Have you ever had an imaginary pet? Many people have, particularly as young children.
Wouldn't you be surprised if you and your real friend left your imaginary pet dogs alone together, and you came back to find a real puppy?
Silly thought, what does it have to do with imaginary numbers?
Watch This
'Note:' For a very detailed explanation of i and the complex numbers, watch the video below, then visit: http://betterexplained.com/articles/avisualintuitiveguidetoimaginarynumbers/
Embedded Video:
 Khan Academy: Introduction to i and imaginary numbers
Guidance
What is the square root of 1?
You may recall running into roots of negatives in Algebra, when attempting to solve equations like:
The definition of "i" :
The use of the word imaginary does not mean these numbers are useless. For a long period in the history of mathematics, it was thought that the square root of a negative number was in fact only within the mathematical imagination, without realworld significance hence, imaginary. That has changed. Mathematicians now consider the imaginary number as another set of numbers that have real significance, but do not fit on what is called the number line, and engineers, scientists, and others solve real world problems using combinations of real and imaginary numbers (called complex numbers) every day.
Imaginary values such as
The uses of i become more apparent when you begin working with increased powers of i, as you will see in the examples below.
Complex numbers
When you combine i's with real numbers, you get complex numbers:
The definition of complex numbers: Complex numbers are of the form
Example A
Simplify
Solutions
Example B
Simplify
Solution
But, we’re not done yet! Since
Example C
Strange things happen when the imaginary constant i is multiplied by itself different numbers of times.
 a) What is i^{2}?
 b) What is i^{3}?
 c) What is i^{4}?
Solutions

a)
∴i2=−1

b)
∴i3=−i

c)
∴i4=1
Concept question wrap up Do you see the application of the crazy analogy from the introduction? Two imaginary pets creating a real puppy is an oddly effective metaphor for the behavior of the powers of i. One i is imaginary, but two i's multiply to be a real number. In fact, every even power of i results in a real number! 

Vocabulary
The imaginary constant i is the square root of 1.
A complex number is the sum of a real number and an imaginary number, written in the form:
Simplifying the radical involves factoring the term(s) under the root symbol, so that perfect squares may be "pulled out", and simplified.
Guided Practice
Simplify the following radicals
1)
2)
3)
4)
Multiply the imaginary numbers
5)
6
7)
Solutions
1) To simplify the radical

9⋅−1−−−−−√ : Rewrite−9 as−1⋅9 
9√⋅−1−−−√ : Rewrite as a product of radicals 
9√⋅i : Substitute−1−−−√→i 
3i : Simplify9√
2) To simplify the radical

12⋅−1−−−−−−√ : Rewrite−12 as−1⋅12 
12−−√⋅−1−−−√ : Rewrite as a product of radicals 
12−−√⋅i : Substitute−1−−−√→i 
3⋅4−−−√⋅i : Factor 12 
23√⋅i : Simplify4√
3) To simplify the radical

17⋅−1−−−−−−√ : Rewrite−17 as−1⋅17 
17−−√⋅−1−−−√ : Rewrite as a product of radicals 
17−−√⋅i : Substitute−1−−−√→i 
17−−√i : Simplify
4) To simplify the radical

−32−−−−√ : Subtract within the parenthesis 
32⋅−1−−−−−−√ : Rewrite−32 as−1⋅32 
32−−√⋅−1−−−√ : Rewrite as a product of radicals 
32−−√⋅i : Substitute−1−−−√→i 
16⋅2−−−−√⋅i : Factor32 
4i2√ : Simplify16−−√
5) To multiply

4⋅3⋅i⋅i : Using the commutative law for multiplication 
12⋅i2 : Simplify 
12⋅−1 : Recalli2=−1 
−12
6) To multiply

4i⋅3 : Simplify16−−√ 
12i
7) To multiply

4√⋅i2−−√⋅4√⋅3√⋅i : Factor 
2⋅i⋅2⋅3√⋅i : Simplify the roots 
43√⋅i2 : Collect terms and simplify 
43√⋅−1 : Recalli2=−1 
−43√
Practice
Simplify:

−49−−−−√ 
−81−−−−√ 
−324−−−−√ 
−121−−−−√ 
−−16−−−−√ 
−−1−−−√ 
−1.21−−−−−√
Simplify:

i8 
i12 
i3 
24i20 
i225 
i1024
Multiply:

i4⋅i11 
5i6⋅5i8 
3−75−−−−√⋅5−3−−−√ 
2−12−−−−√⋅6−27−−−−√ 
−4−10−−−−√⋅5−3−−−√⋅6−18−−−−√