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.
= · = i· = i·4· = 4i .
= · = i·10 = 10i.
= · = i· = i·5· = 5i .
Observe the following:
|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|
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 .
- What is
54÷4 = 13R2 .
Thus, i 54 = - 1 .
- What is
103÷4 = 25R3 .
Thus, i 103 = - i .