TITLE: Explaining Hypercomplex numbers to Children.
QUESTION [9 upvotes]: Imagine a highschool freshman walks up to you and asks you what hypercomplex numbers are.
  Explain to her, in a fair amount of detail, the different types of hypercomplex numbers in a way that any person can understand.

This was something asked to me by one of my friends. I don't know what Hypercomplex numbers are well enough to explain it. Neither does my friend and that's the reason he asked my that question. Wikipedia and other resources have been rather inefficient in delivering even the basic gist of it to us.
We are merely curious to what this extension is basically. Can you please explain it to us?
A simple glance of what it is might suffice.

REPLY [4 votes]: Kids of a young age can understand very easily hypercomplex numbers. Several weeks ago, I demand to my 13 year old boy to find the square root of -1. After some thought, he says that it was impossible. He knew the concept of the real line; I explained then that we can put a line perpendicular to that one and call the numbers on that line the ‘i’ numbers. Then I introduced the idea that multiplying by ‘i’ is a rotation of 90 degrees. When I demanded him to multiply 1 by ‘i’ two times, his eyes staring at me with an admirable joyfulness: he had understood that i^2 was -1.
The last weekend I explained him, and his 11 year old brother, that we can build three different planes in the same fashion. The first, with an ‘i’ letter, is called the complex plane and points in it are ‘complex numbers’. The second, with a perpendicular line labeled with an ‘e’ such that e^2=0 is called dual plane and the points in it are called ‘dual numbers’. Finally we can build a third plane with a Greek letter ‘iota’ on it, such that (iota)^2=1 and points in the plane are called ‘split numbers’. Then, I explained that each point is represented by two numbers, but the rules for multiplying the pairs of numbers are found by the geometry in these planes.
After that, when I mentioned that we could build pairs of pairs, and find rules for adding and multiplying these ‘pairs of pairs’. They were amazed but very interested. All the afternoon we were counting the classes of possible combinations. My 13-yo found that there are 19,683 classes of numbers with 512 real numbers (in pairs). We call all these classes the Quintocento-dodekaions.
p.s.: Two weeks after his first encounter with complex numbers, my son demanded me for the square root of 'i' and with very little effort he found the correct operation to find this number.