Like the x-y plane, there will be 2 perpendicular axis but now the x-axis is labeled as the real axis while the y-axis is labeled as the imaginary axis. How this works is very simple. Suppose we are given the complex number

Knowing that both a and b are real, we plot the value of a on the x-axis since a represents the real part and we plot the value of b on the y-axis since it represents the imaginary part. A graph of this example shows us the geometric relationships of a complex number on the plane.


We define the magnitude of the complex number, denoted by , as the distance from the origin to the complex number. This is equal to  by how we previously calculated the magnitude as such.

It should be obvious by now that  by looking at the graph.

The angle between the ray from the complex number to the origin is denoted by . The term we used to describe this angle is called the argument of z. We can further apply trigonometry identities.

And by substituting the values of a and b into the standard form, we have

which is what we’ll call the polar form of a complex number.

The conjugate of a complex number has the polar form

This is from the knowledge that when we take the negative of the imaginary part, the magnitude  remains the same but the argument changes as the complex number gets ‘reflected on the real axis’

We can also write the inverse of a complex number in polar form.

We will now conclude, without proof which will be done later, the useful Euler’s Formula which states that for all real numbers , we have

It should also be noted that the complex number  has a magnitude of 1.

Euler’s formula is perhaps 1 of the top 5 most recognized formula by mathematicians simply because it beautifully combines the number  with the imaginary number , and cosine and sine functions.