Current Events in Mathematics

Mathematics is the study of numbers and their practicality in solving mysteries of the world. Current events in this subject help to uncover the true meaning of the study. Many who dislike math don’t truly understand what it accomplishes or aims to discover. By focusing on current events, students and professionals alike help to show why math is such a force in society.

Notes on Diffy Qs: Differential Equations for Engineers

Do you see differential equations and think dreadful equations? This doesn’t have to be. Notes on Diffy Qs is an engineer’s guide to differential equations. The book is suited for engineering students taking their first course on “diffy Qs.” Differential

Outstanding Mathematicians (10 Phenomenal Careers)

Furthering math may seem like a daunting task to many; however, for these educators and researchers, that is literally the best part of the job. From solving theorems and proofs to putting math into real-life practice, mathematicians are important members

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.

Math Topics (A Condensed Guide to Mathematics)

No matter who you are, youâ€™ve probably had some math classes in your life. After all, math is an important subject not only for success in technical subject areas, but also for giving your mind a workout and making it

Applied Mathematics Journals (Explained For You)

Applied mathematics bears many connections to various other fields within academic and professional life. Consequently, itâ€™s important to stay up-to-speed on the most recent theoretical and practical developments. Advancements in applied mathematics can affect everything from pharmaceutical research and development

Implicit Differentiation (and Logarithmic Differentiation)

The procedure of implicit differentiation is outlined and many examples are given. Proofs of the derivative formulas for the inverse trigonometric functions are provided and several examples of using them are given. Also detailed is the logarithmic differentiation procedure which

Calculus (Start Here) – Enter the World of Calculus

Welcome, calculus enthusiasts, and enthusiasts-to-be! This page is your introduction to the world of calculus. You may be studying it for the first time, or maybe for the first time since high school. You may be using it in your

Chinese Remainder Theorem

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

Propositional Logic (Truth Tables and Their Usage)

By a mathematical statement (or statement, or proposition) we mean  a declarative sentence that can be classified as either true or false, but not both. For example, the sentences $$1+3=4, \quad 1+3=5, \quad \text{July is not a month}$$ can be accepted

Double Integrals and the Volume Under a Surface

The Volume Under a Surface Consider the rectangle given by $$R=[a,b]\times[c,d]=\{(x,y)\mid a\leq x\leq b, c\leq x\leq d\}.$$ We wish to construct a (regular) partition of $R.$ To do so, let  a=x_0 < x_1 < x_2 < \cdots <

Differentiation Rules (with Examples)

Computing the limit of the difference quotient can be tedious and require ingenuity; fortunately for a large number of common functions there is a better way to compute the derivative. In this section, we detail the power rule and the

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

Euler’s Totient Function and Euler’s Theorem

We prove several properties of Euler’s Totient Function and give many examples. We also discuss solving functional equations and reduced residue systems. One of Euler’s most important theorems is then demonstrated and proven. Introduction to Euler’s Totient Function Definition. For

Congruence Theorems (and Their Proofs)

We discuss congruence theorems on the integers by proving several elementary lemmas. Modular arithmetic and least positive residues is also discussed. Introduction to Congruence Theorems A modern treatment of congruences was introduced by Carl Friedrich Gauss. Congruence, or modular arithmetic,

Inner Products and Orthonormal Bases

Recall that the norm of $x\in \mathbb{R}^n$ defined by $\left|\left |x\right|\right | = \sqrt{x_1^2+x_2^2}$ is not linear. To injective linearity into the discussion we introduce the dot product: for $x,y\in \mathbb{R}^n$ the dot product of $x$ and $y$ is defined

Diagonalization of a Matrix (with Examples)

An $n\times n$ matrix $A$ is called diagonalizable if $A$ is similar to some diagonal matrix $D.$ If the matrix of a linear transformation $T$ with respect to some basis is diagonal then we call $T$ diagonalizable. Diagonalization Theorem Theorem.

Limits (Calculus Starts with Limits)

Limits are used to study the behavior of quantities under a process of change. For example, limits can be used to describe the behavior of a function on its domain. Here we study one-sided limits and two-sided limits with emphasis

Fundamental Theorem of Calculus

We being by reviewing the Intermediate Value Theorem and the Extreme Value Theorem both of which are needed later when studying the Fundamental Theorem of Calculus. The Mean Value Theorem for Integrals and the first and second forms of the

Derivative Definition (The Derivative as a Function)

We begin with the definition of the derivative as a limit of a difference quotient. We then give several examples of how to find the derivative of a function using this definition. Finding an equation of the tangent line is

L ‘Hopital’s Rule and Intermediate Forms

L’Hospital‘s rule is a general method for evaluating an indeterminate form. This rule can be applied to several types of indeterminate forms, using first an appropriate algebraic transformation. The expression “the limit has an indeterminate form” is used to convey

Limit Definition (Precise Definition of Limit)

The history of calculus is interconnected with the history of philosophy and scientific discover. In the 16th Century, French philosopher Rene Descartes combined principles of algebra and geometry by mapping lines and objects using horizontal and vertical planes. By applying

Optimization Problems (Procedures and Examples)

We work through several applied optimization problems emphasizing the important role that the derivative plays. The significance of the Extrema Value Theorem and the First and Second Derivative Tests is also expressed. Optimization Procedures In this section we give a

Parametric Equations and Calculus

Parametric equations are a set of functions of one or more independent variables called parameters and are used to express the coordinates of the points that make up a geometric object such as a curve or surface. We begin by