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

Moonraker
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    The circumference of a circle of radius $r$ is $2\pi r$. In terms of geometric properties of curves, $\mathbb{C}$ is similar to $\mathbb{R}^2$, i.e. you can see circle in $\mathbb{C}$ as if they were circles in $\mathbb{R}^2$. – M.G Jun 02 '16 at 19:49
  • Could you please edit the question to include additional context - for example, what led you to think the circumference might be different? One-sentence posts that merely state a question are discouraged on this site, but you can freely edit the post to improve it. – Carl Mummert Jun 02 '16 at 21:34
  • @Carl Mummert: Sorry for the One-sentence question. As I received the correct answer from M47145 which, however, does not fit for my purpose, I formulated another question, instead of editing this question which received a correct answer. – Moonraker Jun 03 '16 at 08:04

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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.

M47145
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