What is the circumference of a complex unit circle?

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What is the circumference of a unit circle in a complex coordinate system $ e^{i \phi} $ ?

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1 Answer

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It has the same circumference as a unit circle in the regular cartesian x-y plane, namely $2\pi$.

You can see this by thinking of the complex plane as orthogonal to, i.e. at a $90^\circ $ angle relative to, the cartesian plane. The unit circle in the complex plane is just a unit circle in the x-y plane that has been turned $90^\circ$ into the complex plane.

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