Complex Plane

	No, not that complex plane ...
... this complex plane:	a plane for complex numbers! Also called an "Argand Diagram"

Real and Imaginary make Complex

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

When we square a Real Number we get a positive (or zero) result:

2² = 2 × 2 = 4
1² = 1 × 1 = 1
0² = 0 × 0 = 0

What can we square to get −1?

?² = −1

Squaring −1 doesn't work because multiplying negatives gives a positive: (−1) × (−1) = +1, and no other Real Number works either.

So it seems mathematics is incomplete ...

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

i² = −1

And together:

Putting a Complex Number on a 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 go left-right as usual, and have the imaginary number line go up-and-down:

We can then plot a complex number like 3 + 4i : 3 units along (the real axis), and 4 units up (the imaginary axis)
And here's 4 − 2i : 4 units along (the real axis), and 2 units down (the imaginary axis)

Whole New World

Now let's bring the idea of a plane (as seen in Cartesian coordinates, Polar coordinates, and so on) to complex numbers.

It will open up a whole new world of numbers that are more complete and elegant, as we'll see.

Complex Number as a Vector

We can think of a complex number as .

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

And here's the complex number 3 + 4i as a vector:

Adding

We can add complex numbers as vectors, too:

To add the complex numbers 3 + 5i and 4 − 3i : add the real numbers, and add the imaginary numbers separately, like this: (3 + 5i) + (4 − 3i) =(3 + 4) + (5 − 3)i 7 + 2i

Polar Form

Let's use 3 + 4i again:
Here it is in polar form:

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

Those two values (distance and angle) have special names:

• Modulus (r): The distance from the origin to the point. For 3 + 4i, the modulus is 5.
• Argument (θ): The angle measured counterclockwise from the positive Real axis. For 3 + 4i, the argument is 0.927 radians (or about 53.1°)

Let's see how to convert from one form to the other using Cartesian to Polar conversion:

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

And "cos θ + i sin θ" is often shortened to "cis θ", so:

x + iy = r cis θ

3 + 4i = 5 cis 0.927

In some subjects, like electronics, "cis" is used a lot!

Distance

The distance between two complex numbers, z₁ and z₂, is the straight-line distance between their points on the complex plane.

To find the distance, we simply calculate the modulus of their difference:

Why it works: when we subtract z₂ from z₁, we get a new complex number that is the horizontal and vertical gaps between them. Taking the modulus gives us the straight-line distance.

Midpoint

The midpoint between two complex numbers is the exact center point between them. To find it, we simply calculate the average of the two numbers:

Why it works: Adding the two complex numbers and dividing by 2 automatically averages their Real parts (horizontal axis) and Imaginary parts (vertical axis) at the same time.

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 )
• Polar form r cos θ + i r sin θ is often shortened to r cis θ

Next ... learn about Complex Number Multiplication.