Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number

I cannot stop seeing them very related to each other. In one the product is not explicitly defined (it is said that it is the result of a series of cuts) in the other it is stressed that $\epsilon^2=0$ is the defining property.

Both are related to derivatives when evaluated in functions (for example of polynomials or Taylor series) although in one the st symbol is used and in the other $\epsilon$ is used.

Is there a simple relation between these two mathematical constructs? are both the same? is one just a specialization (for a certain operation) case of the other? Is one a field and the other just a ring for example? Is the difference the partial vs. total order?

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    $\begingroup$ They're quite different. Hyperreal analysis is ultimately equivalent to standard analysis, by way of the transfer principle. (Strictly speaking, hyperreal analysis proves "bona fide hyperreal" theorems, but basically no one cares about them.) Smooth infinitesimal analysis is a truly different theory from both. $\endgroup$ – Ian Jun 25 '16 at 18:20
  • $\begingroup$ Also, dual numbers have a nilpotent element, while hyperreals do not -- they are an ordered field. $\endgroup$ – vadim123 Jun 25 '16 at 18:22
  • $\begingroup$ @Ian I guess that answers this question: math.stackexchange.com/questions/341535/… . What do you mean by "nobody cares", is it difficult to deal with hyperreals? what is smooth infinitesimal analysis, a particular application of hyperreals to smooth functions? $\endgroup$ – alfC Jun 25 '16 at 18:25
  • $\begingroup$ @vadim123 So hyperreals is an ordered field and duals are a partially ordered ring? Is that the classification difference to start with? $\endgroup$ – alfC Jun 25 '16 at 18:26
  • $\begingroup$ You really can't think about smooth infinitesimal analysis quite the same way as you think about other structures, because it isn't even founded on the same logic. Its axioms are contradictory in classical logic. For example, the smooth infinitesimal $\mathbb{R}$ is an ordered field, but trichotomy ("for all x,y, either x<y,x>y,or x=y") fails. But it fails in a very weak sense, because you can't prove any counterexamples exist either. So it's not really partially ordered in the classical sense either. $\endgroup$ – Ian Jun 25 '16 at 18:27

You are correct in pointing out that the two constructions are related to each other, because both attempt to extend the real number system to a broader system incorporating infinitesimals. Dual numbers have the advantage that they are much easier to construct. The hyperreals have the advantage that they are useful in analysis, because every function defined on the reals extends naturally to the hyperreals. Similarly, "all" properties of functions and relations similarly extend. This is not the case for the dual numbers which are useful in physics.


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