Nonstandard Analysis (NSA) is a formal framework for calculus with infinitesimals, but also much more.  NSA combines Logic and Mathematics in a very elegant way to yield a new framework for Analysis.  For instance, compare the well-known ε-δ-definition of continuity with the nonstandard version.  Furthermore, if the sequence fa converges to the Dirac delta distribution for a → 0, then the function fε with ε an infinitesimal, is the Dirac Delta distribution.  In general, most objects in Mathematics have simple and intuitive expressions in NSA. 

Despite this elegance, NSA has some shortcomings.  For instance, to study the Dirac Delta distribution fε , we cannot use nonstandard techniques: We have to resort back to ε-δ-analysis.  However, in Relative NSA, there are infinitesimals of different degrees: given any infinitesimal ε, there is ε’ which is infinitely small relative to ε.  In this context, we write 0 << ε’ <<  ε << 1.  Thanks to Logic, all the magic of NSA works relative to every level of magnitude.  For instance, the Dirac Delta distribution fε is continuous and differentiable for increments ε’ that are infinitely small relative to ε.

In general, the introduction of several levels or degrees of infinity gives rise to a more elegant version of NSA.  In particular, Relative NSA does not depend as much on Logic as classical NSA.  The dependence of NSA on Logic is quoted by many as a great hinderance to its main-stream adoption.  For more details on Relative NSA, consult Karel Hrbacek’s recent work.  Note that Relative NSA is also instrumental in obtaining an interpretation of Constructive Analysis in NSA.   

Lastly, let us remember the words of the famous Kurt Gödel: ...there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.