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Non-standard mathematical analysis extends the real number line to include "hyperreals" -- infinitesimals and infinite numbers. Is there (specification for an) implementation of a data type to implement computations using hyperreals? I'm looking for something analogous to the complex number data type you find in Python and Fortran and elsewhere. I actually don't know if such computations are useful: I'm just curious. I've played around with this concept a bit, but since I probably made errors I will spare you-all the details. Reference wikipedia page on hyperreals.

Aaron Watters
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    I guess these three links may interest you. (Although not exactly what you are looking for) 1- http://reference.wolfram.com/mathematica/ref/Limit.html 2- http://reference.wolfram.com/mathematica/ref/Series.html 3 - http://reference.wolfram.com/mathematica/ref/DiracDelta.html – Dr. belisarius Aug 27 '10 at 17:43

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Edit: These are not the hyperreal numbers, but the construction could still be useful for computing derivatives or limits.

Consider quotients of polynomials with real coefficients over the variable w where w denotes the "smallest" infinity (i.e. not a product of smaller infinite numbers).

The polynomials are ordered lexicographically, i.e. the highest power where the polynomials differ determines the ordering. This can be extended in the standard way to quotients of polynomials (like the order on rational numbers, which are quotients of integers).

starblue
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    This sounds like a correct mathematical characterization -- what software/data structure effects such a construction for computations in computer programs? – Aaron Watters Aug 27 '10 at 19:45
  • I know none, but it wouldn't be too difficult to implement it. Reals would be approximated by floating-point numbers, as usual. – starblue Aug 27 '10 at 19:54
  • This could be interesting to have because it could give a mathematically correct semantics for expressions using (ieee-defined) "real" constants such as Inf and -Inf. ("Inf" would be replaced by __many__ new values, such that "Inf * 0" could get the correct value, not only NaN, such as now in octave! – kjetil b halvorsen Aug 27 '13 at 08:59
  • I know that this is 11 years late, but I would like to point out that this is not strictly true. The quotient field of the real polynomial ring is not algebraically closed and would fail to handle transcendental elements like $exp(\omega)$. You could probably approximate hyperreals this way, but you'd still be quite limited. – tox123 Jun 11 '21 at 17:18
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    You are right, I think $exp(\omega)$ is too big for this construction. – starblue Jun 13 '21 at 19:19