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Educational Articles and Videos to Help Everyone With Mathematics

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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 (The Definitive Guide)

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.

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.

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.

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.

Number Theory (Get Started Here)

Number Theory has a long and exciting history. To help understand what Number Theory is all about, in this article, we describe a few basic ideas of Number Theory. From divisibility and mathematical induction to Euler’s theorem and solving polynomial congruence equations, Number Theory can be both highly practical and applicable yet also extremely difficult and abstract. Number Theory also provides us with a playground where students can master proof-writing while learning some very exciting applications of the theory. The Law of Quadratic Reciprocity and the much more recent Tonelli-Shanks algorithm are such examples.

Evaluating Limits Analytically (Using Limit Theorems) [Video]

This video explains the limit theorems, including the Product Rule, Quotient Rule, and the Polynomial Functions Rule, and the Rational Functions Rules. I then work through many examples, including factoring, rationalization, and some limits involving trigonometric functions. In the end, I illustrate the Squeeze Theorem and how to find inequalities when working with the Squeeze Theorem.

We Do Math (The Rise of Mathematics) [Video]

This video takes the viewer on a journey where the focus is on mathematics. At the beginning of the video, I give my opinion that mathematics is “our way of thinking.” From here, we see how mathematics has made profound changes throughout history. Our ability to apply our way of thinking to ourselves keeps changing what mathematics is and thus ourselves.