## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, February 1, 2018 — 4:00 PM EST

**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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1