example is the space of rational numbers. A space is metrizable by a complete metric if and only if it is metrizable

...0 } ^ { \infty }$ with a given function $S ( z )$ in the Schur class. The numbers are defined in terms of a sequence of Schur functions which is constructed ...}$ are called the Schur parameters of $S ( z )$. Every sequence of complex numbers of modulus at most one and having the property that if some term has unit m

into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach consists precisely of the numbers of the form $ \phi ( a) $.

For instance, the set of all natural numbers $ \mathbf Z _ {0} $ ...peration of immediate succession and its iteration. The set of all natural numbers $ >1 $

Here the numbers $ s $ are rational functions of $ x _ {1}, \dots, x _ {n} $,

of complex numbers, on which the moduli of the Gel'fand transforms $ \widehat{a} $ of uniform limits of rational functions on a compactum $ X $

...dea is due to H. Whitney [[#References|[5]]], may be used to calculate the rational cohomology of a simplicial complex $ K $. with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The

adic numbers $ \mathbf Q _ {p} $ ...prescribed poles and zeros lying in a given open subset of the set of all rational points of the curve over $ {\widetilde{\mathbf Q} } _ {p} $.

...t. In particular, the differential ring of everywhere-defined differential rational functions on an affine differential algebraic set is not a differential coo ...for the approximation of differential algebraic functions by differential rational functions. However, the analogue of the Thue–Siegel–Roth theorem has no

...ii [[#References|[5]]] has studied the problem of Riemann: Let $A(t)$ be a rational function of $t$ and let the singularities of the fundamental matrix $X(t)$ where $R(t)$ is a rational function, and the equation

...in the decimal system, while the possible inputs are ordered pairs of such numbers. It is, in general, not assumed that the result is necessarily obtained: th for which the set of its possible inputs is the set of natural numbers.

...(cf. [[Continued fraction|Continued fraction]]): Given a sequence of real numbers $ \{ \mu _ {n} \} $, be the set of all sequences of real numbers $ \{ \mu _ {n} \} $

...permits $s$ real and $2t$ complex imbeddings in the field $\C$ of complex numbers, then ...=\infty$. For a quadratic field $\Q(\sqrt{d})$, where $d$ is a square-free rational integer, $d\ne 1$, one has the formulas

...presentations of $G ( K )$. This makes possible a "modular theory" for the rational representations of these groups analogous to Brauer's modular representatio .... also [[Young tableau|Young tableau]]). The $\lambda$-tableau where these numbers are inserted in order along the rows downward, is called the initial $\lamb

are the spatial wave numbers, which travel in all directions. Computationally, the domain is of finite s For (a1), a wave with wave numbers $ \xi $,

...ages thus defined are referred to as (rational) transductions and inverse (rational) transductions. If $ \mathop{\rm ST}\nolimits $ ...ngs) is again a rational transduction (respectively, a gsm mapping). Every rational transduction $ \tau $

Let $p$ be a prime number and let $k$ be a finite extension of the rational number field $\mathbf{Q}$. A $\mathbf{Z} _ { p }$-extension of $k$ is an ex ...erences|[a17]]] gave a new proof of this using the $\Gamma$-transform of a rational function.

...s been verified under various additional assumptions. Namely, if $F$ has a rational inverse (O.H. Keller) and, more generally, if the field extension $\mathbf{ ...}$ or $\operatorname { deg } F _ { 2 }$ is a product of at most two prime numbers (H. Applegate, H. Onishi). Finally, if there exists one line $l \subset \ma

are rational integers. The modular group can be identified with the quotient group $ real numbers (respectively, integers) and $ ad - bc = 1 $.

...of Kummer was that if one appends to the ordinary numbers certain "ideal" numbers (in the same way as one appends points at infinity in projective geometry), ...Kronecker to extend the theory of divisors to arbitrary rings of algebraic numbers. In Dedekind's theory (1882) the role of the integral elements of a field e

where $p$ runs through all prime numbers. ...f the problem of the [[Distribution of prime numbers|distribution of prime numbers]]. However, the most deeply intrinsic properties of $\zeta(s)$ were discove

...ormed ring]]. For a description of all valuations of the field of rational numbers, see [[#References|[4]]]. of real numbers. In this case the mapping $ x \mapsto \mathop{\rm exp} ( - v ( x) ) $

...ith a "set of points in a coordinate space" . If the coordinates are real numbers, and if the space is two- or three-dimensional, the situation may be clearl In the case of algebraic geometry over the field of complex numbers, every algebraic variety is simultaneously a complex-analytic, differentiab

