### 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.