AcademyComplex Form
Academy
De Moivre Theorem
Level 1 - Math I (Physics) topic page in Complex Form.
De Moivre's Theorem
De Moivre's theorem provides a powerful formula for raising complex numbers to integer powers.
The Theorem
For any real angle \(\theta\) and integer \(n\):
De Moivre
\[(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\]
This can also be written using exponential form:
Exponential Form
\[(e^{i\theta})^n = e^{in\theta}\]
Applications to Powers
De Moivre's theorem simplifies calculating powers of complex numbers in polar form.
If \(z = r(\cos\theta + i\sin\theta)\), then:
Power
\[z^n = r^n(\cos(n\theta) + i\sin(n\theta))\]
For example, to find \((1 + i)^4\):
- Write in polar form: \(1 + i = \sqrt{2} \text{cis} \frac{\pi}{4}\)
- Apply De Moivre: \((\sqrt{2})^4 \text{cis} \pi = 4 \text{cis} \pi = -4\)
Finding Roots
To find the \(n\)-th roots of a complex number \(z = r\text{cis}\theta\):
Roots
\[z_k = r^{1/n} \text{cis}\left(\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, \dots, n-1\]
This gives \(n\) distinct roots evenly spaced around a circle in the complex plane.