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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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.
Using tables of values and graphs of functions, I explain what limits are and how to use them. Then I discuss one-sided limits, two-sided limits, and when limits do not exist. I use several illustrations to develop intuition with limits. Because these methods of estimating limits are not rigorous, we explain how estimating can lead to incorrect results.
I work through several examples of writing a proof by Mathematical Induction (for beginners). I concentrate on cases that demonstrate how to use mathematical induction to prove a statement true for all natural numbers. Afterward, I discuss Strong Induction and show how to use it. Then well-known arithmetic and geometric progressions formulas are proven using induction. Towards the end, I confirm that the Well-Ordering Axiom, Mathematical Induction, and Strong Induction are all logically equivalent.
I begin with 2 by 2 and 3 by 3 linear systems of equations. I then discuss consistent and inconsistent systems of linear equations and work through several examples. The general system of linear equations is defined, and parametrizing a solution set of a linear system is demonstrated. I explain augmented matrices, row operations, and when two linear systems of equations are called equivalent.
Vector functions of real variables are defined and studied. I begin with an unintuitive introduction to vector functions and how they can model motion in space. I discuss the graph of a vector function and how to determine a vector function given geometric information. Operations with vector functions, limits of vector functions, and continuity of vector functions, are also what I explore.
If you already know the Fundamental Theorem of Calculus and find the area under a graph of a function, you are ready to find the area between curves. In this article, I’ll show you how to find the area of a region bounded by two curves. I’ll demonstrate both using vertical strips and horizontal strips.
Do you know what an angle is? I’ll explain the difference between a geometric angle and a trigonometric angle. I’ll go over what angles are, measuring angles, converting between degrees and radians, and the basic types of angles. I then cover the Arc Length formula and the Area of a Sector formula. I work through many examples involving these formulas. Angles and their measures are an essential starting point for any study into trigonometric functions.