Imaginary Numbers
You have seen that some quadratic equations have no real number
solutions.
For example, letâ€™s solve this quadratic equation:
First, we write the equation in the form x^{2} = a.
Next, we use the Square Root Property x
to write two equations: 
x^{2} + 1

= 0 = 1
or 

The solutions,
and
, are not real numbers because there is no
real number whose square is 1.
In order to solve an equation such as x^{2} + 1 = 0, mathematicians defined
a new number, which they represented with the letter i.
Definition  i
The number i is defined as follows:
That is,
i^{2} = 1.
The number i is not a real number. Instead, i is an example of an
imaginary number.
Given the definition of i, we can write the solutions of x^{2} + 1 = 0 as follows:
We check the solutions by replacing x with i or with i in the original
equation:

Check x = i 

Check x = i 
Is
Is
Is 
x^{2} + 1
(i)^{2 }+ 1
1 + 1
0 
= 0 ? = 0 ?
= 0 ?
= 0 ? Yes 
Is
Is
Is
Is 
x^{2} + 1
(i)^{2 }+ 1
i^{2} + 1
1 + 1
0 
= 0 ? = 0 ?
= 0 ?
= 0 ?
= 0 ? Yes 
We can use an imaginary number to rewrite the square root of a negative
number.
Definition â€” Square Root of a Negative Number
If k is a positive real number, then
We can also write the i in front of the radical, like this:
Examples:
Note:
In an expression such as
be sure to
write the i outside the radical symbol.
Example
Simplify:
Solution 

Rewrite
using


Simplify


So,
