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...

# Mathematics Articles

Surprisingly, mathematics has no widely accepted definition. It has been defined in various ways for centuries, such as the science of quantity, logic, or intuition. A few main schools of thought exist today about how to formally define it, 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 truth or lack thereof.

Numbers and symbols aid mathematics as a way of expressing and explaining logic itself, and giving a concrete manifestation of abstract ideas. Through mathematics, we can calculate, measure, count, and analyze both physical objects and theoretical ideas. 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 field through the mathematics articles in this category of Direct Knowledge.

## History

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 makes the theorem 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 proofs have become a key part of math today, and you’ll find proofs for various concepts in the mathematics articles in this category. The ancient Romans focused on applied mathematics, using it for various types of engineering, surveying, bookkeeping, and making calendars. On the other side 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 system used throughout the world today.

Throughout the history of mathematics, long periods of relative non-development would be briefly punctuated by new discoveries. This 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, new discoveries have been made at 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.

## Modern Fields

Within mathematics as a whole exists two main branches: pure, and applied mathematics. However, each of them doesn’t necessarily exist completely 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, including quantity, structure, space, and change. Quantity uses numbers, integers, rational numbers, real numbers, irrational numbers, complex numbers, and more, all combined with arithmetic. Structure is the consequence of relations and operations within sets of numbers. The structure can relate to arithmetic operations, and similarly structured sets can exhibit similar properties. This 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 is where we find the Pythagorean theorem and trigonometry. Finally, the study of change is explored using calculus, where functions are the main 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 include number theory, algebra, combinatorics, and analysis. The pure mathematics articles on these topics often won’t dwell on practical applications. Rather, 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 in real world situations for solving real problems. For example, probability and statistics allow us to use numerical data from experiments or observations in order to make predictions. This 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 calculations 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, using it for addressing how real-world objects behave in the presence of external forces. This is widely applicable to all types of engineering.

Some other applications of mathematics include mathematical modeling, and mathematical programming for developing solutions through the use of algorithms. When just starting out in mathematics, students often won’t encounter many chances to really apply the math they’re learning. But mathematics articles will often find ways to include real word applications in the example problems. This helps give an idea of how the math helps society and how you might be able to use it one day, even if you’re not quite at the level to do it yourself yet.

## Careers in Mathematics

As one of the fundamental subjects, mathematics connects to most fields in some form or another. With a mathematics degree, you could gain another degree in a more specific field, or go straight in to a number of 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 math-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 field is very fast right now, so you can be confident that there’s a position out there waiting for you. You can also be confident that Direct Knowledge will have thorough mathematics articles to help you with some of the toughest areas of math that you might be struggling with.

## 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....

## 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...

## 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 $(X,\geq)$...

## 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....

## 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 $b$...

## 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...

## One-to-One Functions and Onto Functions

One-to-one 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...

## 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 $f$...

## 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...

## 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...

## 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...

## Green’s Theorem (by Example)

Greenâ€™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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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 $A$...

## Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the curl of $\mathbf{V}$ is defined...

## Determinant of a Matrix (Theory and Examples)

Determinants and Trace The determinant function can be defined by essentially two different methods. The advantage of the first definition, one which uses permutations, is that it provides an actual...

## Vector Fields and Gradient Fields

Introduction to Vector Fields Definition. A vector field in $\mathbb{R}^n$ is a function $\mathbf{V}$ that assigns a vector from each point in its domain. A vector field with domain $D$...

## 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...

## Jacobian (Change of Variables in Multiple Integrals)

Jacobians If $x=x(u,v)$ and $y=y(u,v)$ then the Jacobian of $x$ and $y$ with respect to $u$ and $v$ is \begin{equation} \frac{\partial (x,y)}{\partial (u,v)} =J(u,v)=\left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial...