Proof of the irrationality of Euler's number

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The proof of the irrationality of Euler's number can be given in several ways. Evidence of the irrationality of e was first given by Leonhard Euler in 1737, Johann Heinrich Lambert in 1768 (both on continued fraction expansion ) and Joseph Fourier in his lectures at the Ecole Polytechnique in 1815 (an “elementary” proof as evidence of contradiction ). The proof of Fourier was extended by Joseph Liouville to the proof of irrationality of . Some more evidence was given later.

The proof that is even transcendent is more complicated and was first given by Charles Hermite in 1873 .

proof

adoption

Fourier's proof is presented.

We start with Leonhard Euler's representation of Euler's number as a series

.

As can be easily shown, applies .

We now assume that Euler's real number is rational. Then they could be as completely reduced fraction with represent. Da , is not an integer, and thus q> 1. We multiply the series expansion by , which gives us this new series:

Left side

It is as per premise .

Right side, first subtotal

The terms up to the right of the equation are all natural as well , since all denominators to are divisors of the numerator . The sum of these natural numbers is again a natural number.

Right side, second subtotal

The sum of all terms of the term is greater than 0, since all numerators and denominators are different from zero and positive, and also less than 1, as the following consideration shows:

The first link is because , the second link is , the third link is , etc.

The sum of these upper bounds is an infinite, so-called geometrical series and converges:

.

The following applies to the second partial sum , therefore is not a natural number.

Contradiction

The expression leads to the desired contradiction, since the right side , unlike the left side ,, is not a natural number.

Enough

This refutes the prerequisite and it applies , i. i.e., is irrational.

Individual evidence

  1. ^ Euler: De fractionibus continuis dissertatio. Comm. Acad. Sci. Petrop., Volume 9, 1737 (published 1744), 98-137, (E 71), his essay on continued fractions. He also goes into this briefly in his Introductio in analysin infinitorum , Volume 1, 1748. That this was a fully valid proof is shown (using the Riccati differential equation, Euler shows that the continued fraction expansion of e does not terminate), for example, Ed Sandifer in his column How Euler did it , Who proved e is irrational? (MAA online, February 2006), PDF.
  2. Liouville: Sur l'irrationalité du nombre e. J. Math. Pures Appl., 5, 1840, 192.
  3. ^ O. Perron : Irrational Numbers. de Gruyter, Berlin 1910.
  4. E. Krätzel : Number theory. VEB Deutscher Verlag der Wissenschaften, Berlin 1981, pp. 81–83.