In this seminar we wrestle with infinity and its smaller
protege: the infinitesimal. Building off of the foundation provided by
repeating decimals, we explore the following analogy:
Terminating Decimals are to Polynomials
as
Rational Numbers are to ... what?
This seminar uses the entire toolbox of algebra to investigate the infinite and infinitesimal. This document contains examples of questions you should be able to understand to ensure you are prepared for the explorations in this seminar.
This document is an introduction to our first expedition, and you are encouraged to think about the questions before our first meeting. It also contains some additional expeditions that we may undertake throughout the year.
This seminar introduces various rigorous frameworks for thinking about the infinite and the infinitesimal. Using tools students learn in Algebra class and building on ideas introduced in Elementary School around repeating decimals, we discover a backdoor into Calculus. The mysteries on the other side of this door relate to polynomial approximations, which is a powerful and often underdeveloped topic in introductory courses.
The seminar starts with a clever application of the Division Algorithm to uncover the local behavior of rational functions (aka Maclaurin Polynomials/Series). Comparing the geometric series with repeating decimals and lining this up with Zeno’s Paradoxes lay the groundwork for the seminar. Similar polynomial approximations can be established for radical, exponential, logarithmic, and trigonometric functions, all without the messiness of limits or the black box of derivatives, and instead emphasizing the unique properties of these functions.