Justin Laverdure, Department of Pure Mathematics, University of Waterloo
"The Hyperreals (Infinitesimals and the transfer principle)"
Intuitive discussions about calculus among those 'uninitiated' into proper analysis often involve discussions about "infinitely small" quantities. Foundations of calculus, of course, frame these as limiting processes, but can an actual concept of an infinitesimal number be recovered and worked with directly? Turns out the answer is "yes": we may enlarge the real line in such a way that there are infinitesimal quantities, those strictly greater than zero but less than every 1/n.
In fact, the real numbers and the hyperreal numbers are closely related: facts about one can be translated to facts about the other. Thus, understanding the real line helps us to understand the hyperreal line, and interestingly, the converse as well. In particular, certain proofs in analysis admit simpler proofs, once the techniques of "non-standard analysis" have been introduced.
MC 5501