Editing E (mathematical constant) (section)

=== Number theory ===
The real number {{mvar|e}} is [[irrational number|irrational]]. [[Leonhard Euler|Euler]] proved this by showing that its [[simple continued fraction]] expansion does not terminate.<ref>{{cite web|url=http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf|title=How Euler Did It: Who proved {{mvar|e}} is Irrational?|last=Sandifer|first=Ed|date=Feb 2006|publisher=MAA Online|access-date=2010-06-18|url-status=dead|archive-url=https://web.archive.org/web/20140223072640/http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf|archive-date=2014-02-23}}</ref> (See also [[Joseph Fourier|Fourier]]'s [[proof that e is irrational|proof that {{mvar|e}} is irrational]].)

Furthermore, by the [[Lindemann–Weierstrass theorem]], {{mvar|e}} is [[transcendental number|transcendental]], meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with [[Liouville number]]); the proof was given by [[Charles Hermite]] in 1873.<ref>{{cite book | last=Gelfond | first=A. O. | author-link=Alexander Gelfond | translator-last=Boron | translator-first=Leo F. | translator-link=Leo F. Boron | orig-year=1960 | year=2015 | title=Transcendental and Algebraic Numbers | publisher=[[Dover Publications]] |location=New York |series=Dover Books on Mathematics | isbn=978-0-486-49526-2 |mr=0057921 | url={{Google books|408wBgAAQBAJ|Transcendental and Algebraic Numbers|plainurl=yes}} |page=41}}</ref> The number {{mvar|e}} is one of only a few transcendental numbers for which the exact [[Irrationality measure#Irrationality exponent|irrationality exponent]] is known (given by <math>\mu(e)=2</math>).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Irrationality Measure |url=https://mathworld.wolfram.com/IrrationalityMeasure.html |access-date=2024-09-14 |website=mathworld.wolfram.com |language=en}}</ref>

An [[List of unsolved problems in mathematics|unsolved problem]] thus far is the question of whether or not the numbers {{mvar|e}} and {{mvar|π}} are [[Algebraic independence|algebraically independent]]. This would be resolved by [[Schanuel's conjecture]] – a currently unproven generalization of the Lindemann–Weierstrass theorem.<ref>{{Cite book |last1=Murty |first1=M. Ram |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 }}</ref><ref>{{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>

It is conjectured that {{mvar|e}} is [[normal number|normal]], meaning that when {{mvar|e}} is expressed in any [[Radix|base]] the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).<ref>{{cite book|author-link=Davar Khoshnevisan |last=Khoshnevisan |first=Davar |chapter=Normal numbers are normal |year=2006 |title=Clay Mathematics Institute Annual Report 2006 |publisher=[[Clay Mathematics Institute]] |pages=15, 27–31 |chapter-url=http://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf
}}</ref>

In [[algebraic geometry]], a ''[[period (algebraic geometry)|period]]'' is a number that can be expressed as an integral of an [[algebraic function]] over an algebraic [[domain of a function|domain]]. The constant {{pi}} is a period, but it is conjectured that {{mvar|e}} is not.<ref>{{Cite web |last1=Kontsevich |first1=Maxim |last2=Zagier |first2=Don |author-link1=Maxim Kontsevich |author-link2=Don Zagier |year=2001 |title=Periods |url=https://www.ihes.fr/~maxim/TEXTS/Periods.pdf}}</ref>