Proof that e is irrational
Part of a series of articles on the 
mathematical constant e 

Properties 
Applications 
Defining e 

People 
Related topics 
The number e was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that e is irrational, that is, that it can not be expressed as the quotient of two integers.
Contents
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).^{[1]}^{[2]}^{[3]} He computed the representation of e as a simple continued fraction, which is
Since this continued fraction is infinite and rational numbers can't be written as infinite continued fractions, e is irrational. A short proof of the previous equality is known.^{[4]} Since the simple continued fraction of e is not periodic, this also proves that e is not a root of second degree polynomial with rational coefficients; in particular, e^{2} is irrational.
Fourier's proof
The most wellknown proof is Joseph Fourier's proof by contradiction,^{[5]} which is based upon the equality
Initially e is assumed to be a rational number of the form ^{a}⁄_{b}. Note that b couldn't be equal to one as e is not an integer. It can be shown using the above equality that e is strictly between 2 and 3.
We then analyze a blownup difference x of the series representing e and its strictly smaller b^{ th} partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction ^{a}⁄_{b} and the b^{ th} 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.
Suppose that e is a rational number. Then there exist positive integers a and b such that e = ^{a}⁄_{b}. Define the number
To see that if e is rational, then x is an integer, substitute e = ^{a}⁄_{b} into this definition to obtain
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.
We now prove that 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
because all the terms are strictly positive.
We now prove that x < 1. For all terms with n ≥ b + 1 we have the upper estimate
This inequality is strict for every n ≥ b + 2. Changing the index of summation to k = n – b and using the formula for the infinite geometric series, we obtain
Since there is no integer strictly between 0 and 1, we have reached a contradiction, and so e must be irrational. Q.E.D.
Alternate proofs
Another proof^{[6]} can be obtained from the previous one by noting that
and this inequality is equivalent to the assertion that bx < 1. This is impossible, of course, since b and x are natural numbers.
Still another proof^{[7]} can be obtained from the fact that
Generalizations
In 1840, Liouville published a proof of the fact that e^{2} is irrational^{[8]} followed by a proof that e^{2} is not a root of a second degree polynomial with rational coefficients.^{[9]} This last fact implies that e^{4} is irrational. His proofs are similar to Fourier's proof of the irrationality of e. In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third degree polynomial with rational coefficients.^{[10]} In particular, e^{3} is irrational.
More generally, e^{q} is irrational for any nonzero rational q.^{[11]}
See also
 Characterizations of the exponential function
 Transcendental number, including a proof that e is transcendental
 Lindemann–Weierstrass theorem
References
 ↑ Euler, Leonhard (1744). "De fractionibus continuis dissertatio" [A dissertation on continued fractions] (PDF). Commentarii academiae scientiarum Petropolitanae. 9: 98–137.
 ↑ Euler, Leonhard (1985). "An essay on continued fractions". Mathematical Systems Theory. 18: 295–398. doi:10.1007/bf01699475.
 ↑ Sandifer, C. Edward (2007). "Chapter 32: Who proved e is irrational?". How Euler did it. Mathematical Association of America. pp. 185–190. ISBN 9780883855638. LCCN 2007927658.
 ↑ Cohn, Henry (2006). "A short proof of the simple continued fraction expansion of e". American Mathematical Monthly. Mathematical Association of America. 113 (1): 57–62. JSTOR 27641837.
 ↑ de Stainville, Janot (1815). Mélanges d'Analyse Algébrique et de Géométrie [A mixture of Algebraic Analysis and Geometry]. Veuve Courcier. pp. 340–341.
 ↑ MacDivitt, A. R. G.; Yanagisawa, Yukio (1987), "An elementary proof that e is irrational", The Mathematical Gazette, London: Mathematical Association, 71 (457): 217, JSTOR 3616765
 ↑ Penesi, L. L. (1953). "Elementary proof that e is irrational". American Mathematical Monthly. Mathematical Association of America. 60 (7): 474. JSTOR 2308411.
 ↑ Liouville, Joseph (1840). "Sur l'irrationalité du nombre e = 2,718…". Journal de Mathématiques Pures et Appliquées. 1 (in french). 5: 192.
 ↑ Liouville, Joseph (1840). "Addition à la note sur l'irrationnalité du nombre e". Journal de Mathématiques Pures et Appliquées. 1 (in french). 5: 193–194.
 ↑ Hurwitz, Adolf (1933) [1891]. "Über die Kettenbruchentwicklung der Zahl e". Mathematische Werke (in german). 2. Basel: Birkhäuser. pp. 129–133.
 ↑ Aigner, Martin; Ziegler, Günter M. (1998), Proofs from THE BOOK (4th ed.), Berlin, New York: SpringerVerlag, pp. 27–36, ISBN 9783642008559, doi:10.1007/9783642008566.