Mathematics

# Mathematics

Overview

Mathematics is the study of quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

, space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, structure
Structure
Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

, and change
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. Mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s seek out patterns and formulate new conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

s. Mathematicians resolve the truth or falsity of conjectures by mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

s, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

.
Discussion

Recent Discussions
Quotations

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

W.S. Anglin, in Mathematics and History, elucidating the symmetry between the creative and logical aspects of mathematics.

If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.

Roger Bacon Opus Majus, bk.1, ch4.

Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and so ad infinitum,And the great fleas themselves, in turn, have greater fleas to go on,While these again have greater still, and greater still, and so on.

Augustus De Morgan, A Budget of Paradoxes

Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.

Linear Algebra by Fraleigh/Beauregard

Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!

Frederick II of Prussia|Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 221 from Frederick II of Prussia|Frederick to Voltaire, 25 November 1777.
Encyclopedia

Mathematics is the study of quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

, space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, structure
Structure
Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

, and change
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. Mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s seek out patterns and formulate new conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

s. Mathematicians resolve the truth or falsity of conjectures by mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

s, which are arguments sufficient to convince other mathematicians of their validity. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. However, mathematical proofs are less formal and painstaking than proofs in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

. Since the pioneering work of Giuseppe Peano
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...

(1858-1932), David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

(1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...

by rigorous deduction
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

from appropriately chosen axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s and definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

s. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction
Abstraction (mathematics)
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...

and logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

al reasoning, mathematics developed from counting
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...

, calculation
Calculation
A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...

, measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

, and the systematic study of the shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...

s and motions
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

of physical objects. Practical mathematics has been a human activity for as far back as written records
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....

exist. Rigorous arguments
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

first appeared in Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

, most notably in Euclid's
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world
Muslim world
The term Muslim world has several meanings. In a religious sense, it refers to those who adhere to the teachings of Islam, referred to as Muslims. In a cultural sense, it refers to Islamic civilization, inclusive of non-Muslims living in that civilization...

in AD 800, until the Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

, when mathematical innovations interacting with new scientific discoveries
Timeline of scientific discoveries
The timeline below shows the date of publication of major scientific theories and discoveries, along with the discoverer. In many cases, the discoveries spanned several years.-3rd century BC:...

led to a rapid increase in the rate of mathematical discovery that continues to the present day.

The mathematician Benjamin Peirce
Benjamin Peirce
Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics....

(1809-1880) called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

(1879-1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".

Mathematics is used throughout the world as an essential tool in many fields, including natural science
Natural science
The natural sciences are branches of science that seek to elucidate the rules that govern the natural world by using empirical and scientific methods...

, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, medicine
Medicine
Medicine is the science and art of healing. It encompasses a variety of health care practices evolved to maintain and restore health by the prevention and treatment of illness....

, and the social sciences
Social sciences
Social science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...

. Applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

and game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

. Mathematicians also engage in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

## Etymology

The word "mathematics" comes from the Greek μάθημα (máthēma), which means in ancient Greek what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson.

The word máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn.

The word "mathematics" in Greek came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical. In particular, (mathēmatikḗ tékhnē), , meant the mathematical art. In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of "mathematici" meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form (and the less commonly used singular derivative ), goes back to the Latin neuter plural (Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...

), based on the Greek plural , used by Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

(384-322BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...

, which were inherited from the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

## History

The evolution of mathematics might be seen as an ever-increasing series of abstractions
Abstraction (mathematics)
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...

, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals, was probably that of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...

physical objects, prehistoric
Prehistory
Prehistory is the span of time before recorded history. Prehistory can refer to the period of human existence before the availability of those written records with which recorded history begins. More broadly, it refers to all the time preceding human existence and the invention of writing...

peoples also recognized how to count abstract quantities, like time – days, season
Season
A season is a division of the year, marked by changes in weather, ecology, and hours of daylight.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...

s, years. Elementary arithmetic
Elementary arithmetic
Elementary arithmetic is the simplified portion of arithmetic which is considered necessary and appropriate during primary education. It includes the operations of addition, subtraction, multiplication, and division. It is taught in elementary school....

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

, multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

and division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

) naturally followed.

Since numeracy pre-dated writing
Writing
Writing is the representation of language in a textual medium through the use of a set of signs or symbols . It is distinguished from illustration, such as cave drawing and painting, and non-symbolic preservation of language via non-textual media, such as magnetic tape audio.Writing most likely...

