Proof that e is irrational

The proof of the irrationality of Euler's number is feasible with elementary means of analysis as a proof by contradiction. It was first performed in 1737 by Leonhard Euler in the manner described here.

The evidence that is even transcendent, is more complicated and was first performed in 1873 by Charles Hermite.

Evidence

Assumption

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

As can easily be shown, applies.

We now assume that the real Euler number is rational. Then they could be represented as a completely reduced fraction with. There is not a whole number, and thus q > 1, we multiply it by the expansion, by which we obtain this new set:

Left side

It is because by assumption.

Right, first partial sum

The links to on the right side of the equation are also all natural, since all denominators to divider of the counter are. The sum of these integers is a natural number again.

Right side, second partial sum

The sum of all members, from the member is greater than 0, since all the numerator and denominator are different from zero and positive, and also smaller than 1, such as the following consideration is:

The first member is, as the second member, the third member, etc.

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

Therefore, apply to the second partial sum is therefore not a natural number.

Contradiction

The expression results in the desired contrary, since the right-hand side, no natural number other than the left side.

Closing

The condition is refuted and it is considered, that is, is irrational.

• Number Theory
• Proof ( mathematics)