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.