Section 2.4: Complex Numbers

Certain quadratic equations have no real solutions. This is why they created the imaginary unit i which is defined as  where . A set of complex numbers is created by adding real numbers to real multiples of the imaginary unit. When a complex number is written  it is in standard form. 

Addition and Subtraction of Complex Numbers: 
If  and   are two complex numbers in standard form then the definition of addition and subtraction is:

Sum:

Example: 

Difference: 

Example:

Multiplying and Dividing Complex Numbers:
In order to multiply complex numbers in standard form, it is important to use the distributive property or also known as the FOIL method.

Multiplication:

Example:

The equation below is not in standard form:

Because   we can use that to change the equation to standard form

Final answer in standard form:

In division, in order to not have i in the denominator of the equation we use the complex conjugate (conjugate of the denominator) to solve the equation by multiplying it to the numerator and denominator of the original equation.

Division:

Example: 

Multiplying by the complex conjugate:

Answer not in standard form yet

 Final answer in standard form:

Imaginary Number Helpful Facts:

     

The answers of the equations come in full circle after four. This helps if you need to solve a problem like: 

 Because the answers rotate every four, divide the number by 4

 The remainder will tell you the answer of the question.

Because  and the remainder of the equation is 3: