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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:


i1=i  
i2=()2 = - 1  
i3=i2i = - 1(i) = - i  
i4=i3i = - i(i) = - i2 = - (- 1) = 1  
i5=i4i = 1(i) = i  
i6=i5i = - 1  
i7=i6i = - i  
i8=i7i = 1  
i9=i  
  ...    

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


Examples:

  1. What is i54?
    54÷4 = 13R2.
    Thus, i54 = - 1.
  2. What is i103?
    103÷4 = 25R3.
    Thus, i103 = - i.

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