Complex Numbers

Imaginary Numbers

Terms

Problems

Imaginary Numbers

Until now, we have been dealing with real numbers. We have not been able to take the square root of a negative number because the square root of a negative number is not a real number. Instead, the square root of a negative number is an imaginary number--a number of the form , where k < 0 . Imaginary numbers are represented as ki , where i = . For example, = 5i and = i .

We can simplify square roots of negative numbers by factoring out = i and simplifying the resulting root.


Examples:

  1. Simplify .


    = ·  
      = i·  
      = i·4·  
      = 4i .  

  2. Simplify .


    = ·  
      = i·10  
      = 10i.  

  3. Simplify .


    = ·  
      = i·  
      = i·5·  
      = 5i .  

Observe the following:


i 1 = i  
i 2 = ()2 = - 1  
i 3 = i 2 i = - 1(i) = - i  
i 4 = i 3 i = - i(i) = - i 2 = - (- 1) = 1  
i 5 = i 4 i = 1(i) = i  
i 6 = i 5 i = - 1  
i 7 = i 6 i = - i  
i 8 = i 7 i = 1  
i 9 = i  
  ...    

Thus, we can find i n using the following:
  • If n÷4 leaves a remainder of 1, i n = i .
  • If n÷4 leaves a remainder of 2, i n = - 1 .
  • If n÷4 leaves a remainder of 3, i n = - i .
  • If n÷4 leaves no remainder, i n = 1 .


Examples:

  1. What is i 54 ?
    54÷4 = 13R2 .
    Thus, i 54 = - 1 .
  2. What is i 103 ?
    103÷4 = 25R3 .
    Thus, i 103 = - i .