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# 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 x2 = a. Next, we use the Square Root Property x to write two equations: x2 + 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 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:

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 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,

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