Another proof that 0=2

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Time for a math riddle. Haven’t done these in a while. Well, haven’t done any in this blog, when I come to think about it. OK. That was enough thinking. Let’s get down to biusiness.

Take a point on the complex plane. Take one which is on the unit circle:

z=e^{i\theta}

Now replace \theta with \phi = \frac{\theta}{2\pi}. We get:

z=e^{i\theta}=e^{2\pi\i\phi}

Which by the simple laws of arithmetic gives us:

z=e^{2\pi\i\phi}=\left(e^{2\pi\i}\right)^\phi=1^\phi=1

So every point on the unit circle is 1!

As a simple consequence we get 1=-1. Add 1 on both sides and get 2=0.

QED.

Can you spot the error?

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