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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 all hypercomplex number systems that 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 of Minkowski spacetime (whose metric is isomorphic to the metric of 4-dimensional split numbers).

As such, a question emerges, 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 physical axis determined by the laws of algebra?

Anixx
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  • Have you read this? https://physics.stackexchange.com/q/43630/ – DanielC Apr 24 '21 at 23:33
  • @DanielC it seems, they consider any dimension with positive signature a time dimension. Which is wrong (in Newtonian spacetime, for instance, all dimensions have positive metric signature). My feeling is, what makes a dimension a time dimension is not metric signature, but fundamental anisotropy that only real axis has. – Anixx Apr 25 '21 at 02:21
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    Interesting thought. You are right that the time dimension seems different to the other three. In 4D universe that is rotating about an axis, 3 could change and one stay the same, perhaps the time axis is the one that our universe is rotating around! There has also been talk of imaginary time https://en.wikipedia.org/wiki/Imaginary_time that might be of interest. – John Hunter Apr 25 '21 at 08:35
  • @JohnHunter what other things come to mind... We can make more-than-4-dimensional hypercomplex numbers, but in all cases the properties of an axis are determined by the sign of its metric signature (or the sign of square of its basis element). So, the axes would be equal up to isomorphism if those things coincide: for instance, if you take tessarines, the dimension 2 and 4 are isomorphic (both complex), while dimension 3 is hyperbolic – Anixx Apr 25 '21 at 08:44
  • @JohnHunter if you take quaternions, you have 3 complex dimensions. If you take split (hypebolic) numbers, you have 3 split dimensions, if you take higher dimensional hypercomplex numbers, you either have complex (signature positive), hyperbolic (signature negative) or dual (signature zero) dimensions. The signature (the sign before square when we find modulus) has the opposite sign to the square of the basis element. But this is not the case for reals! They have positive signature but also positive square of unity! In this they are unique and have no isomorphic hypercomplex axes! – Anixx Apr 25 '21 at 08:49
  • @JohnHunter only reals have multiplicative unity, which makes them unique. No other axis is isomorphic to reals ever! – Anixx Apr 25 '21 at 08:50
  • As you say, there is something unique about the reals, whether it's linked to the time dimension?...perhaps someone with a good knowledge in this area will do an answer, if not could be worth doing a bounty on it – John Hunter Apr 25 '21 at 09:14

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