# Are $\mathbb{C}$ and $\mathbb{H}$ the only continuous number systems over $\mathbb{R}^{2}$ and $\mathbb{R}^{4}$ respectively?

When I say $$\mathbb{C}$$ is a number system over $$\mathbb{R}^{2}$$, I mean that the field of complex numbers is identified with the set of dilative rotations of the affine real plane ($$\mathbb{R}^{2}$$) characterized by the Pythagorean metric. When I say $$\mathbb{H}$$ is a number system over $$\mathbb{R}^{4}$$, I mean that the skew-field of quaterions is identified with the set of dilative Hermitian rotations over the affine complex plane $$\mathbb{C}^{2}$$ characterized by the Hermitian metric. When I say these are continuous, I mean that they inherit the continuum properties of the real numbers.

It occurs to me that Minkowski space is a geometry over $$\mathbb{R}^{4}$$ characterized by the Minkowski metric. But the continuum of spacetime events does not appear to constitute a number system. Unlike the complex numbers and the quaternions, the Lorentz transformations and space points (spacetime events) are "decoupled".

So I ask, is there some theorem or uncontradicted proposition that says the only continuous number systems expressible as ordered pairs and ordered quadruples of real numbers are the complex numbers and quaternions, respectively?

• See the Cayley-Dickson construction. – WimC Jan 31 at 5:33
• So, I guess the answer is:No. There are, in fact counter examples. However... the counterexamples appear to be variants of the aforementioned number systems (or algebras). – Steven Hatton Jan 31 at 6:05
• If looking at unital associative real algebras without zero-divisors, then $\mathbb{C}$ is the only one of dimension $2$ and $\mathbb{H}$ should be the only one of dimension $4$ (look at $\mathbb{R}[z]$ and the minimal polynomial $f$ of $z$ for various non-real elements $z$ in the algebra, $\mathbb{R}[z]\cong \mathbb{R}[X]/(f(X))$ is commutative with no zeros divisors so it is $\cong \mathbb{C}$, then massage it to obtain the algbera of dimension $4$ must be $\mathbb{C}+j\mathbb{C}$ with $jz = \overline{z} j$) – reuns Jan 31 at 6:09
• Would you regard the octonions as a "continuous number system"? en.wikipedia.org/wiki/Octonion – Lord Shark the Unknown Jan 31 at 7:20
• I mean the full matrix ring $M_2(\mathbb R)$, and I am suggesting that by number system you should mean something like a ring, and if you want, you can assume something like it is a finite-dimensional $\mathbb R$-vector space, i.e., a finite-dimensional algebra over $\mathbb R$. – Kimball Feb 2 at 21:06