Complex Numbers

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

Examples:

A Number is a Combination of Two Numbers?

Can you make up a number from two other numbers? Sure you can! You do it with fractions all the time. The fraction 3/8 is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts". Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number).

Zero

So, a Complex Number has a real and imaginary part.

But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

Adding and Multiplying

To add two complex numbers we add each element separately:

(a,b) + (c,d) = (a+c, b+d)

But to multiply we follow a more interesting rule:

(a,b)(c,d) = (ac-bd, ad+bc)

You can try it yourself: enter (3 + 2i)(1 + 7i) into the Complex Number Calculator.

And an interesting thing is that squaring "i" does indeed produce -1

Imaginary Numbers Exist!

That is a great argument for the existence of the Imaginary Numbers:

When you square the Complex Number 0+i you get -1

So you can square a number to get -1 ... if you use the rules of Complex Numbers