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can someone explain imaginary numbers for me please

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2 Answers

With a "real" number you can square a number and get another number.
    + It will always be positive.
    + And you can take the square root of that 'new' other number
          and you will get the original number back.
"Imaginary" numbers are numbers that when squared give a another number.
    + Except the result is a negative number.
          And according to the rules we have learned for arithmetic you
           cannot take the square root of a negative number.
   + With the rules we know you cannot get the original number back.
 
So we introduce the "Idea" of an imaginary number.
    + With imaginary numbers you can get the original number back.
    +  Because the square of an imaginably number is a negative.
          (not possible under our old rules)
 
So now you can take the square root of a negative number.
    This idea was needed to do some kinds of problems.
Heron of Alexandria (a real old Greek)
    was the first to think of the kind of problem that needed this
    and then he came up with this idea to solve his problem.
 
You have to remember that "real" numbers do not really exists either.
    + They are only ideas that exist in our minds to help us do things
         like count, solve problems, keep track or how rich or poor we are
         help us describe how much of something we have, and solve problems.
So these new "imaginary" numbers don't really exist either.
   + They are just ideas we have to help us solve problems.
 
Old "real" numbers helped us count on a line - a number line.
These new "imaginary" numbers help us count too.
 
Imaginary numbers help us count on a plane.
    We use the X and Y axis to talk about the distance on
    the real number line (X axis) and the distance on
    the imaginary number line (Y axis).
With imaginary numbers we can count in two directions at the same time
      and we can keep track of two things at the same time.
  + Vectors are a good example if you know what they are.
  + Forces, where we keep track of both the size and  direction of the force.
  + Waves, where we keep track of both the size of the wave
        and where in time on the wave we are.
  + Electrical engineers use them All the time to talk about waves;
        like sound waves, light waves, radio waves, electric waves, and more.
 
Another way I describe it is not a real mathematicians way.
But it is fun to think of.
It goes like this:
    Think of an imaginary number like the head on your Root Beer.
    It used to be root beer, and it might be root beer again sometime later,
    but it is not root beer right now, it is imaginary root beer.
 
++++++++++++
The best complete answer I have ever seen is by: Clive Newstead.
http://math.stackexchange.com/questions/199676/what-are-imaginary-numbers/199688#199688
On Stackexchange
Yes, Martii, imaginary numbers can be confusing.  Let's see what we can do.  You should have learned that you can't take the square root of a negative number, since no number times itself gives you a negative result.
 
This is where imaginary numbers come in.  i= the square root of -1.  In other words, i x i= -1.  This lets us take, for example, the square root of -9, which would be 3i.  The square root of -100 would be 10i.
 
To work it the other way, 2i x 2i = -4.  12i x 12i = -144.
 
I hope this makes some sense!