De Moivre's Formula¶

Theorem¶

For any integer $n$ and nonzero complex number $z = r(\cos\varphi + \mathrm{i}\sin\varphi)$ $i sin φ)$ , with $r>0$ $0$ , it holds that:

z^n = r^n(\cos(n\varphi) + \mathrm{i}\sin(n\varphi)).

For positive integers $n$ , the identity also holds at $z=0$ $0$ . Negative powers require $z\ne0$ $0$ .

Interpretation¶

To raise a complex number in polar form to an integer power $n$ , raise its modulus to the power $n$ and multiply its angle by $n$ .

Three points on a circle at angles theta, 2 theta and 3 theta, the powers of cos theta plus i sin theta, each step adding the same angle.

The figure shows the case $r=1$ $1$ : the powers of $\cos\theta+\mathrm{i}\sin\theta$ stay on the unit circle, and each multiplication adds the same angle $\theta$ , so the first, second and third powers lie at the angles $\theta$ , $2\theta$ and $3\theta$ . For $r>1$ the powers also move outwards and for $r<1$ inwards, which gives the spiral in the interactive version.

The formula follows from complex multiplication, which multiplies moduli and adds angles: multiplying $z$ by itself $n$ times gives modulus $r^n$ and angle $n\varphi$ . For negative $n$ one uses reciprocals, since $z^{-1}$ has modulus $1/r$ and angle $-\varphi$ ; this is why $z\neq0$ is required there. In the Examples, $1+\sqrt{3}\mathrm{i}$ has modulus $2$ and angle $\pi/3$ , so its eighth power has angle $8\pi/3$ , the same direction as $2\pi/3$ after subtracting $2\pi$ , and equals $256(-\tfrac{1}{2}+\tfrac{\sqrt{3}}{2}\mathrm{i})=-128+128\sqrt{3}\,\mathrm{i}$ $1283 i$ .

The formula is stated for integers $n$ only. Used with $n=\tfrac{1}{2}$ $\frac{1}{2}$ , it gives different answers for different angles of the same number: $1$ has the angles $0$ and $2\pi$ , and halving them gives $1$ and $-1$ , the two square roots of $1$ . This is why fractional powers of complex numbers have several values. With Euler's formula the theorem reads $(r\mathrm{e}^{\mathrm{i}\varphi})^n=r^n\mathrm{e}^{\mathrm{i}n\varphi}$ $r^{n} e^{i n φ}$ .

Key Results¶

The formula converts integer powers of a complex number into powers of its modulus and multiples of its argument.
It is closely related to Euler's formula.

Examples¶

$(1+\sqrt{3}\mathrm{i})^8=2^8(\cos(8\pi/3)+\mathrm{i}\sin(8\pi/3))$ $i sin (8 π /3))$ .
$(-1+\mathrm{i})^4=(\sqrt{2})^4(\cos(3\pi)+\mathrm{i}\sin(3\pi))=-4$ $- 4$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: de Moivre theorem, de Moivre's theorem Symbols: phi, pi, square root, theta

Last updated: 20 October 2026 03:02