Chinese Remainder Theorem (Examples Included) Fresh chicken eggs in a basket

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 ...
Differentiation Related Rates Applying Implicit Differentiation to rippled drop io n a pond waves

Related Rates (Applying Implicit Differentiation)

In this topic we show how implicit differentiation and the chain rule can be used to calculate the rate of change of one variable in terms of the rate of change of another variable (which may be more easily measured). Introduction to Related Rates The procedure of solving a related rates problem is to find an equation that relates two ...
Notes on Diffy Qs Happy young university students studying with books and laptops in library

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 equations are the language of many sciences, including engineering. Therefore, they are very important to understanding the work that engineers ...
Mathematics Scientist Adding Microscopy To Brain Study App


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 ...
Current Events in Mathematics Close-up Shot of a Hand Holding Chalk and Writing Complex and Sophisticated Mathematical Formula Equation on the Blackboard

Current Events in Mathematics

Current events in mathematics point to a growing field with a diversifying range of utilities and practices. The history of mathematics is an interesting dive into the past, present and future. It's long been one of the most studied disciplines. In fact, it's older than most other fields of study in general. To be fair, it does date back to ...
Confluent Relations using Reduction Relations arrow all pointing together to a single point

Confluent Relations (using Reduction Relations)

We discuss confluent relations; in particular, we prove Newman's Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation. We also give a generalization of Newman's lemma based on the Buchberger-Winkler's Property. Reduction Relations Let $\longrightarrow$ be a relation on $X.$ If there exists $c\in X$ such that $a\stackrel{*}{\longrightarrow} ...
Well Founded Relations Well Founded Induction in a swirling cloud of numbers down to the ground Flying numbers and math symbols

Well-Founded Relations (and Well-Founded Induction)

Well-Founded induction is a generalization of mathematical induction. Well-Founded Induction Definition. Let $\longrightarrow$ be a relation on $X.$ 1) If $A\subseteq X$ and $a\in A,$ then $a$ is called a $\longrightarrow$-minimal element of $A$ if there does not exist $b\in A$ such that $a\longrightarrow b.$ 2) If each nonempty subset of $X$ has a $\longrightarrow$-minimal element, then $\longrightarrow$ is called ...
Partial Order Relations Mappings on Ordered Sets

Partial Order Relations (Mappings on Ordered Sets)

We discuss many properties of ordered sets including Noetherian ordered sets and order ideals. We also detail monotone mappings and isomorphisms between ordered sets. Ordered Sets Throughout we assume $(X,\geq)$ is an ordered set. By this we mean that $X$ is a set and that $\geq$ is binary relation on $X$ that is reflexive, antisymmetric, and transitive. Definition. A subset ...
Equivalence Relations Properties and Closures a binary network graph being handed off

Equivalence Relations (Properties and Closures)

Equivalence Relations We discuss the reflexive, symmetric, and transitive properties and their closures. The relationship between a partition of a set and an equivalence relation on a set is detailed. We then give the two most important examples of equivalence relations. Reflexive, Symmetric, and Transitive Relations Definition. Let $X$ be a set. A relation $R$ on $X$ is called reflexive ...
binary relations graph on a blue background

Binary Relations (Types and Properties)

Let $X$ be a set and let $X\times X=\{(a,b): a,b \in X\}.$ A (binary) relation $R$ is a subset of $X\times X$. If $(a,b)\in R$, then we say $a$ is related to $b$ by $R$. It is possible to have both $(a,b)\in R$ and $(a,b')\in R$ where $b'\neq b$; that is any element in $X$ could be related to any ...
Composition of Functions to do recipes with various soups ingredients and space for text on wooden background

Composition of Functions and Inverse Functions

Composition Theorem. Let $f:X\to Y$ be a function. If $g:Y\to Z$ and $g\circ f$ is injective, then $f$ is injective. Proof. Let $x_1, x_2\in X$. Then $$ f(x_1)=f(x_2) \Longrightarrow (g\circ f)(x_1)=(g\circ f)(x_2) \Longrightarrow x_1=x_2 $$ shows that $f$ is injective. Theorem. Let $f:X\to Y$ be a function. If $g:Y\to Z$ and $g\circ f$ is surjective, then $g$ is surjective. Proof ...
One to One Functions and Onto Functions using pencil and paper

