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So far, in your math career, you've been working with real numbers.  (Even though some of your answers on math tests have been unreal.  haha!)

All the guys that appear on the number line are real numbers (in the real number system):

all numbers on the number line belong to the real number system

You've probably been told (even by me) that you couldn't do sqrt( -4 )  yet because of that negative sign.  The reason is that the answer isn't a real number.  Yes, there IS an answer!  (Try to calm down -- I know this is terribly exciting.)  There is an answer...  and it's imaginary.  No, I'm not making this up.

Here it is:

 

sqrt( 4 ) = 2 ... sqrt( -4 ) = 2i

 

That's an "i" for imaginary number.

So, what's the i?

i = sqrt( -1 )

For the imaginary number system, we just define it this way:

i = sqrt( -1 )