We define the Fibonacci Sequence, develop a formula for the entries. We then use that to establish bounds on the growth of the sequence. We use that to prove a bound on the number of division operations required to compute the Euclidean Algorithm. Finally, we finish by continuing our discussion of the RSA algorithm and introducing the Golden Mean.

In this entry we begin by proving the correctness of the Euclidean Algorithm, then we discuss factorization in rings. We give an example of factoring into irreducibles in two distinct ways. We then define Unique Factorization Domain, Principal Ideal Domain and Euclidean Domain. We then prove that an ED is a PID and a PID is a UFD. We finish by explaining the RSA encryption algorithm.

We continue our investigation into rings and fields. We finish by explaining the Euclidean Algorithm. We also give a python implementation which, for any two positive integers, a and b, returns gcd(a,b) and the pair of integers, s and t, such that as + bt = gcd(a,b).

We give definitions for span, basis and dimension, and then prove that all vector spaces have bases, and that their dimension is well-defined. Then we use that to define the degree of a field extension and prove the tower law for field extensions. After that, we define basic properties of polynomials.

In this entry we cover more basic results about ordinal arithmetic. We also prove the so-called “Fundamental Theorem of Cardinal Arithmetic”, and then we finish with a short discussion about cofinality of cardinals.

Here we dig into ordinal and cardinal arithmetic a little bit. We also establish the famous result (originally due to Cantor) that |S| < |P(S)|, i.e. that the power set of any set is larger than the set.

In this entry, we establish more results about ordinals, including proving that every well-ordered set has the order type of an ordinal. We introduce transfinite induction, Cardinality, and use the Axiom of Choice to prove that every set can be well-ordered, among other results.

In this article we introduce and discuss all of the axioms of ZFC. We introduce Cartesian products, linear and well-orders, and ordinals.

In the previous entry, we introduced Rings and Fields. In this entry we continue our investigation. We will investigate structures that can be built on top of them. This entry will mostly involve grinding through fundamental proofs that are necessary for more interesting results which will come later.

In this entry we introduce rings and fields. We establish some of their basic properties.