# A proof that $e$ or $\pi$ is irrational via algebraic integers

A very brief proof to show that $\sqrt{2}$ is irrational goes like this:

We know that $\sqrt{2}$ is an algebraic integer. If it is rational it should be an integer which is not therefore it is irrational.

Now the question is that can we use a similar proof for showing that $e$ or $\pi$ are irrational?

I mean is there a proof of existence or non-existence of an $n \times n$ matrix $A$ of integers whose eigenvalue is $e$ or $\pi$?

• $e$ and $\pi$ are transcendental...so no such matrix exists. But this is not an elementary result.
– lulu
Oct 24, 2016 at 11:51
• Elementary result? So you mean we can write a proof for non existence?
– user169903
Oct 24, 2016 at 11:52
• here is a good general discussion of some of the ideas. $e$ is easier to work with and a sketch of the proof can be found here.
– lulu
Oct 24, 2016 at 11:53
• there are elementary proofs that $e$ is irrational. here for instance. Transcendence is more subtle.
– lulu
Oct 24, 2016 at 11:55
• Great reference, thanks!
– user169903
Oct 24, 2016 at 11:59