## Pi and e are irrational/transcendental

Two of the most used constants in mathematics and mathematical analysis are the numbers and . These constants, defined in the following, are proven to be irrational. Moreover, these numbers are transcendent, which means that there is no polynomial equation with rational coefficients which have or as a root.

The proof of these facts will be presented below.

First we prove that is irrational. Else, suppose , with positive integers. This means that . Since we have . The right hand side is an integer. The left hand side has two terms. The first one is an integer, and the second one is between and , which is not an integer. Why is that? The first inequality is obvious: a sum of positive terms is greater than one of its terms. For the second inequality, see that . The contradiction assures us of the irrationality of .

Using a similar method, we can prove even more, that is irrational. To do this, assume that , which is equivalent to . Using the same series expansion for as before, and the similar expansion we see that can again be written in the following way: , where are integers ( multiplied with the respective parts of the series with denominator less or equal to ), and and are the “fractional parts” for which we can prove that

For , a similar strategy can be used. See **Proofs from the Book**. But let’s get to serious business and prove that is irrational for all rational nonzero . To do this, we need the following

**Lemma. **For some fixed , let . This function has the following propetrties:

(i) is a polynomial of the form , where are integers.

(ii) For we have .

We can see that , which leads us to . Assume that with positive integers. Then , which is an integer from part (iii) of the lemma.

But the lemma gives us the estimates

Contradiction. Therefore is not rational.

If for some nonzero rational we have , then is also rational. Contradiction.

The above argument works in showing that is not rational. Assuming , define

which satisfies . From the lemma, we get that are integers. Differentiating we get .

Therefore is a positive integer.

Part (ii) of the lemma implies that which can be made arbitrarily small for great enough. This contradicts the fact that is a positive integer. Therefore and then, obviously, are irrational.

A proof of the fact that and are both transcendental can be found here.