##### Like what you saw?

##### Create FREE Account and:

- Watch all FREE content in 21 subjects(388 videos for 23 hours)
- FREE advice on how to get better grades at school from an expert
- Attend and watch FREE live webinar on useful topics

# The Complex Plane - Concept

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Imaginary numbers arise frequently in mathematics, but in order to do much with them we need to know more about the complex plane and the rectangular form of complex numbers. **The complex plane** is a medium used to plot complex numbers in rectangular form, if we think as the real and imaginary parts of the number as a coordinate pair within the complex plane.

Recall that the real numbers can be visualized as points on a line. And we call this the number line. You put 0 in the middle. You decide where 1 is and then you've defined the length 1 and you can put 2, 3, 4 and so on, any of the integers on the number line. And then of course fractions fit in between and irrational numbers fit on the line too. Every real number has a place a unique place on the number line.

Now recall complex numbers are numbers that can be expressed in the form a+bi where a and b, these coefficients are real numbers and i is the special number the square root of -1. How do we visualize complex numbers? We graph them on the plane. And we, the way we do it is we plot a+bi as ab. I want to show you some examples. For example plot 3+2i. a is 3 b is 2 so I would just plot 3 2. So here's 3, 1, 2. That would be the point 3+2i. And sometimes it's nice to draw a line connecting it to the origin, and of course the origin in the complex plain is still zero.

Let's plot another point -2. -2 is a complex number even though it's also a real number. It's -2+0i. So the imaginary part of this number is 0. Let's plot this. -2+0i is -2 0 right here. I'll just write -2 and I can also connect it to the origin.

So it's important to note that numbers that are right on the horizontal axis are real numbers and this is therefore called the real axis.

Now let's take a look at another example. 4i. This can be written in the form 0+4i. So the real part is 0. When we plot this, we plot it as 0 4. 0 4 would be 1, 2, 3, 4 here. So this is 4i. And so just as the horizontal axis is the real axis, the vertical axis is the imaginary axis. So think about the horizontal axis. This this is the number line basically, exactly the same as what we had before, only now we've got numbers above it and numbers below it. All of which are imaginary. None of these numbers are real.

Okay. One more example 1-3i. That would be plotted as 1 -3. So over 1 down 3. So this is 1-3i.

That's it. Thats how we plot complex numbers. We call this the complex plain again the horizontal axis is the real axis. The vertical axis is the imaginary axis.

Please enter your name.

Are you sure you want to delete this comment?

###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

##### Concept (1)

#### Related Topics

- Converting Complex Numbers From Trigonometric Form to Rectangular 3,254 views
- Converting Complex Numbers From Rectangular Form to Trigonometric 12,543 views
- Multiplying Complex Numbers 6,974 views
- Dividing Complex Numbers 6,306 views
- DeMoivre's Theorem 13,210 views
- The Euler Formula 8,676 views
- Finding the Roots of a Complex Number 21,834 views
- More Roots of Complex Numbers 9,629 views
- Introduction to Polar Coordinates 16,841 views
- Converting from Rectangular Coordinates to Polar 34,407 views
- Converting from Polar Coordinates to Rectangular 23,516 views
- The Distance Formula in Polar Coordinates 13,673 views
- Lines in Polar Coordinates 10,637 views
- Symmetry of Polar Graphs 16,120 views
- Graphing Polar Equations 25,234 views
- Families of Polar Curves: Circles, Cardiods, and Limacon 9,971 views
- Families of Polar Curves: Roses 11,045 views
- Families of Polar Curves: Conic Sections 6,783 views
- Trigonometric Form of Complex Numbers 16,524 views

## Comments (0)

Please Sign in or Sign up to add your comment.

## ·

Delete