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?
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.