One-to-One Functions and Onto Functions

Injective Functions Definition. Let $X$ and $Y$ be sets. A function $f:X\to Y$ is called injective if $$ \forall x_1,x_2\in X, f(x_1)=f(x_2)\implies x_1=x_2. $$ Theorem. Let $f:X\to Y$ be a function. If $f$ is injective and $A\subseteq X$, then $f|_A$ is injective. Proof. Let $x_1, x_2\in A$. Then $$ f|_A(x_1)=f|_A(x_2) \Longrightarrow f(x_1)=f(x_2) \Longrightarrow x_1=x_2 $$ shows that $f|_A$ is injective ...
Functions Their Properties and Importance busy life with all the daily functions

Functions (Their Properties and Importance)

Let $X$ and $Y$ be sets, we say $f$ is a function from $X$ to $Y$ if $f$ is a subset of $X\times Y$ such that the domain of $f$ is $X$ and $f$ has the property: if $(x,y)\in f$ and $(x,z)\in f$ then $x=z$. For each $x\in X$, the unique $y\in Y$ such that $(x,y)\in f$ is denoted by ...
Families of Sets Finite and Arbitrarily Indexed venn diagrams

Families of Sets (Finite and Arbitrarily Indexed)

Families of Sets Finite Unions and Intersections Given any $n$ sets $A_1, A_2, \ldots, A_n$, we define their union to be the set $$A_1 \cup A_2 \cup \cdots \cup A_n =\{x\mid x\in A_i \text{ for some } i, 1\leq i \leq n\}. $$ and their intersection to the the set $$ A_1 \cap A_2 \cap \cdots \cap A_n =\{x\mid x\in ...
Set Theory Basic Theorems with Many Examples

Set Theory (Basic Theorems with Many Examples)

We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and Cartesian products. We also demonstrate how to work with families of sets. For a brief discussion of the reviews of (elementary) Halmos' Naive Set Theory read this. A solid understanding of propositional and predicate logic is strongly recommended. To get the most from ...
Canonical Forms and Jordan Blocks Stack of Wooden Blocks with binary code on White Backgroun

Canonical Forms and Jordan Blocks

We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. We demonstrate this with an example and provide several exercises. Jordan Basis A basis of $V$ is called a Jordan basis for $T$ if with respect to this basis $T$ has block diagonal ...
Greens Theorem by Example

Green’s Theorem (by Example)

Green’s Theorem for Simply Connected Regions Green's Theorem is named after the mathematician George Green. Theorem. (Green's Theorem) Let $R$ be a simply connected region with a piecewise smooth boundary curve $C$ oriented counterclockwise and let ${F}= M {i}+ N {j}+0 {k}$ be a continuously differentiable vector field on $R,$ then \begin{align*} \oint_C M dx+Ndy= \iint_R \left( \frac{\partial N}{\partial x}-\frac{\partial ...
Probability Density Functions Famous Gauss curve representing the distribution of probability

Probability Density Functions (Applications of Integrals)

Applications of Integrals We will consider the following applications: average value of a function over a region, mass of a lamina, electric charge, moments and center of mass, moments of inertia, and probability density functions. Average Value Recall the average value of an integrable function of one variable on a closed interval is the integral of the function over the ...
Invariant Subspaces and Generalized Eigenvectors image transformation

Invariant Subspaces and Generalized Eigenvectors

Invariant Subspaces of a Linear Transformation We let $V$ and $W$ denote real or complex vector spaces. Suppose $T \in \mathcal{L}(V)$ and $U$ a subspace of $V$, then we say $U$ is an invariant subspace of $T$ if $u\in U$ implies $Tu\in U.$ Lemma. Suppose $T \in \mathcal{L}(V)$ and $U$ a subspace of $V.$ Then all of the following hold ...
Conservative Vector Fields and Independence of Path

