Powers of Complex Numbers (De Moivre’s Theorem)

Powers of Complex Numbers

De Moivre’s Theorem

De Moivre’s Theorem is a fundamental result in trigonometry and complex numbers, allowing us to raise complex numbers in polar form to integer powers efficiently. This theorem is especially useful in simplifying powers and roots of complex numbers.

• Statement of De Moivre’s Theorem: For any real number θ and integer n, if a complex number is written in polar form as z = r (cos θ + i sin θ), then:

$z^n = [r (\cos \theta + i \sin \theta)]^n = r^n [\cos(n\theta) + i \sin(n\theta)]$

• Key Points:

  • Convert the complex number to polar form before applying the theorem.

  • Multiply the argument (angle) by n and raise the modulus (r) to the power n.

Worked Example

Given: $z = 4(\cos(100°) + i \sin(100°))$

• Find $z^2$:

$z^2 = [4(\cos 100° + i \sin 100°)]^2$

$= 4^2 [\cos(2 \times 100°) + i \sin(2 \times 100°)]$

$= 16 [\cos(200°) + i \sin(200°)]$

• Evaluating the trigonometric values:

$\cos(200°) = -0.94$, $\sin(200°) = -0.34$

$z^2 = 16(-0.94 + i(-0.34))$

$= 16(-0.94 - 0.34i)$

$= -15.04 - 5.44i$

• Generalization: The same process applies for higher powers, simply multiply the angle by n and raise the modulus to n.

Roots of Complex Numbers

To find the nth roots of a complex number, use the formula:

$z_k = r^{1/n} \left[ \cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right) \right]$, where $k = 0, 1, ..., n-1$

• Example: Find the square roots of $z = 4(\cos 100° + i \sin 100°)$

$r = 4$, $\theta = 100°$

$\sqrt{z} = 2 \left[ \cos \left( \frac{100° + 360°k}{2} \right) + i \sin \left( \frac{100° + 360°k}{2} \right) \right]$, $k = 0, 1$

For $k = 0$: $\cos(50°) + i \sin(50°)$ For $k = 1$: $\cos(230°) + i \sin(230°)$

• Roots: $2(\cos 50° + i \sin 50°)$ and $2(\cos 230° + i \sin 230°)$

Summary Table: De Moivre’s Theorem Applications

Operation	Formula	Description
Power	$z^n = r^n [\cos(n\theta) + i \sin(n\theta)]$	Raise modulus to n, multiply angle by n
Root	$z_k = r^{1/n} [\cos(\frac{\theta + 2\pi k}{n}) + i \sin(\frac{\theta + 2\pi k}{n})]$	Take nth root of modulus, divide angle plus $2\pi k$ by n

Additional info: De Moivre’s Theorem is essential for simplifying complex number calculations in trigonometry, especially for powers and roots. It is also foundational for understanding the geometric interpretation of complex numbers on the Argand diagram.