* Pre-Course Evaluation
* Week 1
* natural numbers
* divisibility
* primes
* the sieve of Eratosthenes
* infinitude of primes
* binomial coefficients
* Week 2:
* induction
* sum of the first *n* naturals
* n! > 2^n
* closed form of the Fibonacci numbers
* well ordering principle
* sqrt(2) is irrational
* strong induction
* binary form of natural numbers
* binomial coefficients
* Week 3:
* divisibility algorithm
* greatest common divisors
* fundamental theorem of arithmetic
* primes and products
* parity
* modular arithmetic
* casting out nines
* divisibility by nine
* Week 4:
* greatest common divisors
* Euclidean algorithm
* LamÃ©'s theorem (the runtime of the Euclidean algorithm)
* linear diophantine equations
* modular arithmetic and inverses
* arithmetic modulo a prime
* Wilson's theorem
* Week 5:
* cryptography
* Caesar ciphers
* ROT13 encoding
* one time pad
* RSA cryptosystem
* rational numbers
* rational roots test
* irrational numbers
* test overview
* Week 6:
* complex numbers
* complex conjugate
* Argand diagram
* real part
* imaginary part
* modulus
* argument
* Euler's identity
* trigonometry using Euler's identity
* the quadratic formula
* the Fundamental Theorem of Algebra
* topological proof of the fundamental theorem of algebra
* multiplicity of roots
* polynomial long division
* De Moivre's theorem
* roots of unity
* Week 7:
* Complex numbers
* polar form
* rectangular form
* large powers
* roots of unity
* roots of *z*
* set theory
* set builder notation
* Russell's paradox
* union
* intersection
* disjoint unions
* functions
* pigeon hole principle
* injective
* sujective
* bijective
* infinite sets
* |N|=|Q|
* Cantor diagonalization
* Week 8:
* cardinality
* musical chairs
* power sets
* |X| < |P(X)|
* Hilbert's hotel
* countability
* products of sets
* countable union of countable sets
* the continuum
* decimal expansions
* Week 9:
* Cantor-Bernstein theorem
* finite subsets
* the labelling principle
* Cantor diagonalization
* Euler's phi function
* test review
* Week 10:
* euclidean geometry
* triangles
* angles
* congruence
* isoceles
* equilateral
* side-side-side
* side-angle-side
* angle-side-angle
* aparallel postulate
* PoincarÃ© disk model of hyperbolic geometry
* Week 11:
* construcatibility
* fields
* constructions and fields
* square roots are constructible
* towers of fields
* surds
* conjugation and roots