Polar Form of Complex Numbers

We've seen how a complex number like $3 + 4i$ can be visualized as a point on the complex plane. Typically, we represent that point using an ordered pair: its real part and its imaginary part.

Two Ways to Describe a Point

On a 2D plane, there are two powerful ways to describe the position of a point:

Using the Cartesian form: an ordered pair $(x, y)$ .
Using the Polar form: a distance from the origin (called the modulus) and an angle from the x-axis.

The polar approach treats the complex number like a vector — from the origin to the point — defined by:

r: the modulus (length of the vector)
θ: the argument or angle the vector makes with the positive real axis

Modulus: Distance from the Origin

The modulus of a complex number $z = x + iy$ is the distance from the origin $(0, 0)$ to the point $(x, y)$ .

It's calculated using the Pythagorean theorem where hypotenuse squared is equal to the sum of base squared and perpendicular squared:

Modulus = $|z| = r$

By Pythagoras Theorem:

$r^2 = x^2 + y^2$

$r = \sqrt{x^2 + y^2}$

$|z| = \sqrt{x^2 + y^2}$

Argument: The Angle With the X-Axis

The angle $\theta$ is called the argument of the complex number. It's the angle the line from the origin to the point makes with the real (x) axis.

This angle can be found using:

$\theta = \tan^{-1}\left(\dfrac{y}{x}\right)$

Remember to pay attention to which quadrant the complex number lies in when calculating $\theta$ , since the inverse tangent function only returns values in certain ranges.

Polar to Cartesian Form

If you know the modulus $r$ and angle $\theta$ , you can convert back to Cartesian form using:

$x + iy = r\cos\theta + i r\sin\theta$

We get this from basic trignometry where the side opposite to the angle is hypotenuse times the sine of the angle. Similarly the base of the triangle is hypotenuse times the cosine of the angle. Hypotenuse in this case is the modulus of the complex number.

This form is called the polar form of a complex number, simplified it becomes:

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

This is incredibly useful for multiplication, division, and even raising complex numbers to powers — as we'll see in future lessons.

What is the distance from the origin to a complex number called?

What is the angle between the complex number and the real axis called?

Fill in the blank: x + iy = _____

End of Lesson