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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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    $\begingroup$ Transcendence of $\pi$ shows that squaring the circle is not possible :-) $\endgroup$ – Watson Feb 9 '17 at 23:20
  • $\begingroup$ @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 :) $\endgroup$ – E. Joseph Feb 10 '17 at 9:48

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