...a given sequence $\{ c _ { n } \} _ { n = - \infty } ^ { \infty }$ of real numbers is concerned with finding real-valued, bounded, monotone non-decreasing fun The rational functions $( - z ) P _ { n } ( - z ) / Q _ { n } ( - z )$ are the convergen

over the field of rational numbers equals the number of conjugacy classes of cyclic subgroups of the group. If

are any numbers not simultaneously equal to zero and such that ...ari, "Inequalities for the coefficients of univalent functions" ''Arch. Rational Mech. and Anal.'' , '''34''' : 4 (1969) pp. 301–330</TD></TR><TR><TD v

rational point $ D $, ..., k) $[[#References|[6]]], [[#References|[10]]]. In the general case these numbers are different, but $ { \mathop{\rm ord} } ( D) $

1) The real numbers $t_\text{min}$ and $t_\text{max}$ are the minimum and the maximum of the fu ...^ { 3 }$ intersects the knot $K$ in some number, say $n_j$, of points. The numbers $n_j$ are constants if $( t _ { 1 } , \dots , t _ { m } )$ belongs to a fix

defined over the complex numbers. For smooth projective $ X $ equal (up to a non-zero rational number) to the first non-vanishing coefficient of the $ L $-

...1 }$ and $Z_2$ are said to be numerically equivalent if their intersection numbers are equal, $( D . Z _ { 1 } ) = ( D . Z _ { 2 } ) \in \bf R$ for any Cartie ...here exist an element $v ^ { \prime } \in \overline { N E } ( X / S )$ and numbers $r _ { j } \in {\bf R} _ { \geq 0 }$, which are zero except for finitely ma

In the case of Lie groups over the field $ \mathbf C $ of complex numbers the main result of the local classification is that every simply-connected ...on it, and any analytic homomorphism of $ G $ to an algebraic group is rational. The corresponding algebra of regular functions on $ G $ coincides with

The numbers $c ( n )$ are the Fourier coefficients of $f$. The modular form is a cusp f ...framework for such results. Ramanujan also conjectured that for all prime numbers $p$ one has the inequality $| \tau ( p ) | \leq 2 p ^ { 11 / 2 }$. P. Delig

...not have isolated points (an example is the space $\mathbf{Q}$ of rational numbers). All zero-dimensional spaces are completely regular. Zero-dimensionality i

partly because it contains statements on the theory of numbers which first among them. This was that the set of real numbers on the unit

...w that the kernel is isomorphic to the additive group of $n$-adic rational numbers. Thus, such groups are meta-Abelian (cf. [[Meta-Abelian group|Meta-Abelian ...eferences|[a5]]], while the residually-finite Baumslag–Solitar groups have rational growth [[#References|[a2]]], [[#References|[a6]]] but, as noted earlier, ar

of rational characters of the torus $ T $ , ...e of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple f

...e a countable subset of $ X $, and let $ p_1, p_2, \dots $ be non-negative numbers. Then the function ...initely-additive measure is a measure. For example, if $ X $ is the set of rational points of the segment $ [0,1] $, $ \mathcal{P} $ is the semi-ring of all po

is a rational function of the form $ f ( t) ( 1 - t ) ^ {-} d( A) $, Other homological invariants are the so-called Betti numbers $ b _ {i} ( A) $

of complex numbers the cohomology spaces $ H ^{*} (X _{s} ) $ is the multiplicative group of the field of complex numbers, considered as a real algebraic group, while $$

where $c_j$, $a_{ij}$ and $b_i$ are given numbers. ...e devoted to work in one or other technical mode. The problem is to make a rational distribution of the time spent in working in the various technical modes, t

...rations — arithmetic operations on positive integers and positive rational numbers — can be encountered in the oldest mathematical texts, which indicates th ...plicit numbers. It was assumed, however, that the symbols stood for actual numbers: integers or fractions. A brief table of the contents of one of the best te

real numbers, $ n \geq 2 $, is rational, then there are arbitrarily-small perturbations which will destroy the toru

...an algebraic variety. A morphism of $k$-schemes $\Spec k\to X$ is called a rational point of the $k$-scheme $X$; the set of such points is denoted by $X(k)$. ...complex, or strong, topology on $X(\C)$, the fundamental group, the Betti numbers, etc. The desire to find something similar for arbitrary schemes and the fa

Then the numbers can be described explicitly by rational functions of the values $ \mu $

is a non-increasing sequence of positive numbers and $ \{ f _ {n} \} $, are orthonormal systems. The numbers $ s _ {n} = s _ {n} ( T) $

...ubgroups of classical Lie groups — groups of units of quadratic forms with rational coefficients, groups of units of simple algebras over $\mathbf{Q}$, groups ...ice in $H_\mathbf{R}$ it is necessary and sufficient for $H$ not to permit rational homomorphisms into the group $\mathbf{C}^*$, defined over $\mathbf{Q}$ (thi

