# Polar Form of a Complex Number

Cartesian coordinates are simple enough, but what about *polar* coordinates? We have covered this topic in the past, but we can take things a step further by finding polar forms of complex numbers. But how exactly do we do this, and why might this be an important step? Let's find out:

## Reviewing our terms

Before we get started, let's take a second to review a few important terms:

First of all, how are polar coordinates different compared to Cartesian coordinates? Essentially, a set of Cartesian coordinates is the result of two measures: Up and down. But what might happen if we drew a diagonal line straight from the origin to the coordinates? The endpoint might be the same, but the line now measures a *single* distance from the origin rather than *two* distances (up and down). The single, diagonal line represents our polar coordinates.

Recall that we express our polar coordinates with an angle and a distance. This is notably different compared to Cartesian coordinates, which we represent with x and y, in polar we instead use r (radius) and theta (angle).

While the x and y coordinates represent the distances from the axes, the distance value in a set of polar coordinates represents the distance to the origin (0,0). The angle is measured from the x-axis.

We also need to quickly review the concept of a complex number.

Recall that a complex number takes the following form: $a+bi$

In this expression, a and b are real numbers and i represents the "imaginary unit." This imaginary unit is the square root of -1. We can plot an imaginary unit on the complex plane. Instead of having x and y axes, our new plane has an imaginary axis instead of a y-axis and a real axis instead of an x-axis.

Going back to our complex number $a+bi$, we can plot this as the ordered pair (a,b) on our complex plane.

When we do this, the *distance* from the origin to our complex number from the origin is called the "modulus." We also call this the *absolute value*.

We can express this absolute value as:

$\left|z\right|=|a+bi|$

This also equals:

$\sqrt{(a-0){2}^{}+(b-0){2}^{}}or\sqrt{{a}^{2}+{b}^{2}}$

## Finding polar coordinates for our complex numbers

The rectangular form of our complex number is represented in this format:$z=a+bi$

But we can also represent this complex number in a different way called the "polar form."

Take a look at this diagram:

The y-axis is our imaginary axis (i), while the x-axis is our real axis (a).

Polar coordinates have two values: The angle θ and the distance from the origin (r).

All we need to do is find these two values for our complex number, and we will have our polar coordinates.

But how exactly do we accomplish this? In order to get our values, we need to refer back to the handy Pythagorean Theorem.

Take a look at the above diagram. What kind of shape do we notice? That's right -- a triangle. And whenever we see triangles, we know that we can apply the Pythagorean Theorem to help us solve all kinds of problems. Finding the polar coordinates of a complex number is certainly no exception.

Based on the above diagram, we can plug our values into the Pythagorean Theorem: ${r}^{2}={a}^{2}+{b}^{2}$

Let's also bring our trigonometric ratios into the mix:

We know that the cosine of our angle θ is adjacent/hypotenuse or $a/r$. In other words:

$\mathrm{cos}\theta =a/r$

We also know that the sine of our angle θ is opposite/hypotenuse or $b/r$. In other words: