TITLE: Why hyperreal numbers are built so complicatedly?
QUESTION [8 upvotes]: I have seen approaches at building hyperreal systems by using complicated notions like ultrafilters and the like.
Why not just postulate the existence of infinitesimal element $\varepsilon$ and infinite $\omega=1/\varepsilon$ like we do with complex numbers and build a field system around them?

REPLY [2 votes]: One good reason to use the hyperreals $\mathbb{R}^*$ as constructed using a nonprincipal ultrafilter is that they satisfy a transfer principle (as do other versions of NSA, for example IST).  This means in particular that every function $f$ or relation $R$ on the reals can be "enlarged" to a function $f^*$ or relation $R^*$ on the hyperreals in such a way that a first order statement about $\mathbb{R}$ is true if and only if the corresponding enlarged statement is true of $\mathbb{R}^*$.  
Existence of enlargements is useful when you have some standard function which you want to extend to a nonstandard setting.  If you want to work in $\mathbb{R}((\epsilon))$ for example, the "right" definition of $\cos(\epsilon)$ is clearly $\sum_{n=0}^\infty (-1)^n \epsilon ^{2n} / (2n)!$  (though what $e^{-1/\epsilon}$ should be isn't so clear). But if I have some real function that can't be expressed by a power series, how should it be defined on $\epsilon$?  The transfer principle allows you to dodge this problem.