...ences|[a9]]]), for the aforementioned solutions are given over the complex numbers; ...belong to special classes of functions (cf. [[#References|[a1]]]), such as rational, solitonic, elliptic, bispectral;

is invertible [[#References|[7]]], in particular over the field of rational numbers, and, if $ \Delta (0) = +1 $,

...1,2,3, \ldots \}$, with its subset $P_{\mathbf{N}}$ of all rational prime numbers $\{ 2,3,5,7,\ldots \}$. Here one may define the norm of an integer $|n|$ to ...E.A. Bender and Wormald [[#References|[a1]]] considered the corresponding numbers $\mathcal{P} _ { V } ^ { \# } ( n )$, $\mathcal{P} _ { \text{F} } ^ { \# }

...e natural numbers as well as adding any two natural numbers, since natural numbers can be regarded, for example, as words of the form $ 0 , 0 \mid , 0 \mid ...nce of words. The introduction of more complex structures such as the real numbers and functions on them, etc., are realized in constructive mathematics by th

...the usual order relation is an elementary subsystem of the system of real numbers with the usual order relation. ...the signature of $A$ is $(+,\,.\,,0,1,<,U)$, $|A|$ is the set of all real numbers, $U(A)$ the set of all integers, and $+,\,.\,,0,1,<$, are defined in the u

...[[Lattice of points|Lattice of points]]; [[Geometry of numbers|Geometry of numbers]]). The original insight of E. Bombieri and Iwaniec into the second spacing ...roximated, via [[Taylor series|Taylor series]], by a cubic polynomial with rational quadratic coefficient $a /q$.

adic numbers and let $ \mathbf F _ {p} (( t)) $ fields are also called quasi-algebraically closed. The rational functions in one variable over an algebraically closed field form a $ C _

where $B_{2n}$ are the [[Bernoulli numbers]]. It implies the equality ...ntal functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).

There is a set of rational numbers $a_i$, known as the Seiberg–Witten invariants, which can be obtained by c ...nite number of classes (known as basic classes) $\kappa_i \in H^2(M)$ with rational coefficients $a_i$ (called the Seiberg–Witten invariants), resulting in t

...ity laws gain a statistical expression on the strength of the law of large numbers (probabilities are realized approximately in the form of frequencies, and e The statistical distribution corresponding to this partition is given by the numbers (frequencies) $ n _ {1} \dots n _ {r} $(

Furthermore, the corresponding characteristic numbers, which are elements of the rings $ h ^ {*} ( \mathop{\rm pt} ) $ cf. [[Chern class]]), it follows that the Chern numbers (cf. [[Chern number]]) completely determine the unitary cobordism class of

Irrationality and transcendence of certain numbers. The numbers in question are of the form $\alpha^\beta$ with $\alpha$ an [[algebraic num

...hat is, consuming only polynomial resources) algorithm for factoring large numbers and for computing discrete logarithms. It implied that widely-used public-k ...tary transition operators are rational (or, in general, computable complex numbers), then there is no difference between classical and quantum computability.

be the numbers of linearly independent solutions of the homogeneous integral equation $ associated with it. The numbers $ k $

...$-fold composite (or superposition) of $f_1,\dots,f_n$. For example, every rational function in any number of variables is a composite of the four arithmetic o In 1954 A.G. Vitushkin proved [[#References|[10]]] that if natural numbers $m,n,m_1$, and $n_1$ satisfy the inequality $(m/n)>(m_1/n_1)$, then it is p

...nu$. By allowing $l_{i j}$ to be indeterminates over the field of rational numbers, the generating function version of the matrix tree theorem is obtained [[#

is a finite extension of the field of rational $ p $-adic numbers $ \mathbf Q _ {p} $;

...the left is the first non-trivial coefficient of the $j$-function, and the numbers on the right are the dimensions of the smallest irreducible representations ...me form of moonshine for finite subgroups of the automorphism group of any rational vertex operator algebra obeying a technical (and probably redundant) "C2 c

...l numbers ... using set-theoretic constructions, starting from the natural numbers." <ref>[[Arithmetization]], ''Encyclopedia of Mathematics''. See also Lakof # the creation of the theory of real numbers.

...nalytic cases one can replace the real field $\RR$ by the field of complex numbers $\CC$.</ref>. ...$\l_1/\l_2=-\beta_2/\beta_1$, $\gcd(\beta_1,\beta_2)=1$, is a nonpositive rational number, then the corresponding identity $\left<\beta,\l\right>=0$ implies '

...th reaction, $b_l$ is the "reaction constant" for that reaction, and the numbers $a_{jl}$ are "stoichiometric parameters" , specifying the amount of specie ...s of nonlinear diffusion equations to travelling front solutions" ''Arch. Rational Mech. Anal.'' , '''65''' (1977) pp. 335–361</TD></TR><TR><TD valign="to

He introduced the concept of "rational subgroup" to define the method and the "law of large numbers" imparts the appearance of infinite precision.

