Powers and Roots of Complex Numbers (DeMoivre’s Theorem)

Powers and Roots of Complex Numbers

DeMoivre’s Theorem

DeMoivre’s Theorem is a powerful tool in trigonometry and complex numbers, allowing us to compute powers and roots of complex numbers expressed in trigonometric (polar) form.

• DeMoivre’s Theorem: For any complex number in polar form $z = r(\cos \theta + i \sin \theta)$ and integer $n$,

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

• This theorem is especially useful for simplifying the computation of powers and roots of complex numbers.

Example: Computing a Power

• Given: $z = 2(\cos 15^\circ + i \sin 15^\circ)$

• Find: $z^5$

Solution:

Using DeMoivre’s Theorem:

$z^5 = 2^5 (\cos(5 \times 15^\circ) + i \sin(5 \times 15^\circ))$

$= 32 (\cos 75^\circ + i \sin 75^\circ)$

Example: Division of Complex Numbers in Polar Form

• Given: $z_1 = 8(\cos 65^\circ + i \sin 65^\circ)$, $z_2 = 2(\cos 15^\circ + i \sin 15^\circ)$

• Find: $\dfrac{z_1}{z_2}$

Solution:

• When dividing complex numbers in polar form, divide the moduli and subtract the arguments:

$\dfrac{z_1}{z_2} = \dfrac{8}{2} \left[ \cos(65^\circ - 15^\circ) + i \sin(65^\circ - 15^\circ) \right]$

$= 4 \left[ \cos 50^\circ + i \sin 50^\circ \right]$

Key Points

• Multiplication: Multiply the moduli, add the arguments.

• Division: Divide the moduli, subtract the arguments.

• Powers: Raise the modulus to the power, multiply the argument by the power (DeMoivre’s Theorem).

• Roots: Take the root of the modulus, divide the argument by the root (and consider all possible values by adding $\dfrac{360^\circ k}{n}$ for $k = 0, 1, ..., n-1$).

Additional info:

• These operations are foundational for solving equations involving complex numbers and for applications in engineering, physics, and mathematics.