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How to Graph a Complex Number on the Complex Plane

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  • 0:04 Unrelated Numbers
  • 0:53 Graphing Complex Numbers
  • 3:04 Lesson Summary
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Lesson Transcript
Instructor: Luke Winspur

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Graphing complex numbers is pretty straight forward, but it's not necessarily intuitive. Check out this lesson to learn the vocabulary and the conventions that you'll need.

Unrelated Numbers

Being a math guy, I was also pretty into my science classes during high school. Naturally, the one that I liked the most was the one with the most math in it - physics. It's now been a while since I've reviewed my physics, but I do remember a few things. One of those things that I remember is that electric circuits are defined by two things - their voltage and their current.

Well, this presented a problem for physicists. They would like to have one number that represents both of these pieces of information, but what one number can represent two things? I suppose they could have used coordinates like (x,y). But what if you wanted to multiply or even divide these two numbers? There's no method for dividing coordinates. Is there a better way?

Complex numbers represent two pieces of information, real and imaginary.
Graphing Complex Numbers Graph

Graphing Complex Numbers

Aha, complex numbers! Because they have a real part and an imaginary part, they can represent two pieces of information, and we know how to add, subtract, multiply and divide them as well. This makes complex numbers the ideal way to represent two pieces of information in one number.

But to make complex numbers better than coordinates in every way, we'll need to be able to graph them as well. Because there are two pieces of information in each complex number, we can put this info into a two variable graph. This time, instead of having an x-axis and y-axis, we'll have a real axis and imaginary axis. But even though the axis will be a little different, the process of putting the numbers on the graph is, essentially, the same.

Let's take a look at an example:

Graph -2 + 5i

The axes will still look the same, but now our x-axis is, instead, going to be the real part of the complex number, so we call it the real axis. That makes the y-axis our imaginary axis and will represent the imaginary part of the number. That makes graphing -2 + 5i, basically, the same as graphing (-2,5) on a normal x/y-axis. We go over to -2 on the real axis, and up to positive 5 on the imaginary one, giving us a point right here.

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