What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

Question

In this article that talks about some history of hamilton

http://plus.maths.org/content/curious-quaternions

There is a snippet that says this:

Multiplication is very sneaky. You can only set up rules for multiplication that let you divide in dimensions 1, 2, 4 and 8. This is just a mysterious fact about the universe. Well, if you study maths it's not mysterious because you can see exactly why, but it's mysterious in the sense that when you hear about it first it just sounds completely crazy!

What are they talking about here? What type of division is possible in 2 dimensions but not 3? Could you give me an example of the division in two dimensions?

Answer 1

The complex number $a+bi$ can be identified with the ordered pair $(a,b)$ of reals. So the set $\mathbb{C}$ of complex numbers (these include the reals) can be identified with $\mathbb{R}^2$, the $2$-dimensional space over the reals.

And we can indeed divide a complex number $z$ by a non-zero complex number $w$, and obtain a complex number. The division formula is fairly simple: to calculate $\frac{a+bi}{c+di}$, multiply top and bottom by $c-di$. After a little work we arrive at $\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i$.

This division turns out to have most of the same nice formal properties as ordinary division of real numbers.

Remark: Already if we go on to $4$, we lose some important properties shared by the reals and the complex numbers. For as you know from the article, multiplication in the quaternions is not commutative. The situation gets even worse at $n=8$: in the octonions, multiplication is not associative.

Answer 2

They are talking about the reals, complex numbers, quaternions, and octonions as normed division algebras over the reals. I already have a long missive here but basically, in keeping reals, complex, quaternions, and octonions in that order, any algebra contains all of the previous algebras. Each algebra is the extension of the previous one in a MEANINGFUL way. Specifically multiplication and division from the previous algebra is preserved in the extended algebra AND we can solve more equations in the new extended algebra than we could before. Reals and Complex had been around for a while but Hamilton with an epiphany (after struggling for a while) figured out how to make it work with quaternions and then soon afterwards Graves and Cayley discovered Octonions (independently). It turned out that Octonions are the last in this chain. There is no way to extend them to something 16-th dimensional.

Cool fact, for each of these extensions, the ability to multiply and divide a larger set of numbers and to solve many more equations than we could before, we do have to pay a price. For example multiplication in octonions, multiplication is not commutative and it isn't associative but we can solve equations like

$$xy-yx=1.$$