Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.37 Geometrical interpretation of complex multiplication 4.39 Complex roots of unity

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

4.38 De Moivre’s Theorem

De Moivre’s Theorem

For any real $t$ and integer $n$ ,

(\cos t+i\sin t)^{n}=\cos nt+i\sin nt

Proof. The statement is equivalent to

(e^{it})^{n}=e^{int},

where $n$ on the left-hand side is a power, whereas $nt$ on the right-hand side is an angle. For any complex numbers, the exponential satisfies the multiplication rule

e^{z+w}=e^{z}e^{w};

so De Moivre’s theorem follows from this.