2.4 Complex Numbers

Definition of a complex number: If a and b are real numbers, the number  is a complex number, and it is said to be written in standard form. if , the number


is a real number. If b cannot equal zero, the number

is called an imaginary number. A number of the form


, where b cannot equal zero is called a pure imaginary number


Standard Form:





For Example: Write .

First Step is to expand the product and simplify:



The final step is to write the result in standard form:

Multiplying Complex Numbers:

For example:

The first step you would do is to expand the product and simplify. The next and final step is to combine the real parts and imaginary parts, and then write the answer in standard form.
 

Dividing Complex Numbers: 

Lets divide the following 2 complex numbers:



Step 1) Determine the conjugate of the denominator:

The conjugate of  is 

Step 2) Multiply the numerator and denominator by the conjugate:

Step 3) Simplify:






Adding and Subtracting Complex Numbers:

For example:




The first step is to rewrite the expression without parenthesis. Then, combine the real parts and imaginary parts, and write it in standard form.

Finding the Conjugate:

The conjugate of  is 

The conjugate of  is 


Remember: