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# Complex Numbers

## Contents

#### 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 .