Complex Plane

No, not this complex plane ...
... this complex plane:

It is a plane for complex numbers!

(It is also called an "Argand Diagram")

Real and Imaginary make Complex

A Complex Number is a combination of a Real Number and an Imaginary Number.

Let me explain ...

A Real Number is the type of number you are used to dealing with every day.

Examples: 12.38, ½, 0, -2000

With real numbers we can do things like squaring (multiply a number by itself):

2 × 2 = 4

But what can we square to get -4 (minus 4)?

? × ? = -4

Well, -2 won't work because multiplying negatives gives a positive: (-2) × (-2) = +4, and no other Real Number works either.

So it seems that mathematics is incomplete ...

... but we can fill the gap by imagining there is a number that, when multiplied by itself, gives -1 (call it i for imaginary):

i2 = -1

Now, we can do this:

2i × 2i = 4i2 = 4 × (-1) = -4

An Imaginary Number, when squared gives a negative result

.

Examples: 5i, -3.6i, i/2, 500i

And together:

A Complex Number is a combination of a Real Number and an Imaginary Number

Examples: 3.6 + 4i, -0.02 + 1.2i, 25 - 0.3i, 0 + 2i

The Complex Plane

You may be familiar with the number line:

But where do we put a complex number like 3+4i ?

Let's have the real number line going left-right as usual, and an imaginary number line going up-down:

And we can plot a complex number like 3 + 4i :

It is placed

  • 3 units along (the real axis),
  • and 4 units up (the imaginary axis).
 
     

And here is 4 - 2i :

It is placed

  • 4 units along (the real axis),
  • and 2 units down (the imaginary axis).
 

 

And that is the complex plane:

  • complex because it is a combination of real and imaginary,
  • plane because it is like a geometric plane (2 dimensional).

Whole New World

Now we can bring the idea of a plane (Cartesian coordinates, Polar coordinates, Vectors and more) to complex numbers, and we open up a whole new world of numbers that are more complete and elegant, as you will see.

Complex Number as a Vector

You can think of a complex number as being a vector.


This is a vector.
It has magnitude (length) and direction.

And here is the complex number 3 + 4i

as a Vector:

 

You can add complex numbers as vectors, too:

Here we add the complex numbers 3 + 5i and 4 − 3i as vectors:

Add the real numbers, add the imaginary numbers, like this:

(3 + 5i) + (4 − 3i) = 3 + 4 + (5 − 3)i = 7 + 2i

 

Polar Form

Again, here is the complex number 3 + 4i

As a Vector:

 
     

Here it is again (still as a vector), but

In polar form:

 

So the complex number 3 + 4i can also be shown as distance (5) and angle (0.927 radians).

How do you do the conversions?

Example: the number 3 + 4i

We can do a Cartesian to Polar conversion:

  • r = √(x2 + y2) = √(32 + 42) = √25 = 5
  • θ = tan-1 (y/x) = tan-1 (4/3) = 0.927 (to 3 decimals)

 

We can also take Polar coordinates and convert them to Cartesian coordinates:

  • x = r × cos( θ ) = 5 × cos( 0.927 ) = 5 × 0.6002... = 3 (close enough)
  • y = r × sin( θ ) = 5 × sin( 0.927 ) = 5 × 0.7998... = 4 (close enough)

In fact, a common way to write a complex number in Polar form is

x + iy = r cos θ + i r sin θ = r(cos θ + i sin θ)

And "cos θ + i sin θ" gets shortened to "cis θ"

So 3 + 4i = 5 cis 0.927

 

Summary

The complex plane is a plane with:

  • real numbers running left-right and
  • imaginary numbers running up-down.

To convert from Cartesian to Polar Form:

  • r = √(x2 + y2)
  • θ = tan-1 ( y / x )

To convert from Polar to Cartesian Form:

  • x = r × cos( θ )
  • y = r × sin( θ )

Next ... learn about Complex Number Multiplication.