Conservative Vector Fields and Independence of Path

Independence of Path Definition. The line integral is called independent of path if in a region $D$, if for any two points $P$ and $Q$ in $D$ then the line integral along every piecewise smooth curve in $D$ from $P$ to $Q$ has the same value. Theorem. (Independence of Path) If $V$ is a continuous vector field on the open ...
Diagonalization of a Matrix fluid dynamics amplitude reaches a critical level start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy

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. An $n\times n$ matrix $A$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors. In that case, ...
Line Integrals Theory and Examples

Line Integrals (Theory and Examples)

Definition of a Line Integral Let $C$ be a smooth curve, with parametric equations $x=x(t),$ $y=y(t),$ and $z=z(t)$ for $a\leq t\leq b,$ that lies within the domain of a function $f(x,y,z).$ We say that $C$ is orientable if it is possible to describe direction along the curve for increasing $t.$ Partition $C$ into $n$ sub-arcs, the $k \text{th}$ of which ...
Eigenvalues and Eigenvectors Linear Transformation Matrix and Invertibility Error basically code on screen of the monitor

Eigenvalues and Eigenvectors (Find and Use Them)

Eigenvalues and Eigenvectors Let $\mathbb{F}$ be either the real numbers or the complex numbers. A nonzero vector $v$ in $\mathbb{F}^n$ is called an eigenvector of an $n\times n$ matrix $A$ if $A v$ is a scalar multiple of $v$, that is $A v= \lambda v$ for some scalar $\lambda.$ Note that this scalar $\lambda$ may be zero. The scalar $\lambda$ ...
Mathematics Scientist Adding Microscopy To Brain Study App

About Mathematics

It’s not easy to pinpoint the core definition of math. The subject covers several topics from science, reason, and space. You and I encountered math early in our lives in school, learning everything from arithmetic to algebra. Despite definitions being tough to pinpoint, the importance of answering questions with mathematics creates many jobs and research opportunities.

Mathematics is an abstract field in which numbers help to calculate questions of quantity, time, and space. The area is also a subject to study in and of itself, with mathematical proofs taking years to develop and solve. Mathematics provides answers to many questions humans have about definitions of relativity, physics, and other vital topics.

Jobs in mathematics can often include years of work merely looking for definitions of different phenomena we encounter in the natural world. There are a lot of aspects to math you might not know about yet. If you want to learn more, then starting with articles can be a high jumping-off point.

Mathematics Articles

Surprisingly, mathematics definitions are not widely accepted. Its definition changes in various ways over centuries, such as the science of quantity, logic, or intuition. A few leading schools of thought exist today about how to define it formally, but let’s leave that for more in-depth mathematics articles. In a nutshell, you can consider mathematics a way of finding and using patterns to formulate conjectures that resolve the truth or lack thereof.

Mathematics is a way of expressing and explaining logic itself. Math gives a concrete manifestation of abstract ideas through numbers and symbols. Through mathematics, we can calculate, measure, count, and analyze both physical objects and theoretical concepts. We can study shapes, motions, logic, and physical laws of the universe. Mathematics is vital for many other fields such as natural sciences, medicine, finance, engineering, and even social sciences. Learn more about the theories and applications of the area through the mathematics articles in this category of Direct Knowledge.


As far back as 3000 BC, Mesopotamian states and other societies such as Egypt started using geometry, algebra, and arithmetic. These mathematical tools helped with practical uses such as taxation, trade, and keeping track of time. The most ancient mathematical texts available come from Mesopotamia, including mentions of parts of the Pythagorean theorem. This approach makes the theory the most ancient and widely spread development in mathematics after basic geometry and arithmetic.

