Broadly speaking, __ pure mathematics__ is mathematics that studies entirely abstract concepts.

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

- Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
- Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
- One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics.
- Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

**Pure mathematics** is comprised of 4 major sub-branches:

starts with numbers, first the familiar**Quantitative Mathematics**__natural numbers__and__integers__("whole numbers") and arithmetical operations on them, which are characterized in__arithmetic__.deals with many mathematical objects, such as**Structural Mathematics (Abstract Algebra)**__sets__of numbers and__functions__which exhibit internal structure as a consequence of__operations__or__relations__that are defined on the set.starts with the study of space originating with**Spatial Mathematics**__geometry__– in particular,__Euclidean geometry__, which combines space and numbers,is the branch of**Mathematical Analysis (Change Mathematics)**__mathematics__dealing with__limits__and related theories, such as__differentiation__,__integration__,__measure__,__infinite series__, and__analytic functions__.

__ Quantitative Mathematics__ starts with the quantity of numbers, first the familiar

**Natural Numbers**

In__mathematics__, the**natural numbers**are those used for__counting__(as in "there are*six*coins on the table") and__ordering__(as in "this is the*third*largest city in the country"). In common language, words used for counting are "__cardinal numbers__" and words used for ordering are "__ordinal numbers__". The natural numbers are the basis from which many other number sets may be built by extension: the__integers__, by including (if not yet in) the__neutral element__0 and an__additive inverse__(−*n*) for each nonzero natural number*n*; the__rational numbers__, by including a__multiplicative inverse__(1/*n*) for each nonzero integer*n*(and also the product of these inverses by integers); the__real numbers__by including with the rationals the__limits__of (converging)__Cauchy sequences__of rationals; the__complex numbers__, by including with the real numbers the unresolved__square root of minus one__(and also the sums and products thereof); and so on. These chains of extensions make the natural numbers canonically__embedded__(identified) in the other number systems. Properties of the natural numbers, such as__divisibility__and the distribution of__prime numbers__, are studied in__number theory__. Problems concerning counting and ordering, such as__partitioning__and__enumerations__, are studied in__combinatorics__.**Integers**

An**integer**(from the__Latin__meaning "whole") is a*integer*__number__that can be written without a__fractional component__. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not.**Z**is a__subset__of the set of all__rational__numbers**Q**, in turn a subset of the__real__numbers**R**. Like the natural numbers,**Z**is__countably infinite__. The integers form the smallest__group__and the smallest__ring__containing the__natural numbers__. In__algebraic number theory__, the integers are sometimes qualified as**rational integers**to distinguish them from the more general__algebraic integers__. In fact, the (rational) integers are the algebraic integers that are also__rational numbers__.**Rational Numbers**

In__mathematics__, a**rational number**is any__number__that can be expressed as the__quotient__or__fraction__*p*/*q*of two__integers__, a__numerator__*p*and a non-zero__denominator__*q*. Since*q*may be equal to 1, every integer is a rational number. The__set__of all rational numbers, often referred to as "**the rationals**", the**field of rationals**or the**field of rational numbers**is usually denoted by a boldface**Q**(or__blackboard bold__).**Real Numbers**

In__mathematics__, a**real number**is a value of a__continuous quantity__that can represent a distance along a__line__. The adjective*real*in this context was introduced in the 17th century by__René Descartes__, who distinguished between real and__imaginary____roots__of__polynomials__. The real numbers include all the__rational numbers__, such as the__integer__−5 and the__fraction__4/3, and all the__irrational numbers__, such as √2 (1.41421356..., the__square root of 2__, an irrational__algebraic number__). Included within the irrationals are the__transcendental numbers__, such as__π__(3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as__time__,__mass__,__energy__,__velocity__, and many more. Real numbers can be thought of as points on an infinitely long__line__called the__number line__or__real line__, where the points corresponding to__integers__are equally spaced.**Complex Numbers**

A**complex number**is a__number__that can be expressed in the form*a*+*bi*, where*a*and*b*are real numbers, and*i*is a solution of the equation*x*2 = −1. Because no__real number__satisfies this equation,*i*is called an__imaginary number__. For the complex number*a*+*bi*,*a*is called the*real part*, and*b*is called the*imaginary part*. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world. The complex number system can be defined as the__algebraic extension__of the ordinary real numbers by an imaginary number*i*. This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable*i*, with the rule*i*2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a__field__.