...Aleksandrov–Čech homology functor (over the field $\mathbf{Q}$ of rational numbers; cf. also [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech

algebra; G. Bergman proved that numbers in $ ( 1 , 2 ) $ ...geometry; cf. also [[Analytic space|Analytic space]]; [[Rational function|Rational function]]; [[Analytic set|Analytic set]]; [[Cohomological dimension|Cohomo

...in), where the corresponding dichotomy algebraic versus transcendental (or rational versus irrational) is given a concrete meaning (see also [[#References|[a20

An algebraic vector bundle on a variety defined over the field of complex numbers $ \mathbb{C} $ may be regarded both as an [[Vector bundle, analytic|analyti ...the fibers of the determinant mapping $ \det: U(r,d) \to U(1,d) $ are uni-rational varieties; if $ r $ and $ d $ are co-prime, then $ U(r,d) $ uniquely determ

...aic theory of invariants) that studies algebraic expressions (polynomials, rational functions or families of them) that change in a specified way under non-deg are real or complex numbers, it is converted to the form$$

is a set of real numbers, and suppose that $ \phi : \mathfrak x \times X \times T \rightarrow X $ ...hile the solution of bimatrix games can always brought about by the use of rational operations.

then the subring of rational functions $ K[ x, P ^ {-1} ] $ consist of strictly negative rational numbers. See also [[#References|[a6]]] for this. The roots of $ b( s) $

defines a rational mapping from $ A $ of complex numbers is a [[Complex torus|complex torus]]

is an entire rational function, or a polynomial; if $ a = \infty $ is a rational function. For a transcendental meromorphic function $ f(z) $

...etric with non-positive two-dimensional curvature, then the characteristic numbers of the form is an arbitrary rational Pontryagin–Hirzebruch class of the manifold $ M $

...onverse is false, since a link may be represented by braids with different numbers of strings. In addition, the braids $ \omega \sigma _ {n - 1 } $ ...raic geometry as complements to the discriminant of versal deformations of rational singularities (see [[#References|[12]]], [[#References|[13]]]).

...hroughout we will denote by $q$ the value $p^{r/n}$, which is in general a rational number but is an integer when $F$ is a Frobenius endomorphism. Note that mo ...l$ denote an algebraic closure of the field of [[P-adic number|$\ell$-adic numbers]] where $\ell$ is a prime number distinct from $p$. For $i \in \mathbb{Z}$

are given numbers. Then the solution of the problem at any interior point is expressed by the ...><TR><TD valign="top">[7]</TD> <TD valign="top"> G.I. Petrashen', "On a rational method of solving problems in the dynamical theory of elasticity in the cas

...\phi : X _ { 0 } ( N ) \rightarrow E$. Let $S _ { 0 }$ be the set of prime numbers containing $p$ and the primes where $E$ has bad reduction. For each prime n ...agin, D.Y. Logacev, "Finiteness of Shafarevich–Tate group and the group of rational points for some modular Abelian varieties" ''Algebra i Anal.'' , '''1''' (1

...if the identity \eqref{e:power_series} holds for some sequence of complex numbers $\{a_n\}$), then $f$ is complex-differentiable everywhere in $U$ and indeed ...etitions in the sequence and $\{\mu_n\}$ is a sequence of positive natural numbers, then $f$ can be chosen so that it vanishes at each $z_n$ with order $\mu_n

...case. Kashiwara proved that the roots of the $ b $-function are rational numbers. If $ f : ( \mathbf C ^ {n+1} , 0 ) \rightarrow ( \mathbf C , 0 ) $

...$\{1,2,3,\ldots\}$, with its subset $P_{\mathbb{N}}$ of all rational prime numbers $\{2,3,5,7,\ldots\}$. Here one may define the norm of an integer $|n|$ to b

...entation of the "state space" of a given problem as a domain in a space of numbers $ \mathbf R ^ {n} $. This gives one the possibility to describe by a set of numbers the coordinates of the corresponding point (the coordinate method). In the

and, in particular, the equality of the Betti numbers dimensional Betti numbers are equal, as are the $ p $-

see [[#References|[1]]], [[#References|[2]]]), or its Frobenius, rational or quasi-natural normal form (see [[#References|[4]]]). In contrast to the is the field of complex numbers or, more generally, any algebraically closed field.) Then every one of the

of rational numbers into the Hausdorff space $ \mathbf R $