Freshman Advising Seminar: What is a number?

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
Prime factorization
Algebraic numbers; failure of prime factorization; ideal numbers
p-adic numbers
Cardinals: Cantor
Surreal 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, 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.

  • Financial and Support Services: Financial, wellness, mental health, technology, ... You name it, they can help you: specially designed for you as a first year student.
  • S3 is the standard office offering support of all sorts for all MIT undergraduates. If for some reason you have to miss an exam for example, the first step is normally a visit to S3. It's a very student-oriented office.

    Associate Advisor: Megan Su,


    Cecilia Chen,
    Anas Chentouf,
    Vanessa Deering,
    Paige Dote,
    Kim Eppling,
    Kartikesh Mishra,
    Linh Nguyen,
    Yuru Niu,


    Date Leaders Topic Resources
    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

    Class links (not world accessible)

    Zoom address:, meeting id 920 9939 9985
    Overleaf document:
    Class video recordings:

    Professor Haynes Miller
    Department of Mathematics 2-478
    Massachusetts Institute of Technology
    Cambridge, MA 02139