What is a number? What are the common operations on numbers?
Lecture 0: Video. Covered administrative items, such as syllabus.
Lecture 1a: Video, Prelecture notes, Postlecture notes. Natural numbers, counting, addition, and subtraction.
Lecture 1b: Video, Prelecture notes, Postlecture notes. The number line and subtraction.
Lecture 1c: Video, Prelecture notes, Postlecture notes. Inventing multiplication.
Lecture 1d: Video, Prelecture notes, Postlecture notes. Inventing division and fractions.
Ungraded problems 1 (review of high school arithmetic)
Powers/exponents, Divisibility, and the Euclidean Algorithm
Recommended reading: Gross & Harris Ch 8.
Lecture 2a: Video, Prelecture notes, Postlecture notes. Powers and exponents.
Lecture 2b: Video, Prelecture notes, Postlecture notes. How to solve it.
Lecture 2c: Video, Prelecture notes, Postlecture notes. Divisibility and the Euclidean algorithm.
Ungraded problems 2 (powers, exponents, and division with remainder)
Combinations and Primes
Lecture 3a: Video, Prelecture notes, Postlecture notes. Combinations and the Euclidean algorithm.
Lecture 3b: Video, Prelecture notes, Postlecture notes. Prime numbers.
Lecture 3b2: Video, Handwritten notes. Quiz and homework review.
Recommended reading: Gross & Harris Ch 9-10.
Ungraded problems 3 (Euclidean algorithm and combinations)
Prime patterns, factorization and consequences
Lecture 4a: Video, Prelecture notes, Postlecture notes. Prime patterns.
Lecture 4b: Video, Prelecture notes, Postlecture notes. Fundamental Theorem of Arithmetic
Lecture 4b: Video, Prelecture notes, Postlecture notes. Using factorization for multiplication, division, and counting divisors
Recommended reading: Gross & Harris Ch 11-12.
Ungraded problems 4 (advanced combinations and Eratosthenes sieve)
Relative primes and Euler's totient function
Lecture 5a: Video, Prelecture notes, Postlecture notes. LCM and GCD via factorization.
Lecture 5b: Video, Prelecture notes, Postlecture notes. Relative primes and Euler's totient function.
Lecture 5c: Video, Prelecture notes, Postlecture notes. Irrational and imaginary numbers
Lecture 5d: Video, Prelecture notes, Postlecture notes. Review session: combos and primes
Recommended reading: Gross & Harris Ch 13.
Ungraded problems 5 (factoring, divisors, and relative primes)
What is a number? And modular arithmetic.
Lecture 6a: Video, Prelecture notes, Postlecture notes. Clock arithmetic: beyond counting.
Lecture 6b: Video, Prelecture notes, Postlecture notes. Modular addition and subtraction.
Lecture 6c: Video, Prelecture notes. Postlecture notes. Modular multiplication.
Recommended reading: Gross & Harris Ch 14-15.
Congruences
Lecture 7a: Video, Prelecture notes, Postlecture notes. Congruences and modular arithmetic.
Lecture 7b: Video, Postlecture notes. Interactive bean bag tossing (from Inspiring Mathematics: lessons from the Navajo Nation Math Circles by Dave Auckly, Bob Klein, Amanda Serenevy, and Tatiana Shubin
Lecture 7c: Video, Prelecture notes, Postlecture notes. Review of modular arithmetic computations.
Recommended reading: Gross & Harris Ch 16.
The midterm will cover material from the first half of the semester, at a faster pace than in the quizzes.
A practice midterm is available, in the same format that you should expect; note however that although some of the questions look like the ones on the real midterm, there may be substantial changes to the question testing the same mathematical content.
Division and powers in modular arithmetic
Lecture 8a: Video, Prelecture notes, Postlecture notes. Division in modular arithmetic.
Lecture 8b: Video, Prelecture notes, Postlecture notes. More modular division (via Euclidean algorithm).
Lecture 8c: Video, Prelecture notes, Postlecture notes. Powers via successive squaring in modular arithmetic.
Lecture 8d: Video, Prelecture notes, Postlecture notes. Power patterns interactive
Recommended reading: Gross & Harris Ch 17-18.
Ungraded problems 8 (division in modular arithmetic)
Roots in modular arithmetic
Lecture 9a: Video, Prelecture notes, Postlecture notes. Fermat's Little Theorem
Lecture 9b: (Video didn't record properly), Prelecture notes, Postlecture notes. A few remarks on Pi.
Lecture 9c: Video, Prelecture notes, Postlecture notes. Reciprocals via Fermat's Little Theorem.
Lecture 9d: Video, Prelecture notes, Postlecture notes. Roots in prime modulus arithmetic.
Recommended reading: Gross & Harris Ch 19 and Eugenia Cheng's How to write proofs.
Ungraded problems 9 (applying Fermat's Little Theorem)
Euler's Theorem
Lecture 10a: Video, Prelecture notes, Postlecture notes. Review of Fermat's Little Theorem.
Lecture 10b: Video, Worksheet, Postlecture notes. Non-prime powers interactive.
Lecture 10c: Video, Prelecture notes, Postlecture notes. Euler's Theorem.
Lecture 10d: Video,Prelecture notes, Postlecture notes. Roots in non-prime modulus arithmetic.
Recommended reading: Gross & Harris Ch 20.
Ungraded problems 10 (roots in modular arithmetic)
Codes, primes, and primality tests
Lecture 11a: Video, Prelecture notes, Postlecture notes. Fermat primality test.
Lecture 11b: Video, Prelecture notes, Handout, Postlecture notes. Encryption and codes interactive.
Recommended reading: Gross & Harris Ch 21 and 23.
Public-key cryptography
Lecture 12a: Video, Prelecture notes, Postlecture notes. Public-key cryptography.
Lecture 12b: Video, Handout, Prelecture notes, Postlecture notes. Hybrid cryptosystems interactive
Lecture 12c: Video, Prelecture notes, Postlecture notes. Final exam review
Ungraded problems 12 (hybrid decryption problems)
Recommended reading: Gross & Harris Ch 22.
The final examination will be cumulative and cover all the material from the semester, at a faster pace than in the quizzes. Note that it is cumulative not in the sense that you will specifically be asked questions based on the first half, but instead that that material will form building blocks for the questions on the final. You will not be asked to do the Euclidean algorithm directly; instead, you may be asked to evaluate a fraction in modular arithmetic, which involves as part of it the Euclidean algorithm. You will not be asked to do a powers problem directly; instead, you will need to understand the properties of exponents in order to evaluate modular powers.
For reference, here are two final exams from previous years, 2016 and 2017 (you will need to sign in with your UTORID). You should expect our final exam to be similar, with the following differences: