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.
Best Answer
There is some history behind this.
In the beginning there was ${\mathbb N}=\{1,2,3,\ldots\}$, with $+$, $\cdot$, and $<\>$.
Certain simple equations of the form $a+x=b$ couldn't be solved in ${\mathbb N}$, so they invented ${\mathbb Z}$ containing also $0$ and the negative numbers.
Certain simple equations of the form $ax+b=c$ couldn't be solved in ${\mathbb Z}$, so they invented ${\mathbb Q}$.
Certain simple equations like $x^2=2$ couldn't be solved in ${\mathbb Q}$, nor was there a representant for the area of a unit disk. So they invented ${\mathbb R}$.
Certain simple equations like $x^2+1=0$ couldn't be solved in ${\mathbb R}$, so they invented ${\mathbb C}$, the system of complex numbers. Each complex number can be written in the form $x\>1+y\>i$ with real $x$, $y$ and a special complex number called $i$.
Hamilton then tried in vein to set up a "hypercomplex" number system where each "number" would be of the form $x\>\vec i+y\>\vec j+z\>\vec k$, where $\vec i$, $\vec j$, $\vec k$ are the basis vectors used in elementary vector algebra of ${\mathbb R}^3$. He didn't succeed, but he realized that such a system is possible when the individual hypercomplex numbers are of the form $t\>1+x\>\vec i+y\>\vec j+z\>\vec k$ with $t$, $x$, $y$, $z$ real, and if the operations $+$ and $\cdot$ are appropriately defined. In this way the first true hypercomplex number system, called the quaternions, was born. Apart from the commutativity of multiplication all "rules of algebra" are valid in this system.
It is then natural to ask, for which dimensions $n$ apart from $1$, $2$, $4$ such a system $S$ with "numbers" $\sum_{k=1}^n x_k e_k$, where $\>x_k\in{\mathbb R}$ and the $e_k$ are certain special numbers of $S$, can be set up such that one has an addition and a "reasonable" multiplication in $S$. It is one of the deep theorems of $20^{\rm th}$ century mathematics that there is just one more such system, the Cayley octonions with $n=8$; but associativity of multiplication is no longer present in this system.