Editing Proof that e is irrational (section)

==Euler's proof==
Euler wrote the first proof of the fact that ''e'' is irrational in 1737 (but the text was only published seven years later).<ref>{{cite journal | last = Euler | first = Leonhard | year = 1744 | title = De fractionibus continuis dissertatio | url = http://www.math.dartmouth.edu/~euler/docs/originals/E071.pdf | journal = Commentarii Academiae Scientiarum Petropolitanae | volume = 9 | pages = 98–137 |trans-title=A dissertation on continued fractions}}</ref><ref>{{cite journal | last = Euler | first = Leonhard | title = An essay on continued fractions | journal = Mathematical Systems Theory | year = 1985 | volume = 18 | pages = 295–398 | url = https://kb.osu.edu/dspace/handle/1811/32133 | publication-date = 1985 | doi=10.1007/bf01699475| hdl = 1811/32133 | s2cid = 126941824 | hdl-access = free }}</ref><ref>{{cite book | last1 = Sandifer | first1 = C. Edward | title = How Euler did it | chapter = Chapter 32: Who proved ''e'' is irrational? | url=http://eulerarchive.maa.org/hedi/HEDI-2006-02.pdf |publisher = [[Mathematical Association of America]] | pages = 185–190 | year = 2007 | isbn = 978-0-88385-563-8 | lccn = 2007927658}}</ref> He computed the representation of ''e'' as a [[simple continued fraction]], which is

:<math>e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots, 2n, 1, 1, \ldots]. </math>

Since this continued fraction is infinite and every rational number has a terminating continued fraction, ''e'' is irrational. A short proof of the previous equality is known.<ref>[https://arxiv.org/abs/math/0601660 A Short Proof of the Simple Continued Fraction Expansion of e]</ref><ref>{{cite journal | last = Cohn | first = Henry | journal = [[American Mathematical Monthly]] | volume = 113 | issue = 1 | pages = 57–62 | year = 2006 | title = A short proof of the simple continued fraction expansion of ''e'' | jstor = 27641837 | doi=10.2307/27641837| arxiv = math/0601660 | bibcode = 2006math......1660C }}</ref> Since the simple continued fraction of ''e'' is not [[Periodic continued fraction|periodic]], this also proves that ''e'' is not a root of a quadratic polynomial with rational coefficients; in particular, ''e''<sup>2</sup> is irrational.