, further steps were needed for recording numbers such as tallies or the knotted strings called quipu
Quipu
Quipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...

used by the Inca to store numerical data. Numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....

s have been many and diverse, with the first known written numerals created by Egyptians
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...

in Middle Kingdom
Middle Kingdom of Egypt
The Middle Kingdom of Egypt is the period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty, between 2055 BC and 1650 BC, although some writers include the Thirteenth and Fourteenth dynasties in the Second Intermediate...

texts such as the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...

.
The earliest uses of mathematics were in trading
Trade is the transfer of ownership of goods and services from one person or entity to another. Trade is sometimes loosely called commerce or financial transaction or barter. A network that allows trade is called a market. The original form of trade was barter, the direct exchange of goods and...

, land measurement
Land measurement
Land measurement is the general concept describing the application and theory of measurement of land. Land measurement is an integral quantitative element of Surveying....

, painting
Painting
Painting is the practice of applying paint, pigment, color or other medium to a surface . The application of the medium is commonly applied to the base with a brush but other objects can be used. In art, the term painting describes both the act and the result of the action. However, painting is...

and weaving
Weaving
Weaving is a method of fabric production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. The other methods are knitting, lace making and felting. The longitudinal threads are called the warp and the lateral threads are the weft or filling...

patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...

, "The number of papers and books included in the Mathematical Reviews
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...

database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s and their proofs
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

."

## Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce
Commerce
While business refers to the value-creating activities of an organization for profit, commerce means the whole system of an economy that constitutes an environment for business. The system includes legal, economic, political, social, cultural, and technological systems that are in operation in any...

, land measurement
Land measurement
Land measurement is the general concept describing the application and theory of measurement of land. Land measurement is an integral quantitative element of Surveying....

, architecture
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...

and later astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

invented the path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

using a combination of mathematical reasoning and physical insight, and today's string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

, a still-developing scientific theory which attempts to unify the four fundamental forces of nature
Fundamental interaction
In particle physics, fundamental interactions are the ways that elementary particles interact with one another...

, continues to inspire new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

and applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

. However pure mathematics topics often turn out to have applications, e.g. number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

in cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner...

".
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification
Mathematics Subject Classification
The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...

runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...

and inner beauty
Beauty
Beauty is a characteristic of a person, animal, place, object, or idea that provides a perceptual experience of pleasure, meaning, or satisfaction. Beauty is studied as part of aesthetics, sociology, social psychology, and culture...

. Simplicity
Simplicity
Simplicity is the state or quality of being simple. It usually relates to the burden which a thing puts on someone trying to explain or understand it. Something which is easy to understand or explain is simple, in contrast to something complicated...

and generality are valued. There is beauty in a simple and elegant proof, such as Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's proof that there are infinitely many prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

. G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....

in A Mathematician's Apology
A Mathematician's Apology
A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...

expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

. The popularity of recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...

is another sign of the pleasure many find in solving mathematical questions.

## Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

(1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation
Musical notation
Music notation or musical notation is any system that represents aurally perceived music, through the use of written symbols.-History:...

, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

can be difficult to understand for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

and field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

have been given specialized mathematical meanings. Technical terms such as homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

and integrable
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

have precise meanings in mathematics. Additionally, shorthand phrases such as "iff" for "if and only if" belong to mathematical jargon
Mathematical jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal...

. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s", based on fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proof
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...

s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.

Axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...

. It was the goal of Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...

to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable
Independence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...

formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

## Fields of mathematics

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, and analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, to set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

(foundations
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...

), to the empirical mathematics of the various sciences (applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

), and more recently to the rigorous study of uncertainty
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...

.

### Foundations and philosophy

In order to clarify the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...

, the fields of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

were developed. Mathematical logic includes the mathematical study of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, which deals in an abstract way with mathematical structure
Mathematical structure
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

s and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory
Controversy over Cantor's theory
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found wide acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers....

and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...

which (informally) imply that any formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

, model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, and proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...

, and is closely linked to theoretical
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, as well as to Category Theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

.

