What is a number? In this seminar we'll explore the wonderful richness of
the concept of number - richness in history and culture, from the Maya to
India and the Arab world, and in the astonishing expansion of scope that the
concept has enjoyed in the hands of mathematicians. There is the progression
from natural numbers to integers to rationals to reals; but also the extensions
of the number concept needed to deal with infinities and infinitesimals
by Cantor and Conway among others; the expansion beyond order into complex
and quaternionic numbers; and the flowering into arithmetic via ideas of
Fermat, Gauss, Kummer, and Hensel.
In more detail, the sequence of topics might be
History, language, culture: representations of numbers
Natural numbers: Peano, induction
Rationals and irrationals
Real numbers: Dedekind
Approximation: continued fractions
Complex numbers and quaternions
Algebraic numbers; failure of prime factorization; ideal numbers
The regular class meetings will occur on Wednesdays from 12:00 to 1:00 PM Eastern US time, online. Students will lead discussions of relevant mathematical topics.
We will use Zoom at the passwork-protected address https://mit.zoom.us/j/92099399985, meeting id 920 9939 9985 . It will be important to have a tablet and stylus as well, to facilitate mathematical discussions.
We will scour the web and the MIT library for information about selected topics arising from unpacking the concept of number, and come together weekly to discuss what we've discovered. Each week a pair of students will serve as discussion leaders, in collaboration with the faculty organizer. We will schedule a time to discuss the material with you beforehand. No "Psets" and no "Exams"!
But there will be a deliverable: We will create a manuscript together documenting what we have found, using Overleaf, a tool for collaborative creation of documents in LaTex. Here's a link to this document, available only to members of the class. I'm very excited to see how it develops. Many hands make light work!
This will be a challenging fall semester for everybody! The seminar meetings are one opportunity to share stories, concerns, tricks, and jokes. But if you want to discuss anything at all in private, don't hesitate to be in touch with either Megan or me. We are both here for you. It's our job.
In addition, for your reference, here are two sites with many resources available to you.
Megan Su, email@example.com
Cecilia Chen, firstname.lastname@example.org
Anas Chentouf, email@example.com
Vanessa Deering, firstname.lastname@example.org
Paige Dote, email@example.com
Kim Eppling, firstname.lastname@example.org
Kartikesh Mishra, email@example.com
Linh Nguyen, firstname.lastname@example.org
Yuru Niu, email@example.com
|Wed Sep 2||Ensemble||Numbers in language, history, and culture||Wikipedia: Egyptian fractions|
|Wed Sep 9||Paige and Kim||The natural numbers: Peano||Gabe Perez-Giz: video
Terry Tao: text
|Wed Sep 16||Vanessa and Yuru||Integers, rationals, and irrationals||Terry Tao: text|
|Wed Sep 23||Cecilia and Linh||The real numbers: Dedekind|| Jindrich Zapletal: text
Terry Tao: text
|Wed Sep 30||Anas and Kartikesh||Continued fractions|| Harold Davenport: book
|Wed Oct 7||Paige and Kim||Best rational approximations|| Harold Davenport: book
Evelyn Lamb: SciAm article
Bruce Ikenaga: course notes
|Wed Oct 14||Cecilia and Yuru||Complex numbers, fundamental theorem of algebra|
|Wed Oct 21||Vanessa and Linh||Quaternions|| Grant Sanderson and Glen Eater: Quaterionic explorations
|Wed Oct 28||Anas and Kartikesh||The Euclidean algorithm||Harold Davenport: book|
|Wed Nov 4||Cecilia and Paige||Regaining prime factorization: ideal numbers||Casey Bruck: text|
|Wed Nov 11||--||"Veterans Day"|
|Wed Nov 18||Anas and Kim||p-adic numbers|| Trevor Hyde: Slides
David Madore: Text
|Wed Dec 2||Vanessa and Kartikesh||Cardinal numbers||Wikipedia|
|Wed Dec 9||Linh and Yuru||Surreal numbers||Gretchen Grimm: text|