# Take your math skills to the next level.

Educational Articles and Videos to Help Everyone With Mathematics

## 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.

## 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.

## 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.

## 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.

## Fibonacci Numbers (and the Euler-Binet Formula) [Video]

This article introduces the Fibonacci numbers and shows how to use mathematical induction to prove several Fibonacci identities. I then develop the Euler-Binet Formula involving the golden-ratio. Afterward, I discuss the Fibonacci Prime Conjecture and the growth of the Fibonacci sequence.

## Derivatives and Integrals of Vector Functions (and Tangent Vectors) [Video]

Okay, so after learning what vector functions are by sketching some graphs and using vector functions operations, let’s do some calculus with vector functions. In this video, I explain how to take derivatives and integrals of vector functions. I also discuss tangent vectors in detail.

## Introduction to Functions and Function Notation (Lots of Examples) [Video]

In this video I give an introduction to functions by working through several examples. I illustrate functions through the use of diagrams, lists, and equations. Then I demonstrate how to use function notation by working through several examples. I also work through several examples on finding the domain of a function and finding the zeros of a function.

## Volumes of Solids of Revolution (Disks, Washers, and Shells) [Video]

Finding the volume of the solid generated by rotating a bounded planar region about an axis of rotation is discussed. We cover the disk method, the washer method, and the method of cylindrical shells. We provide several examples of solids generated by revolving around both vertical and horizontal lines.

## 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.

## Trigonometric Functions (A Unit Circle Approach) [Video]

In this introduction to the six trigonometric functions we explain references angles and the unit circle. We also discuss solving right triangles. Many examples are given.

## 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.