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The formula e^{Pi * i) = -1 (see picture below) has been deemed one of the most beautiful mathematical equations, see for example a Quora discussion on the subject. Here Pi = 3.1415... and i is the imaginary number with i^2 = -1.

As I was trying to find if there is a simple relation between e and Pi not involving imaginary numbers, but instead just a finite number of basic operations and functions (not even integrals of course), I came with the following:

  • e^{2 * Pi * i} = 1
  • Take the logarithm on both sides, you get 2 * Pi * i = 0
  • Thus i = 0.

Can you explain this paradox? The solution is straightforward if you are familiar with the theory of functions of complex argument.

It does not seem that you can find the basic relationship I was hoping for between e and Pi, so these two fundamental mathematical constants are not redundant, it seems. Actually, this is still an unsolved mathematical conjecture, and one of the most famous ones.

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