Theoretical computer science
Theoretical computer science
Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

includes computability theory, computational complexity theory
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

, and information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

. Computability theory examines the limitations of various theoretical models of the computer, including the most well known model – the Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute... . Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression Data compression In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use.... and entropy.  Mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics... Set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics... Category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions... Theory of computationTheory of computationIn theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm... #### Quantity The study of quantity starts with number Number A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.... s, first the familiar natural number Natural number In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively... s and integer Integer The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively... s ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic Arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers... . The deeper properties of integers are studied in number theory Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... , from which come such popular results as Fermat's Last Theorem Fermat's Last Theorem In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.... . The twin prime Twin prime A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and... conjecture and Goldbach's conjecture Goldbach's conjecture Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes... are two unsolved problems in number theory. As the number system is further developed, the integers are recognized as a subset Subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment... of the rational number Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number... s ("fractions Fraction (mathematics) A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists... "). These, in turn, are contained within the real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s, which are used to represent continuous Continuous function In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of... quantities. Real numbers are generalized to complex number Complex number A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part... s. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonion Octonion In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H... s. Consideration of the natural numbers also leads to the transfinite number Transfinite number Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these... s, which formalize the concept of "infinity Infinity Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity... ". Another area of study is size, which leads to the cardinal number Cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite... s and then to another conception of infinity: the aleph number Aleph number In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph... s, which allow meaningful comparison of the size of infinitely large sets.  Natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...s IntegerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...s Rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...s Real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...s Complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...s #### Structure Many mathematical objects, such as sets of numbers and functions Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... , exhibit internal structure as a consequence of operations Operation (mathematics) The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation.... or relations Relation (mathematics) In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple... that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory Number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... studies properties of the set of integer Integer The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively... s that can be expressed in terms of arithmetic Arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers... operations. Moreover, it frequently happens that different such structured sets (or structures Mathematical structure In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.... ) exhibit similar properties, which makes it possible, by a further step of abstraction Abstraction Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods.... , to state axiom Axiom In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true... s for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity... , rings Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition... , fields Field (mathematics) In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms... and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra Abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras... . By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory Galois theory In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory... , which involves field theory and group theory. Another example of an algebraic theory is linear algebra Linear algebra Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often... , which is the general study of vector space Vector space A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex... s, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry Geometry Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers .... and algebra Algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures... have very strong interactions in modern mathematics. Combinatorics Combinatorics Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,... studies ways of enumerating the number of objects that fit a given structure.  CombinatoricsCombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,... Number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... Group theoryGroup theoryIn 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... Graph theoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of... Order theoryOrder theoryOrder theory is a branch of mathematics which investigates our 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 gives some basic definitions... #### Space The study of space originates with geometry Geometry Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers .... – in particular, Euclidean geometry Euclidean geometry Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these... . Trigonometry Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves... is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle... . The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries Non-Euclidean geometry Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much... (which play a central role in general relativity General relativity General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics... ) and topology Topology Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing... . Quantity and space both play a role in analytic geometry Analytic geometry Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties... , differential geometry, and algebraic geometry Algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex... . Convex Convex geometry Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,... and discrete geometry Discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,... was developed to solve problems in number theory Geometry of numbers In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by .... and functional analysis Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense... but now is pursued with an eye on applications in optimization and computer science Computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational... . Within differential geometry are the concepts of fiber bundles and calculus on manifold Manifold In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold.... s, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial Polynomial In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents... equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie group Lie group In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure... s are used to study space, structure, and change. Topology Topology Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing... in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology Set-theoretic topology In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of ZFC. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The... , algebraic topology Algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology... and differential topology Differential topology In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :... . In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory Morse theory In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect... . Topology also includes the now solved Poincaré conjecture Poincaré conjecture In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space... . Other results in geometry and topology, including the four color theorem Four color theorem In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color... and Kepler conjecture Kepler conjecture The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic... , have been proved only with the help of computers.  GeometryGeometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers .... TrigonometryTrigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves... Differential geometry TopologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing... Fractal geometryFractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity... Measure theory #### Change Understanding and describing change is a common theme in the natural science Natural science The natural sciences are branches of science that seek to elucidate the rules that govern the natural world by using empirical and scientific methods... s, and calculus Calculus Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... was developed as a powerful tool to investigate it. Functions Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... arise here, as a central concept describing a changing quantity. The rigorous study of real number Real number In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π... s and functions of a real variable is known as real analysis Real analysis Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and 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... , with complex analysis 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 number theory and applied mathematics; as well as in physics,... the equivalent field for the complex number Complex number A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part... s. Functional analysis Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense... focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics Quantum mechanics Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic... . Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equation Differential equation 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... s. Many phenomena in nature can be described by dynamical system Dynamical system A dynamical system is a concept in mathematics where a fixed rule 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... s; chaos theory Chaos theory Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the... makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic Deterministic system (mathematics) In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:... behavior.  CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem... Vector calculus Differential equationDifferential equationA 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...s Dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule 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...s Chaos theoryChaos theoryChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the... Complex analysisComplex analysisComplex 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 number theory and applied mathematics; as well as in physics,... ### Applied mathematics Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional Professional A professional is a person who is paid to undertake a specialised set of tasks and to complete them for a fee. The traditional professions were doctors, lawyers, clergymen, and commissioned military officers. Today, the term is applied to estate agents, surveyors , environmental scientists,... specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation, study, and use of mathematical models in science Science Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe... , engineering Engineering Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of... , and other areas of mathematical practice. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics Pure mathematics Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of... . #### Statistics and other decision sciences Applied mathematics has significant overlap with the discipline of statistics Statistics Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.... , whose theory is formulated mathematically, especially with probability theory Probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single... . Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments Design of experiments In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments... ; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies Observational study In epidemiology and statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator... , statisticians "make sense of the data" using the art of modelling Statistical model A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but... and the theory of inference Statistical inference In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation... – with model selection Model selection Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered... and estimation Estimation Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions... ; the estimated models and consequential predictions should be tested Statistical hypothesis testing A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study . In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold... on new data. Statistical theory Statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that... studies decision problems such as minimizing the risk Risk Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"... (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best Selection algorithm In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list . This includes the cases of finding the minimum, maximum, and median elements. There are O, worst-case linear time, selection algorithms... . In these traditional areas of mathematical statistics Mathematical statistics Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis... , a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost Cost In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost. In this... , under specific constraints: For example, a designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research Operations research Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations... , control theory Control theory Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference... , and mathematical economics Mathematical economics Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity... . #### Computational mathematics Computational mathematics Computational mathematics Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied... proposes and studies methods for solving mathematical problem Mathematical problem A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's... s that are typically too large for human numerical capacity. Numerical analysis Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis .... studies methods for problems in analysis using functional analysis Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense... and approximation theory Approximation theory In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby... ; numerical analysis includes the study of approximation Approximation An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because... and discretization Discretization In mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers... broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithm Algorithm In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning... ic matrix Numerical linear algebra Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, Telecommunication, computational... and graph theory Graph theory In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of... . Other areas of computational mathematics include computer algebra and symbolic computation Symbolic computation Symbolic computation or algebraic computation, relates to the use of machines, such as computers, to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols... .  Mathematical physicsMathematical physicsMathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and... Fluid dynamicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics... Numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis .... Optimization Probability theoryProbability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single... StatisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.... CryptographyCryptographyCryptography is the practice and study of techniques for secure communication in the presence of third parties... Mathematical financeMathematical financeMathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical... Game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others... Mathematical biologyMathematical biologyMathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology... Mathematical chemistryMathematical chemistryMathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena... Mathematical economicsMathematical economicsMathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity... Control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference... ## Mathematics as profession Arguably the most prestigious award in mathematics is the , established in 1936 and now awarded every 4 years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize Nobel Prize The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895... . The Wolf Prize in Mathematics Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts... , instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize Abel Prize The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious... , was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field. A famous list of 23 open problem Open problem In science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved . Some questions remain unanswered for centuries before solutions are found... s, called "Hilbert's problems Hilbert's problems Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics... ", was compiled in 1900 by German mathematician David Hilbert David Hilbert David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of... . This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems Millennium Prize Problems The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...

