CSPE - The Complex Plane Lesson

The Complex Plane

We can graph complex numbers on the complex plane. The complex plane has a horizontal axis that represents the real axis and a vertical axis that represents the imaginary axis.

image of imaginary coordinate plane with the horizontal axis representing the real axis and vertical axis representing the imaginary axis

Let's say you are given the complex number: z = 3 + 2i, you would graph that by going right 3 units on the real axis and up 2 units on the imaginary axis.

image of two lines, one going three units to the right, and the vertical going two units up

Let's think about how we might portray addition and subtraction of complex numbers on the complex plane:

One important number that we want to be able to calculate for a complex number is the modulus, this number is also called the absolute value or magnitude. The modulus is the distance from the origin to the point when a complex number is plotted.

image of two lines, one going three units to the right, and the vertical going two units up with the modulus drawn from the start to the end point

Based on the Pythagorean theorem, we can determine that the modulus, which we will denote as |z| is:

Modulus, Absolute Value or Magnitude of a complex number, z = a + bi, is $LaTeX: \left|z\right|=\left|a+bi\right|=\sqrt[]{a^2+b^2}$ $\left|z\right|=\left|a+bi\right|=\sqrt[]{a^2+b^2}$

So, we know the absolute value of 3 + 2i is equation image indicator $LaTeX: \sqrt[]{13}$ $\sqrt[]{13}$ .

image of two lines, one going three units to the right, and the vertical going two units up with the modulus is the |z|= sq rt of 13 and z = 3+2i

Watch this video to see how we use the Pythagorean Theorem to determine the modulus of a complex number.

Recall last lesson, we learned about the complex conjugate - how might we find a complex conjugate on the complex plane? What do you think the relationship would be?

line z=3+2i with complex conjugate z=3-2i plotted on graph

We can see that the conjugate is reflected across the Real or horizontal axis - since we are changing the sign of the imaginary part this makes sense.

We can also find the distance and midpoint of the segment connecting two points in the complex plane, just like we have done in the x-y plane. Watch this video to see how.

IMAGES CREATED BY GAVS