An introduction to the complex number system

Through your time here at COCC, you’ve existed solely in the real

number system, often represented by a number line.

Just what is a “real” number, anyway? Can you give me an example of a real number?

Just what is a “real” number, anyway? Can you give me an example of a real number?

Just what is a “real” number, anyway? Can you give me an example of a real number?

Just what is a “real” number, anyway? Can you give me an example of a real number?

All of those real numbers (and many, many more) fit into place on the

number line you know and love so well...

But...what lies above and below our beloved number line?

To visualize, mathematicians placed another axis perpendicular to the

real axis...

Look familiar?

The horizontal axis is still

“real”....

Real axis

But...what to call the

vertical axis?

Real axis

Remember, in MTH 065, whenever you found the square root of a

negative number....and just stopped?

What did you write for your solution?

“No Real Number”

Well, the region above and below the real axis is where all of those

“square roots of negative numbers” live.

This axis is called the “imaginary” axis...unfortunately.

“But why, Sean? Why is it called imaginary? Please tell us!”

The “imaginary unit” is cleverly called i and is defined like this:

1i

Real axis

Imaginary axisi is placed right here on our new plane...

...notice that i is not real, so it doesn’t touch the real axis.

Now we have

1i Squaring both sides, we must

also have

2 1i

Real axis

Imaginary axisi2 goes right here...

i2 is not imaginary...it’s real!

Let’s continue the pattern...

1i 2 1i 3 2

( 1)

i i i

i

i

Real axis

Imaginary axisWhere would i3 go?

And, one last example...

4 2 2

( 1) ( 1)

1

i i i

Real axis

Imaginary axis

And so, i4 rejoins the real realm...

Real axis

Imaginary axis

Great...but what about a number that’s over here?

Real axis

Imaginary axis

It’s not oneither axis!

Real axis

Imaginary axis

This type of number is called “complex”; it has both a real and imaginary part.

Real axis

Imaginary axis

Its real coordinate is “– 2” and its

imaginary coordinate is “3”.

Real axis

Imaginary axis

It’s written

– 2 + 3i.

Believe it or not...that last complex number is one of the solutions to our previous quadratic equation!

22( 2) 18x

“Prove it, Rule! We think you’re full of it!”

Real axis

Imaginary axisx = – 2 3i.

Let’s try two more...

2 25 6x x

(3 4) 5x x