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I was learning the proofs of the transcendence of $e$ and $\pi$, and I was wondering what would be the usefulness of these results.

So my questions are:

  • Are there any (relatively simple) application of the transcendence of $e$?

  • More generally, why is it useful to know that a number $x$ is transcendent?

When I talk about transcendence here, I am putting myself in the simplest case: a real number $x$ being transcendent over $\mathbb Q$.

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    Transcendence of $\pi$ shows that squaring the circle is not possible :-) – 2017-02-09
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    @Watson Nice application! It only requires that $\pi$ is not algebraic of degree $2^n$ if I recall correctly. Great use of the transcendence though :) – 2017-02-10

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