Complex Plane

complex plane (flying type) No, not that complex plane ...
... this complex plane:

complex plane (math type)

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:

A Real Number is the type of number we use every day.

Examples: 12.38, ½, 0, −2000

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 does not work because multiplying negatives gives a positive: (−1) × (−1) = +1, 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

An Imaginary Number, when squared gives a negative result

imaginary squared is negative.

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

Putting a Complex on a Plane

You may be familiar with the number line:

number line -10 to +10

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 :

It is placed

  • 3 units along (the real axis),
  • and 4 units up (the imaginary axis).
  complex plane 3+4i

And here is 4 - 2i :

It is placed

  • 4 units along (the real axis),
  • and 2 units down (the imaginary axis).
  complex plane 4-2i


And that is the complex plane:

Whole New World

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

Complex Number as a Vector

We can think of a complex number as a vector.

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

And here is the complex number 3 + 4i

as a Vector:

  complex plane 3+4i vector


You 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
  complex plane vector addition

Polar Form

Let's use 3 + 4i again:   complex plane 3+4i vector

Here it is in polar form:

  complex plane 3-4i is polar 5 at 0.927

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

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

Example: the number 3 + 4i

From 3 + 4i :

And we get distance (5) and angle (0.927 radians)

Back again:

And distance 5 and angle 0.927 becomes 3 and 4 again

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" is often shortened to "cis θ", so:

x + iy = r cis θ

cis is just shorthand for cos θ + i sin θ

So we can write:

3 + 4i = 5 cis 0.927

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


Next ... learn about Complex Number Multiplication.