Later, around 500 BC, the Greeks further refined mathematical methods by introducing rigorous proofs with deductive reasoning. Of course, tests have become a crucial part of math today. You’ll find the proofs for various concepts in the mathematics articles in this category. The ancient Romans focused on applied mathematics, using it for multiple types of engineering, surveying, bookkeeping, and making calendars. On the other end of the world (at the time), the Chinese made contributions such as a place value system and the use of negative numbers. The Hindu-Arabic numeral system evolved in the first 1000 years AD, giving us the method used throughout the world today.

The Increasing Importance of Mathematics

Throughout the history of mathematics questions, long periods of relative non-development would punctuate discoveries. This process started to change in the 15th century. At this time, the Renaissance came into play, allowing mathematical developments to interact with scientific discoveries. Since that time, findings contribute to an increasing pace, including up to modern-day. Mathematics articles in this category are sure to cover these changes and keep you up to date.

Mathematics Articles in Modern Fields

Within mathematics definitions as a whole exists two main branches: pure and applied mathematics jobs. However, each of them doesn’t necessarily live entirely independently, with a distinct line between them. But the following descriptions give a good idea of the differences.

Pure Mathematics

The branch of pure mathematics explores fundamental parts of the field out of intrinsic interest. These parts include quantity, structure, space, and change. The subject of quantity uses numbers, integers, rational numbers, real numbers, irrational numbers, complex numbers, and more, all combined with arithmetic. The structure is the consequence of relations and operations within sets of numbers. The fabric can relate to arithmetic operations, and similarly, structured sets can exhibit similar properties. This concept makes it possible to study larger groups at once or branch into more abstract systems.

The study of space starts with geometry, combining numbers with space. This study is where we find the Pythagorean theorem and trigonometry. Finally, the study of change navigates using calculus, where functions are the central concept used to describe changing quantities. The study of the rate of change (or the change in change) is differential equations, used to describe dynamical systems. Other areas of pure mathematics jobs include number theory, algebra, combinatorics, and analysis. The genuine mathematics articles on these topics often won’t dwell on practical applications. Instead, they just stick to the math itself. These articles are where you can find in-depth explanations and examples of the methods, formulas, and terminology used in math.

Applied Mathematics

The applied branch, on the other hand, aims to use mathematics questions in real-world situations for solving real problems of importance. For example, probability and statistics allow us to use numerical data from experiments or observations to make predictions. This branch is useful in almost every aspect of society. It is the best way we have to predict what might happen in certain situations, allowing us to make the best decisions possible.

Computational sciences use numerical analysis and computer algebra to make complex calculations. These are especially important when counts need to be extremely precise, such as when dealing with advanced technology in medicine or navigation. The physical sciences also depend heavily on applied mathematics jobs, using it to address how real-world objects behave in the presence of external forces. This utility is widely applicable to all types of engineering jobs.

Careers in Mathematics and Related Articles

As one of the fundamental subjects, mathematics connects to the importance of other fields in some form. With a mathematics degree, you could gain another degree in a more specific field, or go straight into several areas. Some professions that rely heavily on mathematics include economist, actuary, financial planner, statistician, investment analyst, professor/teacher, software engineer, accountant, or research scientist.

Of the more specific fields, one can expect to make a median salary of about 88,000 dollars per year after having earned a Master’s degree. Growth in the area is speedy right now, so you can be confident that there’s a mathematics job out there waiting for you. You can also assume that Direct Knowledge will have thorough mathematics articles to help you with some of the toughest areas of the importance of math that you might be struggling with at first.

Mathematics Books

Mathematics applies to virtually every area of importance in our lives and the subject of study. Even so, a lot of people still shy away from the issue, thinking it’s too difficult for them. But anyone can study it with the right assistance. Whether math is a personal interest, a requirement for school, or a way of enhancing what you do professionally, the math books in this category can help you learn the crucial parts of the subject in a way that’s easy to understand.

Mathematics Books Brilliant Young Mathematician is Finishing Writing Complex Equation on the Blackboard

Finding Beauty in Questions of Mathematics

Sometimes, classic mathematics books can be convoluted and hard to figure out. Complicated examples and unnecessary details can keep students from learning subjects that shouldn’t be so difficult. But in reality, math is a method of making the world around us more comfortable to understand, not harder. It gives us a language for describing the indescribable. And that language is beautifully consistent, concise, and pure.

