Powers and Roots of Complex Numbers

Complex Number Basics

A complex number has the form:

$z = a + bi$

where:

• $a$ = real part ($\text{Re}(z)$)
• $b$ = imaginary part ($\text{Im}(z)$)
• $i = \sqrt{-1}$, so $i^2 = -1$

Forms of Complex Numbers

Rectangular Form

$z = a + bi$

Polar (Trigonometric) Form

$z = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta$

where:

• $r = |z| = \sqrt{a^2 + b^2}$ (modulus)
• $\theta = \arg(z) = \arctan(b/a)$ (argument)

Exponential Form

$z = re^{i\theta}$

Converting Between Forms

Rectangular to Polar

$r = \sqrt{a^2 + b^2}$

$\theta = \arctan\left(\frac{b}{a}\right) \quad \text{(adjust for quadrant)}$

Polar to Rectangular

$a = r\cos\theta$

$b = r\sin\theta$

Operations in Polar Form

Multiplication

$z_1 \cdot z_2 = r_1 r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$

Multiply moduli, add arguments.

Division

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$

Divide moduli, subtract arguments.

De Moivre's Theorem

For any integer $n$:

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

Or using cis notation:

$(r\,\text{cis}\,\theta)^n = r^n\,\text{cis}(n\theta)$

Powers of Complex Numbers

To find $z^n$:

1. Convert to polar form
2. Apply De Moivre's theorem
3. Convert back to rectangular if needed

Example: $(1 + i)^8$

1. $r = \sqrt{2}$, $\theta = \frac{\pi}{4}$
2. $z = \sqrt{2}\,\text{cis}(\frac{\pi}{4})$
3. $z^8 = (\sqrt{2})^8\,\text{cis}(8 \cdot \frac{\pi}{4}) = 16\,\text{cis}(2\pi) = 16(1 + 0i) = 16$

Roots of Complex Numbers

The $n$ distinct $n$th roots of $z = r\,\text{cis}\,\theta$ are:

$\sqrt[n]{z} = \sqrt[n]{r} \cdot \text{cis}\left(\frac{\theta + 2\pi k}{n}\right)$

for $k = 0, 1, 2, \ldots, n-1$

Properties of Roots

• There are exactly $n$ distinct $n$th roots
• All roots have the same modulus: $\sqrt[n]{r}$
• Roots are equally spaced on a circle
• Angular separation between consecutive roots: $\frac{2\pi}{n}$

Example: Cube Roots of 8

$z = 8 = 8\,\text{cis}\,0$

Roots: $\sqrt[3]{8} \cdot \text{cis}\left(\frac{0 + 2\pi k}{3}\right)$ for $k = 0, 1, 2$

• $k = 0$: $2\,\text{cis}\,0 = 2$
• $k = 1$: $2\,\text{cis}(\frac{2\pi}{3}) = 2(-\frac{1}{2} + i\frac{\sqrt{3}}{2}) = -1 + i\sqrt{3}$
• $k = 2$: $2\,\text{cis}(\frac{4\pi}{3}) = 2(-\frac{1}{2} - i\frac{\sqrt{3}}{2}) = -1 - i\sqrt{3}$

Roots of Unity

The $n$th roots of 1 are:

$\omega_k = \text{cis}\left(\frac{2\pi k}{n}\right) = e^{2\pi ik/n}$

for $k = 0, 1, 2, \ldots, n-1$

Special Cases

Square roots of 1: $1, -1$

Cube roots of 1: $1, \frac{-1 + i\sqrt{3}}{2}, \frac{-1 - i\sqrt{3}}{2}$

Fourth roots of 1: $1, i, -1, -i$

Powers of i

Power	Value
$i^0$	$1$
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$

Pattern repeats with period 4:

$i^n = i^{n \bmod 4}$