8.5 Polar Form of Complex Numbers

Complex Numbers in Polar Form

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Complex numbers in the plane

Complex numbers live on a 2D coordinate system called the complex plane. The horizontal axis represents real numbers, and the vertical axis represents imaginary numbers.

A complex number $z = a + bi$ gets plotted at the point $(a, b)$. So $3 + 4i$ sits at $(3, 4)$, just like plotting a point in the coordinate plane you already know.

• The absolute value (or modulus) of $z = a + bi$ is the distance from the origin to $(a, b)$:

$|z| = \sqrt{a^2 + b^2}$

This comes straight from the Pythagorean theorem. For $z = 3 + 4i$, the modulus is $\sqrt{9 + 16} = 5$.

• You can also think of complex numbers as vectors from the origin to the point $(a, b)$. This vector interpretation is what makes polar form so natural.

Rectangular vs polar forms

Rectangular form describes a complex number by its horizontal and vertical components:

$z = a + bi$

where $a$ is the real part and $b$ is the imaginary part.

Polar form describes the same number by how far it is from the origin and what angle it makes:

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

This is often shortened to $z = r \, \text{cis} \, \theta$. Here, $r$ is the modulus and $\theta$ is the argument (the angle measured from the positive real axis).

Converting rectangular to polar:

1. Find the modulus: $r = \sqrt{a^2 + b^2}$
2. Find the argument: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$
3. Check the quadrant. If $a < 0$, you need to add $\pi$ to the angle from your calculator, since $\tan^{-1}$ only returns values in Quadrants I and IV.

Converting polar to rectangular:

1. Find the real part: $a = r\cos\theta$
2. Find the imaginary part: $b = r\sin\theta$
3. Write as $z = a + bi$

For example, $z = 4 \, \text{cis} \, \frac{\pi}{3}$ converts to $z = 4\cos\frac{\pi}{3} + 4i\sin\frac{\pi}{3} = 2 + 2\sqrt{3}\,i$.

Multiplication and division in polar form

This is where polar form really pays off. Multiplying and dividing complex numbers in rectangular form involves messy FOIL-ing and rationalizing denominators. In polar form, the rules are clean:

• Multiplication: Multiply the moduli, add the arguments.

$z_1 z_2 = r_1 r_2 \, \text{cis}(\theta_1 + \theta_2)$

• Division: Divide the moduli, subtract the arguments.

$\frac{z_1}{z_2} = \frac{r_1}{r_2} \, \text{cis}(\theta_1 - \theta_2)$

Geometrically, multiplying by a complex number scales the distance from the origin (by $r_2$) and rotates the point (by $\theta_2$). This is a big conceptual takeaway.

Powers and roots of complex numbers

Powers follow directly from the multiplication rule. If you multiply $z$ by itself $n$ times, you raise the modulus to the $n$th power and multiply the angle by $n$:

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

This result is called De Moivre's Theorem, and it works for any positive integer $n$.

Roots go the other direction. The $n$th roots of $z = r \, \text{cis} \, \theta$ are:

$\sqrt[n]{z} = \sqrt[n]{r} \, \text{cis}\left(\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, 2, \ldots, n-1$

Steps for finding $n$th roots:

1. Take the $n$th root of the modulus: $\sqrt[n]{r}$
2. Divide the argument by $n$ to get the first root ($k = 0$)
3. Add $\frac{2\pi}{n}$ to get each successive root
4. Stop after $n$ roots total

All $n$ roots have the same modulus, so they sit equally spaced on a circle of radius $\sqrt[n]{r}$. For instance, the three cube roots of $8$ are spaced $\frac{2\pi}{3}$ apart (120°) on a circle of radius 2.

Applications of De Moivre's Theorem

De Moivre's Theorem states:

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

Beyond computing powers and roots, this theorem is useful for:

• Solving equations like $z^4 = -16$ by converting to polar form and applying the root formula
• Deriving trig identities by expanding the left side using the binomial theorem and matching real and imaginary parts

For example, expanding $(\cos\theta + i\sin\theta)^2$ and equating real parts gives you the double angle formula $\cos(2\theta) = \cos^2\theta - \sin^2\theta$.

Geometry of polar complex numbers

The polar form makes geometric meaning visible:

• Modulus $r$ = distance from the origin
• Argument $\theta$ = angle with the positive real axis
• Multiplying by a complex number = rotating and scaling
• $n$th roots = $n$ equally spaced points on a circle
• The unit circle ($r = 1$) is a special case where every point has the form $\text{cis} \, \theta$, connecting directly to the trig values you already know

Polar form and trigonometry connections

Polar form ties complex numbers back to trigonometry in a deep way. The most famous connection is Euler's formula:

$e^{i\theta} = \cos\theta + i\sin\theta$

This means the polar form $z = r(\cos\theta + i\sin\theta)$ can also be written as:

$z = re^{i\theta}$

You may not use the exponential form heavily in this course, but it's worth knowing it exists. It's the reason the multiplication and division rules work so neatly: when you multiply exponentials, you add exponents.

Euler's formula also provides an elegant way to derive trig identities. Multiplying $e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha+\beta)}$ and expanding both sides gives you the angle addition formulas:

$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$