Main content

### Course: Precalculus > Unit 3

Lesson 1: The complex plane# Plotting numbers on the complex plane

Understand the roles of real and imaginary parts in forming coordinates. Learn how to plot complex numbers on a unique two-dimensional grid known as the complex plane. Discover the real and imaginary parts of complex numbers, and see how they correspond to the horizontal and vertical axes. Created by Sal Khan.

## Want to join the conversation?

- So when graphing on the complex plane, the imaginary value is in units of i? For example, if you had to graph 7 + 5i, why would you only include the coeffient of the i term? Is it because that the imaginary axis is in terms of i? Thank you :)(35 votes)
- Yes, you would graph it at 7 (as x) and 5 (as y).(8 votes)

- Could there ever be a complex number written, for example, 4i + 2? And a graph where the x axis is replaced by "Im," and the y axis is "Re"?(13 votes)
- The coordinate grid we use is a construct to help us understand and see what's happening. Technically, you can set it up however you like for yourself. The reason we use standard practices and conventions is to avoid confusion when sharing with others. Imagine the confusion if everyone did their graphs differently.(36 votes)

- Given that there is point graphing, could there be functions with i^3 or so? Or is the extent of complex numbers on a graph just a point?(8 votes)
- i^3 is i*i*i=i^2 * i = - 1 * i = -i. So I don't see what you mean by i to the third. And we represent complex number on a plane as ordered pair of real and imaginary part of a complex number. Though there is whole branch of mathematics dedicated to complex numbers and functions of a complex numbers called complex analysis, so there much more to it.(15 votes)

- Does _i_ always go on the y axis?(8 votes)
- It's just an arbitrary decision to put _i_ on the y-axis. But yes, it always goes on the y-axis.(7 votes)

- Is there any video over the complex plane that is being used in the other exercises? where complex numbers are written as cos(5/6pi) + sin(5/6pi) ?(9 votes)
- I'm not sure if there is a lesson that is related to that. I apologize!(2 votes)

- Does a point on the complex plane have any applicable meaning? Or is it simply a way to visualize a complex number?(4 votes)
- Both!

We can use complex numbers to solve geometry problems by putting them on the complex plane. This is a common approach in Olympiad-level geometry problems.(7 votes)

- so anything with an i is imaginary(6 votes)
- However, 'i' is always imaginary if it is
*not*i^2(3 votes)

- How does the complex plane make sense? I don't understand how imaginary numbers can even be represented in a two-dimensional space, as they aren't in a number line. I've heard that it is just a representation of the magnitude of a complex number, but the "complex plane" makes even less sense than a complex number.(4 votes)
- So when you were in elementary school I'm sure you plotted numbers on number lines right? Well complex numbers are just like that but there are two components: a real part and an imaginary part. So if you put two number lines at right angles and plot the components on each you get the complex plane! :D(4 votes)

- Can complex numbers only be plotted on the complex plane with the use of cartesian and polar coordinates only?(4 votes)
- You can make up any coordinate system you like, e.g. you could say the point (a, b) is where you arrive by starting at the origin, then traveling a distance a along a line of slope 2, and a distance b along a line of slope -1/2. This is the Cartesian system, rotated counterclockwise by arctan(2).

But the Cartesian and polar systems are the most useful, and therefore the most common systems.(5 votes)

- I'd really like to know where this plane idea came from, because I never knew about this.(2 votes)
- A guy named Argand made the idea for the complex plane, but he was an amateur mathematician and he earned a living maintaining a bookstore in Paris.(6 votes)

## Video transcript

Move the orange dot
to negative 2 plus 2i. So we have a
complex number here. It has a real part, negative 2. It has an imaginary
part, you have 2 times i. And what you see
here is we're going to plot it on this
two-dimensional grid, but it's not our
traditional coordinate axes. In our traditional
coordinate axis, you're plotting a real x value
versus a real y-coordinate. Here on the horizontal
axis, that's going to be the real part
of our complex number. And our vertical axis is going
to be the imaginary part. So in this example,
this complex number, our real part is the
negative 2 and then our imaginary part
is a positive 2. And so that right over
there in the complex plane is the point negative 2 plus 2i. Let's do a few more of these. So 5 plus 2i. Once again, real part is
5, imaginary part is 2, and we're done. Let's do two more of these. 1 plus 5i. 1-- that's the real part--
plus 5i right over that Im. All right, let's do
one more of these. 4 minus 4i. Real part is 4, imaginary
part is negative 4. And we're done.