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## Lemma: Unit Circle

Let $$x\in\mathbb R$$ be any real number and let $$z$$ be the complex number, for which $$x$$ is the imaginary part, i.e. $$x=\Im(z)\Longleftrightarrow z:=ix$$.

The distance of the complex exponential function from the point of origin is equal $$1$$, formally

$|\exp(ix)|=1\quad\quad\text{for all }x\in\mathbb R.$

Geometrically, the complex numbers $$\exp(ix)$$ form a figure called the unit circle:

### Fun questions For which values of $$x$$ does $$\exp(ix)$$ reach the complex number $$i$$? For which values of $$x$$ does $$\exp(ix)$$ reach the complex number $$1$$?

| | | | | created: 2016-02-29 22:58:10 | modified: 2020-06-14 16:39:05 | by: bookofproofs | references: [581]