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Transcendental number
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{{Short description|In mathematics, a non-algebraic number}} {{Redirect|U-number|thermal conductivity|R-value (insulation)#U-value}} In [[mathematics]], a '''transcendental number''' is a [[real number|real]] or [[complex number]] that is not [[algebraic number|algebraic]]: that is, not the [[Zero of a function|root]] of a non-zero [[polynomial]] with [[integer]] (or, equivalently, [[rational number|rational]]) [[coefficient]]s. The best-known transcendental numbers are {{mvar|[[Pi|π]]}} and {{mvar|[[e (mathematical constant)|e]]}}.<ref>{{cite web |first=Cliff |last=Pickover |title=The 15 most famous transcendental numbers |website=sprott.physics.wisc.edu |url=http://sprott.physics.wisc.edu/pickover/trans.html |access-date=2020-01-23}}</ref><ref>{{cite book |last1=Shidlovskii |first1=Andrei B. |date=June 2011 |title=Transcendental Numbers |publisher=Walter de Gruyter |isbn=9783110889055 |page=1}}</ref> The quality of a number being transcendental is called '''transcendence'''. Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare: indeed, [[almost all]] real and complex numbers are transcendental, since the algebraic numbers form a [[countable set]], while the [[set (mathematics)|set]] of [[real numbers]] {{tmath|\R}} and the set of [[complex number]]s {{tmath|\C}} are both [[uncountable set]]s, and therefore larger than any countable set. All '''transcendental real numbers''' (also known as '''real transcendental numbers''' or '''transcendental irrational numbers''') are [[irrational number]]s, since all [[rational numbers]] are algebraic.<ref name=numbers>{{cite book |last1=Bunday |first1=B. D. |last2=Mulholland |first2=H. |title=Pure Mathematics for Advanced Level |date=20 May 2014 |publisher=Butterworth-Heinemann |isbn=978-1-4831-0613-7 |url=https://books.google.com/books?id=02_iBQAAQBAJ |access-date=21 March 2021 |language=en}}</ref><ref>{{cite journal |last1=Baker |first1=A. |title=On Mahler's classification of transcendental numbers |journal=Acta Mathematica |date=1964 |volume=111 |pages=97–120 |doi=10.1007/bf02391010 |s2cid=122023355 |doi-access=free }}</ref><ref>{{cite arXiv |last1=Heuer |first1=Nicolaus |last2=Loeh |first2=Clara |title=Transcendental simplicial volumes |date=1 November 2019 |class=math.GT |eprint=1911.06386 }}</ref><ref>{{cite encyclopedia |title=Real number |department=mathematics |url=https://www.britannica.com/science/real-number |access-date=2020-08-11 |encyclopedia=Encyclopædia Britannica |lang=en}}</ref> The [[Converse (logic)|converse]] is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, [[Irrational number#Algebraic|algebraic irrational]], and transcendental real numbers.<ref name=numbers/> For example, the [[square root of 2]] is an irrational number, but it is not a transcendental number as it is a [[Zero_of_a_function#Polynomial_roots|root of the polynomial]] equation {{math|''x''<sup>2</sup> − 2 {{=}} 0}}. The [[golden ratio]] (denoted <math>\varphi</math> or <math>\phi</math>) is another irrational number that is not transcendental, as it is a root of the polynomial equation {{math|''x''<sup>2</sup> − ''x'' − 1 {{=}} 0}}.
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