10.5 Polar Form of Complex Numbers

Complex numbers expand our mathematical toolkit beyond real numbers by combining a real part and an imaginary part on a 2D plane. This lets us solve equations (like $x^2 + 1 = 0$) that have no real solutions and opens the door to powerful geometric interpretations of arithmetic. The polar form, in particular, turns multiplication and exponentiation into simple operations on distances and angles.

Complex Numbers and the Complex Plane

Complex numbers on the plane

The complex plane (also called an Argand diagram) is a 2D coordinate system built for complex numbers. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

A complex number $z = a + bi$ gets plotted as the point $(a, b)$:

• $a$ (the real part) determines the horizontal position
• $b$ (the imaginary part) determines the vertical position

So $3 + 2i$ sits at the point $(3, 2)$, while $-1 - 4i$ sits at $(-1, -4)$. Think of it like a standard x-y plane, but the y-axis now tracks the imaginary component.

Absolute value of complex numbers

The absolute value (or modulus) of a complex number $z = a + bi$ is written $|z|$ and calculated as:

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

This is just the distance from the origin to the point $(a, b)$ on the complex plane, straight from the Pythagorean theorem. It measures the magnitude of the complex number regardless of its direction, and it's always a non-negative real number.

For example, $|3 + 4i| = \sqrt{9 + 16} = 5$.

Polar form of complex numbers

Instead of describing a complex number by its horizontal and vertical components (rectangular form), you can describe it by how far it is from the origin and what angle it makes. That's the polar form:

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

or equivalently, using Euler's formula:

$z = re^{i\theta}$

• $r$ is the magnitude (modulus): $r = \sqrt{a^2 + b^2}$
• $\theta$ is the argument, the angle measured counterclockwise from the positive real axis

To find $\theta$, use $\theta = \tan^{-1}\!\left(\frac{b}{a}\right)$, but you need to adjust based on which quadrant the point falls in. The calculator gives you an angle assuming the point is in Quadrant I or IV, so if the point is in Quadrant II or III (meaning $a < 0$), add $\pi$ to the result.

The angle $\theta$ between $-\pi$ and $\pi$ is called the principal argument.

Rectangular vs polar forms

Rectangular → Polar:

1. Calculate the magnitude: $r = \sqrt{a^2 + b^2}$
2. Calculate the angle: $\theta = \tan^{-1}\!\left(\frac{b}{a}\right)$, adjusting for the quadrant

Polar → Rectangular:

1. Calculate the real part: $a = r\cos\theta$
2. Calculate the imaginary part: $b = r\sin\theta$

Multiplication and division in polar form

This is where polar form really pays off. Multiplication and division become straightforward:

• Multiplication: Multiply the magnitudes, add the angles.

$\left(r_1 e^{i\theta_1}\right)\left(r_2 e^{i\theta_2}\right) = r_1 r_2 \, e^{i(\theta_1 + \theta_2)}$

• Division: Divide the magnitudes, subtract the angles.

$\frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} \, e^{i(\theta_1 - \theta_2)}$

Compare that to multiplying two complex numbers in rectangular form (FOIL, combining terms, tracking $i^2 = -1$). Polar form makes it much cleaner.

Powers and roots in polar form

Powers: To raise $z = re^{i\theta}$ to the $n$th power, raise the magnitude to the $n$th power and multiply the angle by $n$:

$z^n = r^n e^{in\theta}$

Roots: To find the $n$th roots of $z = re^{i\theta}$, take the $n$th root of the magnitude and divide the angle by $n$. But there are $n$ distinct roots, found by adding multiples of $\frac{2\pi}{n}$:

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

Each value of $k$ gives a different root. These $n$ roots are evenly spaced around a circle of radius $\sqrt[n]{r}$ centered at the origin.

Applications of De Moivre's Theorem

De Moivre's Theorem states that for a complex number $z = r(\cos\theta + i\sin\theta)$ and any integer $n$:

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

This is the trigonometric version of the power rule above. It's useful for:

• Computing powers of complex numbers without repeated multiplication
• Finding roots of complex numbers (by setting $n$ to a fraction and accounting for all $k$ values)
• Solving polynomial equations that have complex solutions, such as $z^4 = -16$

Visualization on the complex plane

Geometric intuition makes complex operations click:

• Addition works like vector addition. Place the tail of one vector at the head of the other; the resultant vector is the sum.
• Multiplication by $e^{i\theta}$ rotates a complex number by angle $\theta$ counterclockwise around the origin, without changing its magnitude. Multiplication by a positive real number scales the magnitude without changing the angle.
• Powers simultaneously rotate and scale. Squaring a number doubles its angle and squares its distance from the origin.
• $n$th roots split the full circle into $n$ equal parts. The roots of any complex number form a regular polygon inscribed in a circle centered at the origin.

Complex Exponential and the Unit Circle

Euler's formula connects complex exponentials to trig functions:

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

The set of all complex numbers of the form $e^{i\theta}$ traces out the unit circle (the circle of radius 1 centered at the origin). Every point on the unit circle has magnitude 1 and is fully determined by its angle $\theta$.

The exponential form $re^{i\theta}$ is compact and especially convenient for multiplication, division, and powers because it combines magnitude and angle into a single expression. Whenever you see periodic behavior or rotations in math or science, complex exponentials are likely working behind the scenes.