# Did anyone ever propose a hypercomplex numbers system with more than one anisotropic axis?

The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, natural exponent of real numbers is always positive, etc.

On the other hand, the imaginary axis is totally symmetric against zero: every truth about $$i$$ is also true about $$-i$$ up to isomorphism. The same is true for other axes in hypercomplex number systems I know, including split-complex, dual numbers, bicomplex numbers, tessarines, quaternions, split-quaternions...

I also know that in physical space we also have one anisotropic axis (time) and three symmetric spatial axes. This is true both for classical spacetime or Minkowski spacetime (whose metric is isomorphic to the metric of 4-dimensional split numbers).

As such, a question emerged, whether the real axis is naturally linked to the time dimension, because it is the only non-symmetric axis? Is the existence of no more than one asymmetric axis determined by the laws of mathematics?

Or it is possible to invent a ring over reals that would naturally include more than one asymmetric axis?

Did anyone propose such system?

• This is not special for real numbers. For an arbitrary field $K$, if $a$ in $K$ is not a perfect square (no solution to $b^2 = a$ in $K$) then on the system of numbers $x+y\sqrt{a}$ with $x, y \in K$, the conjugation mapping sending $x+y\sqrt{a}$ to $x - y\sqrt{a}$ preserves addition and multiplication. More generally, if $f(t)$ is irreducible of degree $d$ with coefficients in $K$ and $r$ and $s$ are roots of $f(t)$ (outside of $K$), the system of numbers $a_0 + a_1r + \cdots + a_{d-1}r^{d-1}$ behaves just like the system of numbers $a_0 + a_1s + \cdots + a_{d-1}s^{d-1}$ where $a_i \in K$.
– KCd
Apr 25 '21 at 0:45
• @KCd you said this is not special for real numbers, but the rest of your comment in spirit says that all other hypercomplex axes have symmetry. Apr 25 '21 at 1:33
• @KCd by the way, split-complex and dual numbers are also symmetric, even though they do not satisfy your first condition. Apr 25 '21 at 1:35
• There are no intrinsic "axes" since vector spaces usually don't have a canonical basis: the fields generated over $\mathbf Q$ by $\sqrt{2}$ and by $1+\sqrt{2}$ are the same. What is intrinsic are field automorphisms, which may look simpler in some bases than others (like the $\mathbf R$-basis $\{1,i\}$ instead of $\{2-i,3+5i\}$ for $\mathbf C$), but they are field automorphisms regardless of how we write them in formulas. Quaternion algebras (Hamilton's quaternions are a special case) can be defined in a formula-free way as 4-dimensional central simple algebras, not singling out special axes.
– KCd
Apr 25 '21 at 2:59
• @KCd Well, at least all hypercomplex munbers have one canonical basis element: multiplicative unity. This makes the real axis special. Apr 25 '21 at 3:13

This won't be much of a "historical" answer (although my external links probably have interesting history in them), but hopefully it'll be mathematically useful, at least if construed as a long comment. I'll use Einstein notation throughout, except in blue expressions.

Consider a system of arithmetic on a finite-dimensional vector space over $$\Bbb R$$ (or your favourite subfield, e.g. $$\Bbb Q$$) with basis $$e_i$$ satisfying $$a_ie_i+b_ie_i=(a_i+b_i)e_i$$ and $$a_ie_ib_je_j=u_{ijk}a_ib_je_k$$ for coefficients $$u_{ijk}$$. We can write this as $$ab=c$$ with $$c_k=u_{ijk}a_ib_j$$. Axis $$l$$ exhibits the symmetry you describe iff $$a_i^\prime=\color{blue}{(1-2\delta_{il})a_l}$$ satisfies $$(ab)^\prime=a^\prime b^\prime$$, i.e.$$\color{blue}{[(1-2\delta_{il})(1-2\delta_{jl})-1+2\delta_{kl}])u_{ijk}}=0.$$You've noted the $$u_{ijk}$$ for complex numbers satisfy this condition for $$l=1$$ but not $$l=0$$, where $$e_0:=1,\,e_1:=i$$. @Conifold's point is $$u_{ijk}=\color{blue}{\delta_{ij}\delta_{ik}}$$ satisfies the condition for all $$l$$.

I also know that in physical space we also have one anisotropic axis (time) and three symmetric spatial axes.

I may not be able to walk into the past, but the equations describing nature are (largely) time-symmetric. (If that seems like a contradiction, bear in mind this is a big topic, sadly not addressed by the line of analysis you tried).

a question emerged, whether the real axis is naturally linked to the time dimension

The real mathematical reason time is dissimilar to space is the signs of real eigenvalues of the metric tensor. (There are two conventions regarding which sign time gets; in the linked article, the chosen convention makes time's eigenvalue negative, but I'll use the opposite convention hereafter in explaining the relevance.) Infinitesimal paths have squared length $$ds^2=dt^2-dx^2-dy^2-dz^2$$ in special relativity, if we use units where $$c=1$$, so $$\sqrt{ds^2}$$ is real for paths that don't exceed the speed of light, i.e. are causal. In more general geometries than Minkowski, the details become more complicated, but the signs of metric tensor eigenvalues are unchanged, and effectively allow us to count time and space dimensions.

Is the existence of no more than one asymmetric axis determined by the laws of mathematics?

As noted above, no. As for multiple time dimensions, that's also doable, albeit for unrelated reasons you can guess it in light of the paragraph above.