In** **__ structural mathematics (Abstract Algebra)__, many mathematical objects, such as

**Combinatorics**

Combinatorics__mathematics__primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of__finite____structures__. It is closely related to many other areas of mathematics and has many applications ranging from__logic__to__statistical physics__, from__evolutionary biology__to__computer science__, etc. Subfields are:*Enumerative combinatorics**Analytic combinatorics*__Analytic combinatorics__*Partition theory**Graph theory**Design theory**Finite geometry**Order theory**Matroid theory**Extremal combinatorics**Probabilistic combinatorics**Algebraic combinatorics**Combinatorics on words**Geometric combinatorics**Topological combinatorics**Arithmetic combinatorics**Infinitary combinatorics*

**Number Theory**

Number theory**arithmetic**, is a branch of__pure mathematics__devoted primarily to the study of the__integers__. Number theorists study__prime numbers__as well as the properties of objects made out of integers (e.g.,__rational numbers__) or defined as generalizations of the integers (e.g.,__algebraic integers__). Integers can be considered either in themselves or as solutions to equations (__Diophantine geometry__). Questions in number theory are often best understood through the study of__analytical__objects (e.g., the__Riemann zeta function__) that encode properties of the integers, primes or other number-theoretic objects in some fashion (__analytic number theory__). One may also study__real numbers__in relation to rational numbers, e.g., as approximated by the latter (__Diophantine approximation__). The older term for number theory is*arithmetic*.**Group Theory**

In__mathematics__and__abstract algebra__,**group theory**studies the__algebraic structures__known as__groups__. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as__rings__,__fields__, and__vector spaces__, can all be seen as groups endowed with additional__operations__and__axioms__. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.__Linear algebraic groups__and__Lie groups__are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as__crystals__and the__hydrogen atom__, may be modelled by__symmetry groups__. Thus group theory and the closely related__representation theory__have many important applications in__physics__,__chemistry__, and__materials science__. Group theory is also central to__public key cryptography__.**Graph Theory**

In__mathematics__,**graph theory**is the study of, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of*graphs*,*vertices**nodes*, or points that are connected by edges, arcs, or lines. Graphs can be directed or undirected. Graphs are one of the prime objects of study in__discrete mathematics__.**Order Theory**

**Order theory**is a branch of__mathematics__which investigates the intuitive notion of order using__binary relations__. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the__order theory glossary__.**Elementary algebra**

The part of algebra that is usually taught in elementary courses of mathematics.**Abstract algebra**

In which__algebraic structures__such as__groups__,__rings__and__fields__are__axiomatically__defined and investigated.**Linear algebra**

In which the specific properties of__linear equations__,__vector spaces__and__matrices__are studied. Algebraic structures, with their associated__homomorphisms__, form__mathematical categories__.__Category theory__is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.**Universal Algebra**

**Universal algebra**(sometimes called**general algebra**) is the field of__mathematics__that studies__algebraic structures__themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.**Boolean algebra**A branch of algebra abstracting the computation with the

__truth values__*false*and*true*.**Commutative algebra**

**Commutative algebra**is the branch of__algebra__that studies__commutative rings__, their__ideals__, and__modules__over such rings. Both__algebraic geometry__and__algebraic number theory__build on commutative algebra. Prominent examples of commutative rings include__polynomial rings__, rings of__algebraic integers__, including the ordinary__integers__, and*p*__-adic integers__.**Computer algebra**

The implementation of algebraic methods as__algorithms__and__computer programs__.**Homological algebra**

The study of algebraic structures that are fundamental to study__topological spaces__.**Algebraic number theory**

In which the properties of numbers are studied from an algebraic point of view.**Algebraic geometry**

A branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.**Algebraic combinatorics**

In which algebraic methods are used to study combinatorial questions.**Relational algebra**

A set of__finitary relations__that is__closed__under certain operators.**Algebra over a field****or more generally****algebra over a ring****.**

Many classes of algebras over a field or over a ring have a specific name:**Algebra In****measure theory****Algebra In****category theory****Algebra In****logic**

In** **__ spatial Mathematics__ the study of space originates with

**Geometry**

**Geometry**(from the__Ancient Greek__: γεωμετρία;"earth",*geo-*"measurement") is a branch of*-metron*__mathematics__concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a__geometer__.**Trignonometry**

**Trigonometry**(from__Greek__, "triangle" and*trigōnon*, "measure"*metron*__[1]__) is a branch of__mathematics__that studies relationships involving lengths and__angles__of__triangles__. The field emerged in the__Hellenistic world__during the 3rd century BC from applications of__geometry__to__astronomical studies__.**Differential Geometry**

**Differential geometry**is a__mathematical__discipline that uses the techniques of__differential calculus__,__integral calculus__,__linear algebra__and__multilinear algebra__to study problems in__geometry__. The__theory of plane and space curves__and__surfaces__in the three-dimensional__Euclidean space__formed the basis for development of differential geometry during the 18th century and the 19th century.**Topology**

In__mathematics__,**topology**(from the__Greek__τόπος,*place*, and λόγος,*study*) is concerned with the properties of space that are preserved under__continuous__deformations, such as stretching,__crumpling__and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called__open sets__, that satisfy certain properties, turning the given set into what is known as a__topological space__. Important topological properties include__connectedness__and__compactness__.**Fractal Geometry**

In mathematics, a**fractal**is an__abstract object__used to describe and simulate naturally occurring objects. Artificially created fractals commonly exhibit similar patterns at increasingly small scales. It is also known as**expanding symmetry**or**evolving symmetry**. If the replication is exactly the same at every scale, it is called a__self-similar__pattern. An example of this is the__Menger sponge__. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the__Mandelbrot set__. Fractals also include the idea of a detailed pattern that repeats itself.**Measure Theory**

In__mathematical analysis__, a**measure**on a__set__is a systematic way to assign a number to each suitable__subset__of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the__Lebesgue measure__on a__Euclidean space__, which assigns the conventional__length__,__area__, and__volume__of__Euclidean geometry__to suitable subsets of the*n*-__dimensional__Euclidean space**R***n*. For instance, the Lebesgue measure of the__interval__[0, 1] in the__real numbers__is its length in the everyday sense of the word – specifically, 1.

