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| {{DISPLAYTITLE:Proof that ''e'' is irrational}}
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| {{E (mathematical constant)}}
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| The number ''e'' was introduced by [[Jacob Bernoulli]] in 1683. More than half a century later, [[Leonhard Euler|Euler]], who had been a student of Jacob's younger brother [[Johann Bernoulli|Johann]], proved that ''e'' is irrational, that is, that it can not be expressed as the quotient of two integers.
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| ==Euler's proof==
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| Euler wrote the first proof of the fact that ''e'' is irrational in 1737 (but the text was published only seven years later).<ref>{{cite journal | last = Euler | first = Leonhard | date = 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 | volume = 18 | pages = 295–398 | url = https://kb.osu.edu/dspace/handle/1811/32133 | publication-date = 1985}}</ref><ref>{{cite book | last1 = Sandifer | first1 = C. Edward | title = How Euler did it | chapter = Chapter 32: Who proved ''e'' is irrational? | 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
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| :<math>e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots, 2n, 1, 1, \ldots]. \,</math>
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| Since this continued fraction is infinite, ''e'' is irrational. A short proof of the previous equality is known.<ref>{{cite journal | last = Cohn | first = Henry | journal = [[American Mathematical Monthly]] | volume = 113 | issue = 1 | pages = 57–62 | publisher = [[Mathematical Association of America]] | date = 2006 | title = A short proof of the simple continued fraction expansion of ''e'' | jstor = 27641837}}</ref> Since the simple continued fraction of ''e'' is not [[Periodic continued fraction|periodic]], this also proves that ''e'' is not a root of second degree polynomial with rational coefficients; in particular, ''e''<sup>2</sup> is irrational.
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| ==Fourier's proof==
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| The most well-known proof is [[Joseph Fourier]]'s [[reductio ad absurdum|proof by contradiction]],<ref>{{Cite book | last1 = de Stainville | first1 = Janot | date = 1815 | title = Mélanges d'Analyse Algébrique et de Géométrie | trans_title = A mixture of Algebraic Analysis and Geometry | publisher = Veuve Courcier | pages = 340–341}}</ref> which is based upon the equality
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| : <math>e = \sum_{n = 0}^{\infty} \frac{1}{n!}\cdot</math>
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| Initially ''e'' is assumed to be a rational number of the form <sup>''a''</sup>⁄<sub>''b''</sub>. We then analyze a blown-up difference ''x'' of the series representing ''e'' and its strictly smaller {{nowrap|''b''<sup> th</sup>}} partial sum, which approximates the limiting value ''e''. By choosing the magnifying factor to be the [[factorial]] of ''b'', the fraction <sup>''a''</sup>⁄<sub>''b''</sub> and the {{nowrap|''b''<sup> th</sup>}} partial sum are turned into integers, hence ''x'' must be a positive integer. However, the fast convergence of the series representation implies that the magnified approximation error ''x'' is still strictly smaller than 1. From this contradiction we deduce that ''e'' is irrational.
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| Suppose that ''e'' is a [[rational number]]. Then there exist positive integers ''a'' and ''b'' such that ''e'' = <sup>''a''</sup>⁄<sub>''b''</sub>. Define the number
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| :<math>
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| x = b!\,\biggl(e - \sum_{n = 0}^{b} \frac{1}{n!}\biggr)\!
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| </math>
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| To see that if ''e'' is rational, then ''x'' is an integer, substitute ''e'' = <sup>''a''</sup>⁄<sub>''b''</sub> into this definition to obtain
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| :<math>
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| x = b!\,\biggl(\frac{a}{b} - \sum_{n = 0}^{b} \frac{1}{n!}\biggr)
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| = a(b - 1)! - \sum_{n = 0}^{b} \frac{b!}{n!}\,.
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| </math>
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| The first term is an integer, and every fraction in the sum is actually an integer because ''n'' ≤ ''b'' for each term. Therefore ''x'' is an integer.
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| We now prove that {{nowrap|0 < ''x'' < 1}}. First, to prove that ''x'' is strictly positive, we insert the above series representation of ''e'' into the definition of ''x'' and obtain
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| :<math>x = b!\,\biggl(\sum_{n = 0}^{\infty} \frac{1}{n!} - \sum_{n = 0}^{b} \frac{1}{n!}\biggr) = \sum_{n = b+1}^{\infty} \frac{b!}{n!}>0\,,\!</math>
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| because all the terms are strictly positive.
