De Moivre's Theorem

complex-numbers

Definition

De Moivre's Theorem

De Moivre’s Theorem states that for all complex numbers $z$ and all natural numbers $n$, the following equation holds:

$$\forall x\in \mathbb{Z},n\in \mathbb{N}:(\cos z+i\sin z{)}^n=\cos (nz)+i\sin (nz)$$

This theorem is named after Abraham de Moivre.

Power of a Complex Number

Using this theorem, we can simplify the calculation of raising a complex number to a power of a natural number:

$$[|z|,\arg z{]}^n=[n\cdot |z|,n\cdot \arg z]$$