# 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.