Construction of the Transreal Numbers and Algebraic Transfields [pdf]

Briefly, this is very similar to nonstandard analysis and the hyperreal number system. The paper appears to follow along the trails considered by NSA and HR textbooks previously. Those works are fully rigorous, so this paper seems a little redundant in the logic portion.

I even attempted to code those systems in C/python/java.

So, essentially the transreal numbers treat NaN as a "value"?

"The main difference between transreal arithmetic and IEEE floating-point arithmetic is thus that nullity compares equal to nullity, whereas NaN does not compare equal to NaN."

https://en.wikipedia.org/wiki/James_A._D._W._Anderson

I'm not 100% comfortable with the idea that n/0 retains the sign of n, 0 is neutral - not negative or positive. Although a division by a positive infinitesimal would retain the sign of the numerator, with zero I'd expect a result that was either both positive and negative, or neither.

As far as I got into the paper, the proofs seemed somewhat circular and no significant value (from a mathematical standpoint) was covered.

The comparison to the use of a negative square root was unreasonable - since even if i is not a meaningful value, it still can act as a useful intermediary, whose weirdness goes away at the end.