### Non-Mathematician Rediscovers R^{n}, C^{n}

#### Posted by John Baez

I did a lot of historical reading about Hamilton, quaternions and the like when writing my review article about the octonions. When Hamilton invented the quaternions in 1843, it was a big deal: people hadn’t realized you could just *make up* new rules for multiplication. A lot of people joined in the fun, inventing their own hypercomplex number systems. Most of these systems aren’t terribly interesting. Hamilton’s pal John T. Graves found one of the really special ones: the octonions. Clifford found a whole bunch. But in general, most mathematicians now prefer to study hypercomplex number systems *en masse* rather than individually. They’re now called “real algebras”, and there are lots of nice general theorems about them.

This does not prevent amateurs from continuing to invent hypercomplex number systems and become excited about them. An amusing example was brought to my attention by David Farrell:

- Rodney Rawlings, Non-mathematician devises hypercomplex numbers, with possible implications for mathematics and the philosophy of science.

Brace yourself: Rawlings links his discovery to the Objectivist philosophy of Ayn Rand!

A few quotes from the above ‘press release’ (apparently written by Rawylings himself) will convey the spirit of the thing:

Toronto ON, Canada, December 25, 2007 — A Toronto, Ontario, writer and editor has arrived at a system of creating hypercomplex numbers — numbers that extend the complex number system to more dimensions — using only high school algebra, as viewed through the lens of Ayn Rand’s philosophy of Objectivism. He contends that this has implications for mathematics and the philosophy of science.

Rodney Rawlings calls his multidimensional numbers “RADN numbers” — for “rotating any-dimensional numbers,” because they have a property of rotation exactly analogous to that of the complex numbers. They are also commutative and associative like them.

He says that he arrived at this result by asking himself what exactly numbers are, how they arise in the human mind, and what their relationship to reality is. But these questions were only so fruitful because he used a correct philosophy, he claims — Ayn Rand’s. Any other philosophy, such as the currently influential one of Karl Popper, he says, would not have led to such a result. “This has two implications: first, that Rand’s philosophy has a strong element of truth, at least in the area of epistemology; and second, that the type of numbers I discovered must have a special significance, seeing as how they are intimately related to the basic nature of numbers.”

[…]

Rawlings contends that the RADN program (which, he hastens to add, is not a new one but already known to mathematicians under a different name) must have a unique status among the hypercomplexes, because of the way he, a non-mathematician, arrived at them by means of extremely simple algebra absent any of the tools of modern analysis, but armed with a philosophy that takes a particular and unconventional view of the nature of concepts and of mathematics.

Accordingly, Rawlings decided to write up his thoughts and reasoning in an essay entitled “Understanding Imaginaries Through Hidden Numbers,” which he is currently offering at a low price…

Perhaps Objectivism encourages scientists to offer their findings only to people willing to pay for them. Luckily, we don’t need to buy Rawling’s book to discover which algebra he has reinvented, since elsewhere he has admitted that they’re the multicomplex numbers.

I hadn’t heard this term before! It turns out the **multicomplex numbers** $\mathbb{MC}_n$ are the associative real algebra freely generated by an $n$th root of $-1$. When $n$ is even, this algebra is just $\mathbb{C}^{n/2}$: a direct sum of $n/2$ copies of the complex numbers. When $n$ is odd, it’s $\mathbb{R}^n$: a direct sum of $n$ copies of the real numbers.

To anyone familiar with mathematics, these algebras are the mathematical equivalent of white paint or the C major scale: very fundamental, very important, very thoroughly worked over, very familiar and bland, very hard to say anything new about. But, to anyone first discovering them, they must seem incredibly exciting!

I offer this item mainly as further evidence for the powerful hold mathematics has on our imagination. It draws us with irresistible force towards the simplest and most beautiful patterns. We can’t resist it. These patterns will be discovered over and over, across the universe, for as long as intelligent life ekes out an existence in any corner. Let’s enjoy it!

Puzzle: what’s the slickest, most elegant proof that the hypercomplex numbers are isomorphic to the algebras I described: $\mathbb{R}^n$ and $\mathbb{C}^{n/2}$?

## Re: Non-Mathematician Rediscovers Rn, Cn

Hang on – are you sure that the multicomplexes are isomorphic to $\mathbb{R}^n$ in the odd case? I would have thought $\mathbb{R} \times \mathbb{C}^{(n-1)/2}$.

I don’t know about simplest and most elegant, but off the cuff I’d factor $x^n + 1$ into irreducible polynomials over $\mathbb{R}$ and use the Chinese remainder theorem. I’m feeling a little lazy about spelling it all out… and I hope someone will come up with something slicker.