2.4 Complex Numbers

Some quadratic equations have no real solutions.

For example:   has no real solution because there is no real number x that can be squared to produce -1.

So, mathematicians created an expanded system of numbers using the imaginary unit i.

Imaginary Unit:  where .

By adding real numbers to this imaginary unit, you obtain a set of complex numbers.

          If a and b are real numbers, the number a + bi is a complex number.
       
          Standard form of a complex number is a + bi, where a is the real part of the complex
          number and the number bi (where b is a real number) is called the imaginary part.

          

          If  , the number a  + bi is called an imaginary number. 

Pictured below are the categories of complex numbers. 

Not pictured: Imaginary numbers are outside the domain of "real" but still inside "complex" 

Real Numbers: Rational or irrational numbers 

Irrational Numbers: Any number that cannot be expressed as a ratio of two integers 

Rational Numbers: Recognized as terminating/repeating decimals; Defined as able to be written as a ratio of two integers                                  

Integers: Positive and negative whole numbers 

Whole Numbers: All the integers, including 0 

Natural Numbers: The positive integers 

Equality of Complex Numbers 

Two complex numbers a + bi and c + di, written in standard form, are equal to each other if and only if a = c and b = d. 

Addition and Subtraction of Complex Numbers

Sum:

          

          Example: 

Difference: 

          Example: 

The additive identity in a complex number system is zero. 

          The additive inverse of the complex number a + bi is -(a + bi) = -a -bi. 

Multiplication of Complex Numbers 

Use a similar procedure to multiplying two polynomials and combining like terms (FOIL method). 

          

          Example: 

                                                       

                                                       

                                                       

Division of Complex Numbers 

To rid an equation of its complex denominator: use complex conjugates. 

(This is called realizing the denominator).

          The conjugate of  is .

          Example: 

                                        

                                       

                                       

                                         
 Don't forget to write your final answer in standard form! 

Tips about i 

    

  

      

      

     
      
      
      

And so on.....

To determine the value of i to any power, use long division, and whatever the remainder is, you look to the set of the first four values of i and choose accordingly.