Mathematics is an all-encompassing term that people use to describe how people think about the world around us and ourselves. Because of the generality and abstraction, there is no universal definition of what math is. For instance, as William Thurston said, “mathematics is not about numbers, equations, computations or algorithms; it’s about understanding.”
In this category, I have articles and videos in mathematics. Topics include logic, elementary set theory, discrete mathematics, analysis, and geometry. I also discuss category theory, abstract algebra, topology, and various areas in applied math. Initially, I am working on Number Theory, Calculus 1, Calculus 2, Calculus 3, Linear Algebra, Introduction to Proofs, and Precalculus. I publish the videos on YouTube and Facebook, and I post the video descriptions here in this category. The full articles that accompany the videos are also what I publish here in this category.
You’re trying to prove a statement is true using mathematical induction, but then you realize that the induction hypothesis doesn’t seem strong enough. What do you do? It turns out that there is more than one way to do induction. Let’s learn what strong induction is.
In this video, I am going to explain what arithmetic and geometric progressions are. I’ll also give several basics examples. Then I will prove the summations formulas for both the arithmetic and geometric progressions. In the end, I’ll discuss how these progressions arise in nature.
In this video, I discuss various forms of mathematical induction, including strong induction. I establish that the base case of an induction argument is required. Also, I explain the differences between the second-order logic induction principle and the first-order statement.
In this video, I’m going to work through five examples using mathematical induction. The first two examples involve summations, and the last three examples have an inequality. In each of these examples, I work through specials cases before wiring a proof so that you can get a feel for the result before actually proving the result holds using mathematical induction.
In this video, I’m going to talk about the Well-Ordering Axiom. First, I’ll discuss the ordering and why it’s reflexive, antisymmetric, and transitive. Then I explain the Well-Ordering Axiom followed by some basic examples. After that, I’ll discuss whether it should be an axiom or a theorem.
In this video, I’m going to cover three first examples using mathematical induction. So I’m going to assume that you have never seen mathematical induction before. So this is the first step, in a series of steps, to learn how to use and what mathematical induction is. Of course, to do that, we need to start with what the natural numbers are.
In this video, I discuss the six basic trigonometric functions and their graphs. I cover transformations such as a change in period, phase shift, amplitude, and vertical and horizontal shifts. I work on five examples in detail.
Okay, so you understand how to check if a quadratic congruence is solvable, but how do you find the solutions? In this article, I cover the Tonelli-Shanks algorithm by working through several examples. I also give a complete solution to a general quadratic congruence equation.
There are so many applied math journals out there; how can you keep track of them all? How do they relate to each other, what types of research does each one represent? Who should use applied mathematics journals to publish their work? And how do they work? In this article, I go over these questions by listing many applied mathematics journals and going over each one.
In this article, I cover topics related to solving quadratic congruences. In particular, I explain quadratic residues and the Legendre symbol in detail.
