Complex Numbers
Imaginary Numbers
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:
- Simplify
.
= 
·
= i· 
= i·4· 
= 4i .
- Simplify
.
= ·
= i·10 = 10i.
- 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:
- What is
i
54
?
54÷4 = 13R2 .
Thus, i 54 = - 1 . - What is
i
103
?
103÷4 = 25R3 .
Thus, i 103 = - i .
