Mathematics
Linear Algebra (2026)
6 videos
This is a standard university-level linear algebra course.
Discrete Math (2025)
22 videos
This is a standard college-level course in Discrete Mathematics. The course is being taught to high school students, and so an entire year is spent on the course (as opposed to a single semester), allowing more time to be spent on each topic.
Calculus 1 (2024)
22 videos
This is a standard college-level Calculus 1 course. The textbook being used is Thomas Calculus ISBN-13: 978-0-321-87896-0.
Geometry (2024)
18 videos
This course is intended as a high school-level Euclidean Geometry course. The course begins by briefly covering Propositional Logic, Predicate Logic with Identity, and Set Theory before beginning Geometry proper using Hilbert's axioms (or close approximations). Upon finishing Planer Geometry, we will briefly cover Solid Geometry and Analytic Geometry. Time permitting, I will then lay the foundations of Trigonometry, followed by briefly looking at the Axioms of Arithmetic.
Intermediate Algebra (2023)
22 videos
A full Intermediate Algebra course, taught over a year to high school students, covering the ground between a first algebra course and college algebra. The course opens with integers, rationals, order of operations, and the properties of algebra, then works through solving linear equations, absolute value equations, and applications; graphs and equations of lines; inequalities and absolute value inequalities; and systems of equations. The second half turns to properties of exponents, operations on polynomials, factoring, rational expressions, and radicals — including radicals and exponents together — before taking up quadratic equations by radicals, by exponents, and by the quadratic formula, along with their applications. It closes with functions and function operations, logarithms, compound interest, and an introduction to the trigonometric functions. Every topic gets the time it needs, and the test and midterm reviews are included as their own lectures.
General Topology (Point Set Topology)
22 videos
This is intended as an undergraduate-level, first-semester, General Topology course. We begin with Set Theory, and work our way through the three Cs of Topology (Continuity, Connectedness, and Compactness). The course is a continuation of my Logic playlist, and so it starts with Set Theory and a review of Relations and Functions, then gets into Topology proper starting on Lecture 5. Logic playlist: https://www.youtube.com/playlist?list=PLJWhXkxQzDF58EHPCuxHS-iaMpom9qTqg
College Algebra
12 videos
A college-level College Algebra course that does not start where most algebra courses start. Before any algebra, we build the logic the subject is written in — propositional logic, then predicate logic — and then state the ten field axioms and prove the basic theorems of arithmetic from them. Only then does the algebra begin, on a foundation the student has actually seen laid. From there the course works through the standard material: real numbers and decimals, the four operations, absolute value, inequalities and the real number line, exponents, monomials and polynomials, products of binomials, factoring, rational expressions, radicals and rational exponents, and rationalizing denominators. It then takes up sets and linear inequalities in one variable, coordinates in the plane, inclination and slope, functions and relations, and equations of lines — with a look at the basics of trigonometry along the way. The course is in progress.
Real Analysis (2020)
26 videos
A first course in Real Analysis, equivalent to an Analysis 1 course — the rigorous rebuilding of single-variable calculus from the axioms up. We open with preliminaries: logic, sets, functions, and then the field and positivity axioms for the real numbers, along with the distribution of the integers and rationals and the standard inequalities and identities. From there the course follows the usual arc. Convergent sequences: the definition of convergence, sequences and sets, the Monotone Convergence Theorem, and the Sequential Compactness Theorem. Continuous functions: continuity, the Extreme Value and Intermediate Value Theorems, uniform continuity, the epsilon-delta criterion, images and inverses, monotone functions, and limits. Differentiation: the algebra of derivatives, differentiating inverses and compositions, the Mean Value Theorem and Cauchy's, and Leibniz notation. Integration: upper and lower integrals, the Archimedes-Riemann Theorem, additivity, monotonicity and linearity, continuity and integrability, and both Fundamental Theorems of Calculus. Proofs throughout — this is the course where the theorems from calculus finally get argued for.