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For each [[irrational number]] ''x'' take a [[sequence]] of [[rational number]]s {''x_k ...ics)|singleton]] to be open, and using as a [[neighborhood base]] for each irrational number ''x'', the sets <math> U_n(x) = \{ x_k : k \ge n \} \cup \{x\}.</mat ...

...n to irrational numbers that are not quadratic by using either [[quadratic irrational]]s or simply [[rational number]]s. It is named after [[Harold Davenport]] Given a number α which is either rational or a quadratic irrational, we can find unique integers ''x'', ''y'', and ''z'' such that ''x'', ''y'' ...

...]] on a [[Diophantine approximation]]. The theorem states that for every [[irrational number]] ''ξ'' there are infinitely many [[coprime integers|relatively prim The condition that ''ξ'' is irrational cannot be omitted. Moreover the constant <math>\sqrt{5}</math> is the best ...

...restrictions on possible values of [[L2 cohomology|<math>l^2</math>-Betti numbers]]. ...is possible for <math>l^2</math>-Betti numbers to be [[irrational numbers|irrational]]. ...

{{Short description|Irrational numbers which appear to be rational}} A '''schizophrenic number''' or '''mock rational number''' is an [[irrational number]] which displays certain characteristics of [[rational number]]s. It ...

...periodic precisely when the original number is a cubic [[Irrational number|irrational]]. A standard way of writing real numbers is by their [[decimal representation]], such as: ...

{{short description|Special type of irrational number}} ...''' (sometimes spelled '''Bruno''' or '''Bryuno''') is a special type of [[irrational number]] named for Russian mathematician [[Alexander Bruno]], who introduce ...

{{Short description|Irrational system of points and lines}} ...for which every combinatorially equivalent realization has at least one [[irrational number]] as one of its coordinates. It can be constructed from the diagonal ...

...ast=Niven |first=Ivan |author-link=Ivan Niven |year=1956 |title=Irrational Numbers |url=https://archive.org/details/irrationalnumber00nive |url-access=registr ...hmer |first=Derrick H. |year=1933 |title=A note on trigonometric algebraic numbers |journal= The American Mathematical Monthly|jstor=2301023 | volume = 40 |nu ...

<!--[[Category:Algebraic numbers]]-->[[Category:Quadratic irrational numbers]] ...

{{short description|Property of an irrational number}} ...r solutions to the Markov Diophantine equation|Markov number|a set of real numbers|Markov spectrum}} ...

...bda</math> be any [[irrational number]]. The theorem states that the real numbers <math>\lambda\omega(n)</math> are asymptotically uniformly distributed modu [[Category:Theorems about prime numbers]] ...

{{Short description|Concept in the geometry of numbers}} In the [[geometry of numbers]], the '''Klein polyhedron''', named after [[Felix Klein]], is used to gene ...

...hat is, it is generally the sum of [[unit fraction]]s. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain s If only finitely many numbers are included, the key issue is usually to find a simple expression for the ...

exists (that is, it [[Limit (mathematics)|converges]]) and is an [[irrational number]].<ref name="guy">{{citation |last=Guy | first=Richard K. | author-l | title = Expression of real numbers with the help of infinite series ...

...mmon Core State Standards Initiative]], the meaning of the product of real numbers steps through a series of notions generally beginning with repeated additio Once the natural (or whole) numbers have been defined, and understood as a means to count, a child is introduce ...

...supremum.svg|thumb|300px|Every non-empty subset <math>M</math> of the real numbers <math>\mathbb{R}</math> which is bounded from above has a least upper bound ...mbers]], and it is also intimately related to the construction of the real numbers using [[Dedekind cut]]s. ...

...ov spectrum''', devised by [[Andrey Markov]], is a complicated set of real numbers arising in [[Markov number|Markov Diophantine equations]] and also in the t ...lue. Such 1/''c'' make up the '''Lagrange spectrum''' ''L'', a set of real numbers at least {{radic|5}} (which is the smallest value of the spectrum). The for ...

...06.</ref> where the [[rational function]] is conformally conjugate to an [[irrational rotation]] of the standard [[annulus (mathematics)|annulus]]. and an [[irrational number]] <math>\theta</math>, such that ...

Fix a positive [[irrational number]] ''α'' with [[continued fraction]] expansion [''a''₀; '' ...tion]] of positive integers as a sum of distinct non-consecutive Fibonacci numbers. ...