Proof that e is irrational

mathematical constant $e$
Part of a series of articles on the

Properties
Natural logarithm Exponential function
Applications
compound interest Euler's identity Euler's formula half-lives exponential growth and decay
Defining $e$
proof that $e$ is irrational representations of $e$ Lindemann–Weierstrass theorem
People
John Napier Leonhard Euler
Related topics
Schanuel's conjecture

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.

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

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \ldots, 2n, 1, 1, \ldots]. \,

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, e² is irrational.

Fourier's proof

The most well-known proof is Joseph Fourier's proof by contradiction,^[5] which is based upon the equality

e = \sum_{n = 0}^{\infty} \frac{1}{n!}\cdot

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.

2 = 1 + \tfrac{1}{1!} \ \ < \ \ e = 1 + \tfrac{1}{1!} + \tfrac{1}{2!} + \tfrac{1}{3!} + \cdots \ \ < \ \ 2 + \tfrac{1}{2} (1 + \tfrac{1}{3} + \tfrac{1}{3^2} + \tfrac{1}{3^3} + \cdots) \ \ < \ \ 3

We then analyze a blown-up 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

x = b!\,\biggl(e - \sum_{n = 0}^{b} \frac{1}{n!}\biggr)\!

To see that if e is rational, then x is an integer, substitute e = ^a⁄_b into this definition to obtain

x = b!\,\biggl(\frac{a}{b} - \sum_{n = 0}^{b} \frac{1}{n!}\biggr) = a(b - 1)! - \sum_{n = 0}^{b} \frac{b!}{n!}\,.

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

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\,,\!

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

\frac{b!}{n!} =\frac1{(b+1)(b+2)\cdots(b+(n-b))} <\frac1{(b+1)^{n-b}}\,.\!

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

x =\sum_{n = b+1}^\infty \frac{b!}{n!} < \sum_{n=b+1}^\infty \frac1{(b+1)^{n-b}} =\sum_{k=1}^\infty \frac1{(b+1)^k} =\frac{1}{b+1} \biggl(\frac1{1-\frac1{b+1}}\biggr) = \frac{1}{b} < 1.

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

(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,

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

\frac1e=e^{-1}=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\cdot

Generalizations

In 1840, Liouville published a proof of the fact that e² is irrational^[8] followed by a proof that e² is not a root of a second degree polynomial with rational coefficients.^[9] This last fact implies that e⁴ 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, e³ is irrational.

More generally, e^q is irrational for any non-zero rational q.^[11]

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 978-0-88385-563-8. 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. doi:10.2307/27641837. 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, doi:10.2307/3616765, 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: Springer-Verlag, pp. 27–36, doi:10.1007/978-3-642-00856-6, ISBN 978-3-642-00855-9 .

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