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:
Proofs: 1
Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6
Propositions: 7