This article covers much about the mathematical constant e, Euler's number, concluding with the result that it is irrational.

## Introduction

The mathematical constant e was first found by Bernoulli with the formula

We will use this formula to determine a new formula for e and then we will use it to prove e's irrationality.

## Lemmas

**Lemma 1. **The sequence increases.

**Proof. **We need to show

which is equivalent to

Simplifying, we get

If we change parameters and set , we get

which simplifies to

which, of course, holds.

**Lemma 2. **The sequence has an upper bound of .

**Proof. **It follows directly:

**Corollary 3. **The sequence converges.

Proof. Follows directly from Lemmas 1 and 2.

**Lemma 4.** The expression is equal to .

**Proof. **Since the first expression is equal to due to the Bolzano-Weierstrass theorem, it immediately follows.

**Lemma 5. **

**Proof.** It is obvious:

**Lemma 6. **The Taylor series expansion for is

Proof. We know an expression for , so we will differentiate it to obtain a result. It is obvious that this expression is 1 when k=0, so if we make the definition , . Now we differentiate our expression.

As we can see, , so all of the other derivatives will be 1 when evaluated at 0. This yields the following Taylor series for our function:

And the proof is complete.

**Corollary 7. **An infinite sum representation of e is

**Proof. **We take x=1 in the previous lemma to obtain this.

## The proof of the theorem

We will use a proof by contradiction. Let's assume e is rational and for , it can be written as p/q. Observe the following equalities:

Since the expression on the RHS is a positive integer, so must be the expression on the LHS. This yields:

The last term of the last sum must be an integer because the sum equals a positive integer and its first term is a positive integer. This is important, the contradiction follows from here. Let's call this integer R. Now, observe these:

Therefore:

Now also note that q > 0, so all the terms in R are strictly positive, therefore R > 0. So we have 0 < R < 1, but we earlier established that R was a positive integer. As there are no integers between 0 and 1, we have a contradiction. Hence, it is impossible to express e as a ratio of two integers, so it must be irrational. Proof complete.