**In **__ change mathematics__, understanding and describing change is a common theme in the

**Calculus**

It has two major branches,__differential calculus__(concerning rates of change and slopes of curves), and__integral calculus__(concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the__fundamental theorem of calculus__. Both branches make use of the fundamental notions of__convergence__of__infinite sequences__and__infinite series__to a well-defined__limit__**Vector Calculus**

**Vector calculus**, or**vector analysis**, is a branch of mathematics concerned with__differentiation__and__integration__of__vector fields__, primarily in 3-dimensional__Euclidean space__The term "vector calculus" is sometimes used as a synonym for the broader subject of__multivariable calculus__, which includes vector calculus as well as__partial differentiation__and__multiple integration__. Vector calculus plays an important role in__differential geometry__and in the study of__partial differential equations__. Vector calculus studies various__differential operators__defined on scalar or vector fields, which are typically expressed in terms of the__del__operator, also known as "nabla". Del is used to calculate the__gradient__,__curl__,__divergence__, and__Laplacian__of various object like__Scalar fields__,__Vector fields__and__Vectors and pseudovectors__**Differential Equations**

A**differential equation**is a__mathematical____equation__for an unknown__function__of one or several__variables__that relates the values of the function itself and its__derivatives__of various__orders__. Differential equations play a prominent role in__engineering__,__physics__,__economics__,__biology__, and other disciplines.**Dynamical Systems**

In__mathematics__, a**dynamical system**is a system in which a__function__describes the__time__dependence of a point in a__geometrical space__. Examples include the__mathematical models__that describe the swinging of a clock__pendulum__,__the flow of water in a pipe__, and__the number of fish each springtime in a lake__. At any given time, a dynamical system has a__state__given by a__tuple__of__real numbers__(a__vector__) that can be represented by a__point__in an appropriate__state space__(a geometrical__manifold__). The*evolution rule*of the dynamical system is a function that describes what future states follow from the current state. Often the function is__deterministic__, that is, for a given time interval only one future state follows from the current state. However, some systems are__stochastic__, in that random events also affect the evolution of the state variables.**Real analysis**

Real analysis**theory of functions of a real variable**) is a branch of mathematical analysis dealing with the__real numbers__and real-valued functions of a real variable. In particular, it deals with the analytic properties of real__functions__and__sequences__, including__convergence__and__limits__of__sequences__of real numbers, the__calculus__of the real numbers, and__continuity__,__smoothness__and related properties of real-valued functions.**Complex Analysis**

**Complex analysis**, traditionally known as the**theory of functions of a complex variable**, is the branch of__mathematical analysis__that investigates__functions__of__complex numbers__. It is useful in many branches of mathematics, including__algebraic geometry__,__number theory__,__analytic combinatorics__,__applied mathematics__; as well as in__physics__, including the branches of__hydrodynamics__,__thermodynamics__, and particularly__quantum mechanics__. By extension, use of complex analysis also has applications in engineering fields such as__nuclear__,__aerospace__,__mechanical__and__electrical engineering__.**Functional analysis**

Functional analysis__vector spaces__endowed with some kind of limit-related structure (e.g.__inner product__,__norm__,__topology__, etc.) and the__linear operators__acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of__spaces of functions__and the formulation of properties of transformations of functions such as the__Fourier transform__as transformations defining__continuous__,__unitary__etc. operators between function spaces. This point of view turned out to be particularly useful for the study of__differential__and__integral equations__.**Measure theory**

A**measure**on a__set__is a systematic way to assign a number to each suitable__subset__of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the__Lebesgue measure__on a__Euclidean space__, which assigns the conventional__length__,__area__, and__volume__of__Euclidean geometry__to suitable subsets of the {\displaystyle n-dimensional Euclidean space. For instance, the Lebesgue measure of the__interval__[0,1] in the__real numbers__is its length in the everyday sense of the word – specifically, 1.**Numerical analysis**

**Numerical analysis**is the study of__algorithms__that use numerical__approximation__(as opposed to general__symbolic manipulations__) for the problems of mathematical analysis (as distinguished from__discrete mathematics__).**Chaos Theory**

**Chaos theory**is a branch of__mathematics__focusing on the behavior of__dynamical systems__that are highly sensitive to__initial conditions__. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of__chaotic complex systems__, there are underlying patterns, constant__feedback loops__, repetition,__self-similarity__,__fractals__,__self-organization__, and reliance on programming at the initial point known as*sensitive dependence on initial conditions*. The__butterfly effect__describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in China can cause a hurricane in Texas. Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This happens even though these systems are__deterministic__, meaning that their future behavior is fully determined by their initial conditions, with no__random__elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as**deterministic chaos**, or simply**chaos**.

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