Editing Pi (section)

=== Transcendence ===
{{See also|Lindemann–Weierstrass theorem}}[[File:Squaring the circle.svg|thumb|alt=A diagram of a square and circle, both with identical area; the length of the side of the square is the square root of pi|Because {{pi}} is a [[transcendental number]], [[squaring the circle]] is not possible in a finite number of steps using the classical tools of [[Compass-and-straightedge construction|compass and straightedge]].|left]]
In addition to being irrational, {{pi}} is also a [[transcendental number]], which means that it is not the [[solution (equation)|solution]] of any non-constant [[polynomial equation]] with [[rational number|rational]] coefficients, such as <math display="inline">\frac{x^5}{120}-\frac{x^3}{6}+x=0</math>.{{sfn|Arndt|Haenel|2006|p=6}}{{efn|The polynomial shown is the first few terms of the [[Taylor series]] expansion of the [[sine]] function.}} This follows from the so-called [[Lindemann–Weierstrass theorem#Transcendence of e and π|Lindemann–Weierstrass theorem]], which also establishes the transcendence of [[E (mathematical constant)|the constant ''{{mvar|e}}'']].

The transcendence of {{pi}} has two important consequences: First, {{pi}} cannot be expressed using any finite combination of rational numbers and square roots or [[nth root|''n''-th roots]] (such as <math>\sqrt[3]{31}</math> or <math>\sqrt{10}</math>). Second, since no transcendental number can be [[Constructible number|constructed]] with [[Compass-and-straightedge construction|compass and straightedge]], it is not possible to "[[squaring the circle|square the circle]]". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.{{sfn|Posamentier|Lehmann|2004|p=25}} Squaring a circle was one of the important geometry problems of the [[classical antiquity]].{{sfn|Eymard|Lafon|2004|p=129}} Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.<ref>{{cite book |last=Beckmann |first=Petr |author-link=Petr Beckmann |title=History of Pi |publisher=St. Martin's Press |year=1989 |orig-year=1974 |isbn=978-0-88029-418-8 |page=37}} {{pb}} {{cite book |last1=Schlager |first1=Neil |last2=Lauer |first2=Josh |title=Science and Its Times: Understanding the Social Significance of Scientific Discovery |publisher=Gale Group |year=2001 |isbn=978-0-7876-3933-4 |url-access=registration |url=https://archive.org/details/scienceitstimesu0000unse |access-date=19 December 2019 |archive-url=https://web.archive.org/web/20191213112426/https://archive.org/details/scienceitstimesu0000unse |archive-date=13 December 2019 |url-status=live}}, p. 185.</ref>

An [[List of unsolved problems in mathematics|unsolved problem]] thus far is the question of whether or not the numbers ''{{mvar|π}}'' and ''{{mvar|e}}'' are [[Algebraic independence|algebraically independent]] ("relatively transcendental"). This would be resolved by [[Schanuel's conjecture]]<ref>{{Cite book |last1=Murty |first1=M. Ram |author-link1=M. Ram Murty |url=https://link.springer.com/book/10.1007/978-1-4939-0832-5 |title=Transcendental Numbers |last2=Rath |first2=Purusottam |date=2014 |publisher=Springer |language=en |doi=10.1007/978-1-4939-0832-5 |isbn=978-1-4939-0831-8}} {{pb}} {{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}</ref> – a currently unproven generalization of the Lindemann–Weierstrass theorem.<ref>{{mathworld|title=Lindemann-Weierstrass Theorem|id=Lindemann-WeierstrassTheorem|ref=none}}</ref>