Welcome to a very special city named, Complex City.  This is special because all the occupants of this place are numbers.  There are the Reals which occupy the south and east of the city and  in the north and west are the Imaginary numbers.  My name is Counstable Squared, and I am in charge of keeping the peace in this city.  It's a huge job, but some number has to do it.  I'm here to tell you that we have had our first larceny in our community, and it has made all the residents fearful of their existence.  If it could happen to cute little "Square root negative two", it could happen to any of us.
 
 
 

 Your Task

Here's how the larceny occurred.  Square root negative two had a date for a multiplication and an addition problem with Square root negative eight.  She was picked up on time and they proceeded to the addition first.  The two of them started to add and bingo, that's when it happened.  Someone or something stole the negative from little two and left an "i" in it's place.  As Counstable of Complex City, I am putting you in charge of finding the culprit of this heinous crime.  Little Square root negative two will never be the same without her negative.  Below is a list of questions to be answered by you so that I may explain to Square root negative two what actually happened.
What does "i" mean and what is it's history?
Where did the negative in front of Square root negative two go? Explain.
Is she the same now with the "i"? Explain.
Can she ever add with Square root negative eight? Explain.
Can she ever multiply with Square root negative eight? Explain.
Now that you know the facts, you must join the investigation and solve this problem.  First follow the links below to investigate all the fine aspects of this case.  You will be required to give me a  word processed document answering the questions above.
 

Continue your investigation!