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That just wraps up a mystery inside another. Why would tan(x) be irrational if x is rational? That seems like an even bigger mystery than why π is irrational.

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It's all mysteries. The point of "simple" is to create a series of simple steps, each step should be straight forward and easy to understand.

Terse != simple

Thats easy if you remember arctan(y/x) is the angle from x axis to (x,y)

I fail to see how this helps. Actual proof is more complicated, it is based on continued fractions.

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