", was published in 2000. Solution of each of these problems carries a \$1 million reward, and only one (the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

) is duplicated in Hilbert's problems.

## Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". In the original Latin Regina Scientiarum, as well as in German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science
Natural science
The natural sciences are branches of science that seek to elucidate the rules that govern the natural world by using empirical and scientific methods...

follows the rise of Baconian science
Baconian method
The Baconian method is the investigative method developed by Sir Francis Bacon. The method was put forward in Bacon's book Novum Organum , or 'New Method', and was supposed to replace the methods put forward in Aristotle's Organon...

, which contrasted "natural science" to scholasticism
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...

, the Aristotelean method
Organon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...

of inquiring from first principles
First principles
In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption. In mathematics, first principles are referred to as axioms or postulates...

. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology
Experimental psychology
Experimental psychology is a methodological approach, rather than a subject, and encompasses varied fields within psychology. Experimental psychologists have traditionally conducted research, published articles, and taught classes on neuroscience, developmental psychology, sensation, perception,...

, biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, or physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Many philosophers believe that mathematics is not experimentally falsifiable
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...

, and thus not a science according to the definition of Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

. However, in the 1930s Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...

convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, hypothetico
Hypothesis
A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...

-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably Imre Lakatos
Imre Lakatos
Imre Lakatos was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...

, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

of assumptions. Intuition
Intuition (knowledge)
Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...

and experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

ation also play a role in the formulation of conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

s in both mathematics and the (other) sciences. Experimental mathematics
Experimental mathematics
Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...

continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

.

The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts
Liberal arts
The term liberal arts refers to those subjects which in classical antiquity were considered essential for a free citizen to study. Grammar, Rhetoric and Logic were the core liberal arts. In medieval times these subjects were extended to include mathematics, geometry, music and astronomy...

; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities
University
A university is an institution of higher education and research, which grants academic degrees in a variety of subjects. A university is an organisation that provides both undergraduate education and postgraduate education...

divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...

.

• Definitions of mathematics
Definitions of mathematics
Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions...

• Dyscalculia
Dyscalculia
Dyscalculia is a specific learning disability involving innate difficulty in learning or comprehending simple arithmetic. It is akin to dyslexia and includes difficulty in understanding numbers, learning how to manipulate numbers, learning maths facts, and a number of other related symptoms...

• Iatromathematicians
Iatromathematicians
Iatromathematicians was a school of physicians in 17th century Italy who tried to apply laws of mathematics and mechanics in order to understand functioning of the human body...

• List of mathematics competitions
• Mathematical anxiety
Mathematical anxiety
Mathematical anxiety is anxiety about one's ability to do mathematics independent of skill.-Math anxiety:Math anxiety is a phenomenon that is often considered when examining students’ problems in mathematics. Mark H. Ashcraft, Ph.D. defines math anxiety as “a feeling of tension, apprehension, or...

• Mathematical game
Mathematical game
A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game...

• Mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

• Mathematics and art
Mathematics and art
Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid, the Parthenon, the Colosseum...

• Mathematics education
Mathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....

• Pseudomathematics
Pseudomathematics
Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models...

• Benson, Donald C., The Moment of Proof: Mathematical Epiphanies, Oxford University Press
Oxford University Press
Oxford University Press is the largest university press in the world. It is a department of the University of Oxford and is governed by a group of 15 academics appointed by the Vice-Chancellor known as the Delegates of the Press. They are headed by the Secretary to the Delegates, who serves as...

, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
• Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
• Davis, Philip J.
Philip J. Davis
Philip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...

and Hersh, Reuben
Reuben Hersh
Reuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...

, The Mathematical Experience
The Mathematical Experience
The Mathematical Experience is a 1981 book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective...

. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.
• Gullberg, Jan
Jan Gullberg
Jan Gullberg, was a Swedish writer on popular Science and Medical topics. He is the author of Mathematics: From the Birth of Numbers, published by W. W. Norton in 1997 . The book was positively reviewed in Scientific American, but more reservedly in New Scientist.-References:...

, Mathematics — From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X.
• Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics
Encyclopaedia of Mathematics
The Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....

. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online.
• Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover Publications
Dover Publications
Dover Publications is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche. It publishes primarily reissues, books no longer published by their original publishers. These are often, but not always, books in the public domain. The original published editions may be...

, 2003, ISBN 0-486-43268-8.

• Free Mathematics books Free Mathematics books collection.
• Encyclopaedia of Mathematics
Encyclopaedia of Mathematics
The Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....

online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
• HyperMath site at Georgia State University
• FreeScience Library The mathematics section of FreeScience library
• Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.)
• Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

), integral, and other mathematical equations.
• Cain, George: Online Mathematics Textbooks available free online.
• Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.
• Mathematical Structures, list information about classes of mathematical structures.
• Mathematician Biographies. The MacTutor History of Mathematics archive
MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland...

Extensive history and quotes from all famous mathematicians.
• Metamath. A site and a language, that formalize mathematics from its foundations.
• Nrich, a prize-winning site for students from age five from Cambridge University
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

• Open Problem Garden, a wiki
Wiki
A wiki is a website that allows the creation and editing of any number of interlinked web pages via a web browser using a simplified markup language or a WYSIWYG text editor. Wikis are typically powered by wiki software and are often used collaboratively by multiple users. Examples include...

of open problems in mathematics
• Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article exchange with Wikipedia. Uses TeX
TeX
TeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....

markup.
• Some mathematics applets, at MIT
• Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
• Patrick Jones' Video Tutorials on Mathematics
• Citizendium: Theory (mathematics).