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 every rational number has a terminating continued fraction, 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 \ \ <\ \ 1+(1+{\tfrac {1}{2}}+{\tfrac {1}{2^{2}}}+{\tfrac {1}{2^{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!}}={\frac 1{(b+1)(b+2)\cdots (b+(n-b))}}<{\frac 1{(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 }{\frac 1{(b+1)^{{n-b}}}}=\sum _{{k=1}}^{\infty }{\frac 1{(b+1)^{k}}}={\frac {1}{b+1}}{\biggl (}{\frac 1{1-{\frac 1{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+{\frac 1{b+2}}+{\frac 1{(b+2)(b+3)}}+\cdots <1+{\frac 1{b+1}}+{\frac 1{(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

{\frac 1e}=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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