# Proof that e is irrational

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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 cannot 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

- [math]\displaystyle{ 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.^{[4]}^{[5]} 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 well-known proof is Joseph Fourier's proof by contradiction,^{[6]} which is based upon the equality

- [math]\displaystyle{ e = \sum_{n = 0}^{\infty} \frac{1}{n!}\cdot }[/math]

Initially *e* is assumed to be a rational number of the form ^{a}⁄_{b}. Note that *b* could not be equal to 1 as *e* is not an integer. It can be shown using the above equality that *e* is strictly between 2 and 3:

- [math]\displaystyle{ 2 = 1 + \tfrac{1}{1!} \lt e = 1 + \tfrac{1}{1!} + \tfrac{1}{2!} + \tfrac{1}{3!} + \cdots \lt 1 + \left (1 + \tfrac{1}{2} + \tfrac{1}{2^2} + \tfrac{1}{2^3} + \cdots \right ) = 3. }[/math]

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

- [math]\displaystyle{ x = b!\left(e - \sum_{n = 0}^{b} \frac{1}{n!}\right). }[/math]

To see that if *e* is rational, then *x* is an integer, substitute *e* = ^{a}⁄_{b} into this definition to obtain

- [math]\displaystyle{ x = b!\left (\frac{a}{b} - \sum_{n = 0}^{b} \frac{1}{n!}\right) = a(b - 1)! - \sum_{n = 0}^{b} \frac{b!}{n!}. }[/math]

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

- [math]\displaystyle{ x = b!\left(\sum_{n = 0}^{\infty} \frac{1}{n!} - \sum_{n = 0}^{b} \frac{1}{n!}\right) = \sum_{n = b+1}^{\infty} \frac{b!}{n!}\gt 0, }[/math]

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

- [math]\displaystyle{ \frac{b!}{n!} =\frac1{(b+1)(b+2)\cdots(b+(n-b))} \le \frac1{(b+1)^{n-b}}. }[/math]

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

- [math]\displaystyle{ x =\sum_{n = b+1}^\infty \frac{b!}{n!} \lt \sum_{n=b+1}^\infty \frac1{(b+1)^{n-b}} =\sum_{k=1}^\infty \frac1{(b+1)^k} =\frac{1}{b+1} \left (\frac1{1-\frac1{b+1}}\right ) = \frac{1}{b} \lt 1. }[/math]

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^{[7]} can be obtained from the previous one by noting that

- [math]\displaystyle{ (b+1)x=1+\frac1{b+2}+\frac1{(b+2)(b+3)}+\cdots\lt 1+\frac1{b+1}+\frac1{(b+1)(b+2)}+\cdots=1+x, }[/math]

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 ^{[8]}^{[9]} can be obtained from the fact that

- [math]\displaystyle{ \frac{1}{e} =e^{-1}=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\cdot }[/math]

Define [math]\displaystyle{ s_{n} }[/math] as follows:

- [math]\displaystyle{ s_{n}=\sum_{k=0}^{n} \frac{(-1)^{k}}{k!}. }[/math]

Then:

- [math]\displaystyle{ e^{-1}-s_{2n-1}=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} -\sum_{k=0}^{2n-1} \frac{(-1)^{k}}{k!} \lt \frac{1}{(2n)!}, }[/math]

which implies:

- [math]\displaystyle{ 0\lt (2n-1)! \left (e^{-1}-s_{2n-1} \right ) \lt \frac{1}{2n} \leq \frac{1}{2} }[/math]

for any integer [math]\displaystyle{ n \geq 2. }[/math]

Note that [math]\displaystyle{ (2n-1)!s_{2n-1} }[/math] is always an integer. Assume [math]\displaystyle{ e^{-1} }[/math] is rational, so, [math]\displaystyle{ e^{-1}=\tfrac{p}{q} }[/math] where [math]\displaystyle{ p, q }[/math] are co-prime and [math]\displaystyle{ q \neq 0. }[/math] It's possible to appropriately choose [math]\displaystyle{ n }[/math] so that [math]\displaystyle{ (2n-1)!e^{-1} }[/math] is an integer i.e. [math]\displaystyle{ n \geq \tfrac{q+1}{2}. }[/math] Hence, for this choice, the difference between [math]\displaystyle{ (2n-1)!e^{-1} }[/math] and [math]\displaystyle{ (2n-1)!s_{2n-1} }[/math] would be an integer. But from the above inequality, that's impossible. So, [math]\displaystyle{ e^{-1} }[/math] is irrational. This means that [math]\displaystyle{ e }[/math] is irrational.

## Generalizations

In 1840, Liouville published a proof of the fact that *e*^{2} is irrational^{[10]} followed by a proof that *e*^{2} is not a root of a second degree polynomial with rational coefficients.^{[11]} 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.^{[12]} In particular, *e*^{3} is irrational.

More generally, *e*^{q} is irrational for any non-zero rational *q*.^{[13]}

## 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".
*Commentarii academiae scientiarum Petropolitanae***9**: 98–137. http://www.math.dartmouth.edu/~euler/docs/originals/E071.pdf. - ↑ Euler, Leonhard (1985). "An essay on continued fractions".
*Mathematical Systems Theory***18**: 295–398. doi:10.1007/bf01699475. https://kb.osu.edu/dspace/handle/1811/32133. - ↑ 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. - ↑ A Short Proof of the Simple Continued Fraction Expansion of e
- ↑ 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. - ↑ de Stainville, Janot (1815).
*Mélanges d'Analyse Algébrique et de Géométrie*. 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 - ↑ Penesi, L. L. (1953). "Elementary proof that
*e*is irrational".*American Mathematical Monthly*(Mathematical Association of America)**60**(7): 474. doi:10.2307/2308411. - ↑ Apostol, T. (1974). Mathematical analysis (2nd ed., Addison-Wesley series in mathematics). Reading, Mass.: Addison-Wesley.
- ↑ Liouville, Joseph (1840). "Sur l'irrationalité du nombre
*e*= 2,718…" (in french).*Journal de Mathématiques Pures et Appliquées*. 1**5**: 192. - ↑ Liouville, Joseph (1840). "Addition à la note sur l'irrationnalité du nombre
*e*" (in french).*Journal de Mathématiques Pures et Appliquées*. 1**5**: 193–194. - ↑ Hurwitz, Adolf (1933) [1891]. "Über die Kettenbruchentwicklung der Zahl
*e*" (in german).*Mathematische Werke*.**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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