Imaginary Numbers
You have seen that some quadratic equations have no real number
solutions.
For example, let’s solve this quadratic equation:
First, we write the equation in the form x2 = a.
Next, we use the Square Root Property x
to write two equations: |
x2 + 1
![](./articles_imgs/165/algebr1.gif) |
= 0 = -1
or |
![](./articles_imgs/165/algebr2.gif) |
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 x2 + 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,
i2 = -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 x2 + 1 = 0 as follows:
![](./articles_imgs/165/algebr6.gif)
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 |
x2 + 1
(i)2 + 1
-1 + 1
0 |
= 0 ? = 0 ?
= 0 ?
= 0 ? Yes |
Is
Is
Is
Is |
x2 + 1
(-i)2 + 1
i2 + 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 — Square Root of a Negative Number
If k is a positive real number, then
![](./articles_imgs/165/algebr7.gif)
We can also write the i in front of the radical, like this:
![](./articles_imgs/165/algebr8.gif)
Examples:
![](./articles_imgs/165/algebr9.gif)
Note:
In an expression such as
be sure to
write the i outside the radical symbol.
Example
Simplify:
![](./articles_imgs/165/algebr11.gif)
Solution |
![](./articles_imgs/165/algebr11.gif) |
Rewrite
using
![](./articles_imgs/165/algebr12.gif) |
![](./articles_imgs/165/algebr13.gif) |
Simplify
![](./articles_imgs/165/algebr14.gif) |
![](./articles_imgs/165/algebr15.gif) |
So,
![](./articles_imgs/165/algebr16.gif)
|