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| We now prove that ''x'' < 1. For all terms with {{nowrap|''n'' ≥ ''b'' + 1}} we have the upper estimate
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| :<math>\frac{b!}{n!}
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| =\frac1{(b+1)(b+2)\cdots(b+(n-b))}
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| \le\frac1{(b+1)^{n-b}}\,.\!
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| </math>
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| This inequality is strict for every ''n'' ≥ ''b'' + 2. Changing the index of summation to ''k'' = ''n'' – ''b'' and using the formula for the [[Geometric series#Infinite geometric series|infinite geometric series]], we obtain
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| :<math>
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| x
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| =\sum_{n = b+1}^\infty \frac{b!}{n!}
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| < \sum_{n=b+1}^\infty \frac1{(b+1)^{n-b}} | |
| =\sum_{k=1}^\infty \frac1{(b+1)^k}
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| =\frac{1}{b+1} \biggl(\frac1{1-\frac1{b+1}}\biggr)
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| = \frac{1}{b}
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| \le 1.
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| </math>
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| Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so ''e'' must be irrational. [[Q.E.D.]]
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| ==Alternate proofs==
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| Another proof<ref>{{Citation | last1 = MacDivitt | first1 = A. R. G. | last2 = Yanagisawa | first2 = Yukio | title = An elementary proof that ''e'' is irrational | journal = [[The Mathematical Gazette]] | volume = 71 | issue = 457 | pages = 217 | year = 1987 | publisher =[[Mathematical Association]] | place = London | jstor = 3616765}}</ref> can be obtained from the previous one by noting that
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| :<math>(b+1)x=1+\frac1{b+2}+\frac1{(b+2)(b+3)}+\cdots<1+\frac1{b+1}+\frac1{(b+1)(b+2)}+\cdots=1+x,</math>
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| and this inequality is equivalent to the assertion that ''bx'' < 1. This is impossible, of course, since ''b'' and ''x'' are natural numbers.
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| Still another proof<ref>{{cite journal | last = Penesi | first = L. L. | date = 1953 | title = Elementary proof that ''e'' is irrational | journal = [[American Mathematical Monthly]] | publisher = [[Mathematical Association of America]] | volume = 60 | issue = 7 | pages = 474 | jstor = 2308411 }}</ref> can be obtained from the fact that
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| :<math>\frac1e=e^{-1}=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\cdot</math>
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| ==Generalizations==
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| In 1840, [[Joseph Liouville|Liouville]] published a proof of the fact that ''e''<sup>2</sup> is irrational<ref>{{cite journal | last = Liouville | first = Joseph | journal = [[Journal de Mathématiques Pures et Appliquées]] | title = Sur l'irrationalité du nombre ''e'' = 2,718… | series = 1 | volume = 5 | pages = 192 | year = 1840 | language = french}}</ref> followed by a proof that ''e''<sup>2</sup> is not a root of a second degree polynomial with rational coefficients.<ref>{{cite journal | last = Liouville | first = Joseph | journal = [[Journal de Mathématiques Pures et Appliquées]] | title = Addition à la note sur l'irrationnalité du nombre ''e'' | series = 1 | volume = 5 | pages = 193–194 | year = 1840 | language = french}}</ref> This last fact implies that ''e''<sup>4</sup> is irrational. His proofs are similar to Fourier's proof of the irrationality of ''e''.
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| More generally, ''e''<sup>''q''</sup> is irrational for any non-zero rational ''q''.<ref>{{Citation | last1=Aigner | first1=Martin | author1-link = Martin Aigner | last2=Ziegler | first2=Günter M. | author2-link=Günter M. Ziegler | title=[[Proofs from THE BOOK]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1998|pages=27–36|isbn=978-3-642-00855-9|doi=10.1007/978-3-642-00856-6|edition=4th}}.</ref>
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| ==See also==
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| *[[Characterizations of the exponential function]]
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| *[[Transcendental number]], including a [[Transcendental_number#Sketch_of_a_proof_that_e_is_transcendental|proof that ''e'' is transcendental]]
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| *[[Lindemann–Weierstrass theorem]]
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| ==References==
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| <references/>
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| [[Category:Diophantine approximation]]
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| [[Category:Exponentials]]
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| [[Category:Article proofs]]
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| [[Category:E (mathematical constant)]]
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