Is there an implementation of a hyperreactive data type for performing calculations in non-standard analysis?

Question

Non-standard mathematical analysis extends the line of real numbers to include “hyperreality” - infinitesimal and infinite numbers. Is there a (specification for) data type implementation for implementing computations using hyperrealizations? I am looking for something similar to the complex number data type that you will find in Python and Fortran and elsewhere. Actually, I don’t know if such calculations are useful: I'm just curious. I played around with this concept a bit, but since I probably made mistakes, I will spare you all the details. Wikipedia hyperlink page link .

Answer 1

I think you could do it with the following construct:

Hyperreal numbers are parts of polynomials with real coefficients over the variable w , where w denotes the "least" infinity (that is, not the product of the smaller infinite numbers).

Polynomials are ordered lexicographically, i.e. the highest degree to which polynomials differ determines order. This can be extended in a standard way to numerical polynomials (for example, order on rational numbers that are parts of integers).