A lot of students might scoff at that last statement, but it is true. All you need are the right tools to help you understand mathematics definitions at a fundamental level. The mathematics books in this category provide those tools. Practical examples make the concepts clear as day and save you study time by focusing on understanding the concepts at a fundamental level.

Which Math Books are Right for You?

The books found here span a wide range of topics that are perfect for many ages and levels. And any college student knows that math is generally required as one of the core classes in college, even if you’re majoring in something else entirely. But that just shows that university course planners know math has relevance and importance in all fields. So embrace the subject with a positive attitude and see how the math books here can add to your study.

For example, math has had a long relationship with sculptors and painters over time. They often used geometry to create shapes and patterns with the most visually pleasing aspects possible or to make structures in architecture, both attractive and robust, in terms of structure. Music, too, has a deep relationship with math. Numbers and their properties relate to elements of music such as rhythm and meter, pitch, tempo, pulse, and more. The golden ratio and Fibonacci numbers are even central parts of the work of some composers.


As most students know, calculus is a big deal in mathematics jobs and theory. It has played an important role and continues to be crucial for fields like engineering and science. Hence why most students need to take it in some form or another, and the math books here include a whole series that focuses on it. There are three levels included in this category that spans calculus onetwo, and three (multivariable). They follow typical class structures reasonably well, making them high supplements for college courses or tools for AP exam prep.

Proofs, Number Theory, and More

Calculus isn’t the only math you find in university, especially if you’re majoring in math or science. In addition to Calculus, you’ll find other advanced math books here on topics like Proofs (An Introduction to Mathematical Arguments). You’ll also find some subjects which are great both for university and advanced high school students. High schoolers just starting to get into more advanced math can find books on topics like linear algebra and number theory that can help them with AP exams for college credit. Or those already in college can use them as supplemental material for current classes. Each of them starts with the basics and guides students with thorough examples and practice.

Nothing to Lose

Forget anything anyone ever said about not being a “math person.” All it takes to get comfortable in the field is practice and dedication. From there, whatever your passion, mathematics can complement it. Extend your learning process wherever you are in the world, by using mathematics in your career.

Mathematics FAQ

If you have more questions about the importance of mathematics, look no further! These popular topics can help explain mathematics definitions and meanings more clearly.

What is the full meaning of mathematics?

The full meaning of mathematics is the study, understanding, and knowledge of abstract elements of the world. Math allows people to explore human life and its relation to the planet. Although abstract concepts use math, the idea of math is simply the application of numbers as a language for complex problems and limits of human knowledge.

Why is math important in life?

Math unlocks answers to many of life’s most important questions and mysteries. To solve several scientific problems, individuals must know how to use math as a tool to reach reasonable conclusions. Without math, a variety of core elements of human life would go unknown and unexplored. Math is vital in building a rational understanding of the world.

Why is math so hard?

Math is hard because the subject is abstract. Mathematics uses numbers as a vehicle to explore several phenomena regarding space, physics, and time. To improve at math, individuals must study different theories, approaches, and proofs to manage the topic’s difficulty. By doing so, math becomes less complicated and more accessible.

Is math real or invented?

Math is a tool used to explore questions and problems of reality. Some believe math is not practical because various uses of math seem human-made. Humans discover math through critical thinking and the use of numbers to exploring abstract concepts. The process is not one of the inventions but of discovery to solve some of the world’s most convoluted problems.


  • Created by professionals with the intent of helping others become professionals, Direct Knowledge began by simply pulling together the best articles and media it could find. The company was officially founded on November 10, 2017 by a professor who wanted to share his knowledge with anyone. With an academic background and a passion for professional and individual development, the Direct Knowledge mission started with one passion: grow knowledge and offer a comprehensive place to learn, discuss, and network.

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