Imaginary Numbers
Definition

A number that when squared (multiplied by itself) gives a negative result. 
Try
Let's try squaring some numbers to see if we can get a negative result:
No luck! Always positive, or zero.
It seems like you cannot multiply a number by itself to get a negative answer ...

... but imagine that there is such a number (call it i for imaginary) that could do this:
Would it be useful, and what could we do with it? 
Well, by taking the square root of both sides we get this:
Which means that i is the answer to the square root of 1.
And that is very useful ... by simply accepting that i exists we can solve things where we need to take the square root of a negative number.
Example: what is the square root of 9 ?
Answer: √(9) = √(9 × 1) = √(9) × √(1) = 3 × √(1) = 3i
So long as we keep that little "i" there to remind us that we still
need to multiply by √1 we are safe to continue with our solution!

Unit Imaginary Number
The "unit" Imaginary Number (the equivalent of 1 for Real Numbers) is √(1) (the square root of minus one).
In mathematics we use i (for imaginary) but in electronics they use j (because "i" already means current, and the next letter after i is j). 
Examples of Imaginary Numbers
i 
12.38i 
i 
3i/4 
0.01i 
i/2 
Imaginary Numbers are not "Imaginary"
Actually Imaginary Numbers were once thought to be impossible, and so they were called "Imaginary" (to make fun of them).
But then people researched them more and discovered they were actually useful and important becaue they filled a gap in mathematics ... but the "imaginary" name has stuck.
And that is also how the name "Real Numbers" came about (real is not imaginary).
Useful
Here are 2 cases where they are useful:
Electricity


AC (Alternating Current) Electricity changes between positive and negative in a sine wave.
If you combine two AC currents they may not match properly, and it can be very hard to figure out the new current.
But using imaginary numbers and real numbers together makes it a lot easier to do the calculations. 


And the result may be "Imaginary" current, but it could still hurt you! 
Quadratic Equation

The Quadratic Equation can give results that include imaginary numbers ... 
... and maybe after more calculations the i value gets cancelled out (or become real numbers because they get squared), leaving an answer that is all real. 
Interesting Property
The Unit Imaginary Number, i, has an interesting property. It "cycles" through 4 different values each time you multiply:

So:
 i × i = 1,
 then 1 × i = i,
 then i × i = 1,
 then 1 × i = i (back to i again!)

Conclusion

The unit imaginary number, i, equals the square root of minus 1
Imaginary Numbers are not "imaginary", they are real and useful, and you may need to use them one day! 
