# Mathematics - Articles, Videos, and Commentary

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.

## Using Transformations to Graph Trig Functions (5 Examples) [Video]

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.

## Tonelli-Shanks Algorithm (by Example)

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.

## Applied Mathematics Journals (Explained For You)

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.

## Math Topics (A Condensed Guide to Mathematics)

I wrote this article for those with a beginning interest in mathematics. Here I explain some of the primary areas of mathematics by grouping them into math topics and then discussing each one. This organization is only a broad overview, though I hope it helps you find your specific area of interest.

## Euler’s Totient Function and Euler’s Theorem

Many people have celebrated Euler’s Theorem, but its proof is much less traveled. In this article, I discuss many properties of Euler’s Totient function and reduced residue systems. As a result, the proof of Euler’s Theorem is more accessible. I also work through several examples of using Euler’s Theorem.

## About Mathematics and Why It Is Essential

Mathematics uses numbers as a language to explore some of the world’s most complex theories and problems. Children in school associate math with difficulty or confusion. With proper study, the subject can become an exciting way to view the world. Read all about the topic and its many branches.

## Fermat’s Theorem (and Wilson’s Theorem)

Maybe you have heard of Wilson’s Theorem? But did you know that’s is converse also holds. In this article, I prove both Wilson’s Theorem, its converse, and Fermat’s Theorem. Then you will also see many examples using Fermat’s theorem.

## Classical Mathematics (a Look Into Its History and Fields)

In this article, I take a look at classic mathematics. I discuss intuitionism and constructivism and the uses of classical mathematics throughout time. Then, from the Islamic Golden age to European developments, I review some of its histories. I also briefly explain some of the elementary fields of classic mathematics. In the end, mathematics concerns itself with the search for truth.

## Chinese Remainder Theorem

This definitive guide covers proofs, examples, algorithms, applications, and the Chinese Remainder Theorem history. It also includes links to additional resources such as online articles, courses, books, and tutors to help students learn from various sources. Professionals can also use these resources to increase their knowledge of the field or help structure courses for their students.

## Applications of Congruence (in Number Theory)

So you probably know a divisibility test for 2, 3, and 5. But what about 7, 11, 13, or even larger primes? In this article, I go over divisibility tests. Including how to create your own. I also discuss the Days of the Week problem, where you are to determine the day of the week from a given date very quickly.

## Polynomial Congruences with Hensel’s Lifting Theorem

The idea behind solving polynomial congruence equations is that we can reduce a congruence equation to an equivalent system of congruence equations using prime factorization. We then 1) solve each equation modulo a prime number (by brute force), 2) use Hensel’s Lifting theorem, and then 3) piece together the solutions using the Chinese Remainder Theorem. We provide several nontrivial examples many of which are workable by hand.

## Linear Congruences and Their Solvability

In this article, you will learn what linear congruences are and when they are solvable. How to solve them will also be covered in detail. I discuss an ad hoc method, using the Euclidean algorithm, and using the inverse of an integer.

## Congruence Theorems (and Their Proofs)

In this article, I discuss modular congruence. I demonstrate the congruence is an equivalence relation, and I prove several lemmas concerning the basic properties of congruence. Towards the end, I go over modular arithmetic and its properties.

## Diophantine Equations (of the Linear Kind)

In this article, I discuss what Diophantine Equations are and the difficulting of solving them. Then, I detail how to solve two-variable linear diophantine equations. Towards the end, I solve a multi-variable linear Diophantine equation concerning pennies, dimes, and quarters.

## Fundamental Theorem of Arithmetic

In this article, I prove one of the most celebrated theorems in all of mathematics. But first, I explain why this theorem is both fundamental and unique. I also explore some applications and discuss the least common multiple and their connection to greatest common divisors.

## Euclidean Algorithm (by Example)

The Euclidean Algorithm is to find the greatest common divisor of two given integers. In this article, you will see this critical algorithm proven in detail. Further, I will show you how to use these computations to solve linear congruence equations and linear Diophantine equations. While this algorithm has been around a while, it is the key to much success.

## Greatest Common Divisors (and Their Importance) [Video]

We discuss several simple lemmas for greatest common divisors and linear combinations. I then prove Bezout’s identity to show that the greatest common divisor of two integers is the smallest linear combination. We also work through several elementary facts concerning relatively prime integers, and I present many examples.

## Prime Numbers (Theorems and the Infinitude of Primes) [Video]

We discuss the difference between composites and prime numbers and then show how to find prime numbers using Eratosthenes’ sieve. We then prove the infinitude of prime numbers and show that every natural number has a prime divisor by assuming the well-ordering axiom.

## Divisibility (and the Division Algorithm) [Video]

The notion of divisibility is motivated and defined. We work through many examples and prove several simple divisibility lemmas –crucial for later theorems. We also discuss linear combinations, and I present the division algorithm with its proof. Afterward, I demonstrate the importance of the division algorithm through examples.