Mathematics is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Quantity is a kind of property which exists as magnitude or multitude Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Benjamin Peirce called it "the science that draws necessary conclusions". Benjamin Peirce (ˈpɜrs purse) April 4, 1809 – October 6, 1880) was an American Mathematician who ^[2] Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. A mathematician is a person whose primary area of study and research is the field of Mathematics. ^[3][4] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject A definition is a statement of the meaning of a Word or Phrase. ^[5]

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. 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 Logic is the study of the principles of valid demonstration and Inference. Reasoning is the cognitive process of looking for Reasons for beliefs conclusions actions or feelings Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In Physics, motion means a constant change in the location of a body Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Mathematics in China emerged independently by the 11th century BC Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere The Timeline below shows the date of publication of major scientific theories and discoveries along with the discoverer ^[6]

Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Medicine is the art and science of healing It encompasses a range of Health care practices evolved to maintain and restore Human Health by the The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Economics is the social science that studies the production distribution, and consumption of goods and services. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application ^[7]

Contents 1 Etymology 2 History 3 Inspiration, pure and applied mathematics, and aesthetics 4 Notation, language, and rigor 5 Mathematics as science 6 Fields of mathematics 6.1 Quantity 6.2 Structure 6.3 Space 6.4 Change 6.5 Foundations and philosophy 6.6 Discrete mathematics 6.7 Applied mathematics 7 Common misconceptions 7.1 Mathematics and physical reality 8 See also 9 Notes 10 References 11 External links

Etymology

The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical". English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. ^[8] In English, however, the noun mathematics takes singular verb forms. It is often shortened to math in English-speaking North America and maths elsewhere.

History

Main article: History of mathematics

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation --> Abstraction is the process or result of generalization by reducing the information The first abstraction was probably that of numbers. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting Stone Age Paleolithic See also Paleolithic, Recent African Origin, Early Homo sapiens, Early human migrations "Paleolithic" For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of A day (symbol d is a unit of Time equivalent to 24 Hours and the duration of a single Rotation of planet Earth with respect to the A season is one of the major divisions of the Year, generally based on yearly periodic changes in Weather. A year (from Old English gēr) is the time between two recurrences of an event related to the Orbit of the Earth around the Sun Arithmetic (addition, subtraction, multiplication and division), naturally followed. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data. A tally (or tally stick) was an ancient memory aid device to record and document numbers quantities or even messages Quipu or khipu (sometimes called talking knots) were recording devices used in the Inca Empire and its predecessor societies in the Andean Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The Indus Valley civilization developed the modern decimal system, including the concept of zero. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin

From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. The Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. Commerce is a division of trade or production which deals with the exchange of goods and services from producer to final consumer Land measurement is the general concept describing the application and theory of Measurement of land. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study These needs can be roughly related to the broad subdivision of mathematics into the studies of quantity, structure, space, and change.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1. Bulletin of the American Mathematical Society (often abbreviated as Bull Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 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 theorems and their proofs. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true "^[9]

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. Many Mathematicians derive aesthetic pleasure from their work and from Mathematics in general At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Commerce is a division of trade or production which deals with the exchange of goods and services from producer to final consumer Land measurement is the general concept describing the application and theory of Measurement of land. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study For example, Richard Feynman invented the Feynman path integral using a combination of mathematical reasoning and physical insight, and today's string theory continues to inspire new mathematics. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings 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. The 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. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a In 1960 the Physicist Eugene Wigner published an article titled " The Unreasonable Effectiveness of Mathematics in the Natural Sciences " arguing that the "

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Operations Research (OR in North America South Africa and Australia and Operational Research in Europe is an interdisciplinary branch of applied Mathematics and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

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 and inner beauty. Aesthetics or esthetics ( also spelled æsthetics) is commonly known as the study of sensory or sensori-emotional values sometimes called NOTICE TO WOULD-BE-ROMEOS*************** Simplicity and generality are valued. Simplicity is the property condition or quality of being simple or un-combined There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 A Mathematician's Apology is a 1940 essay by British mathematician G Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. Recreational mathematics is an umbrella term referring to Mathematical puzzles and Mathematical games.

Notation, language, and rigor

Main article: Mathematical notation

Most of the mathematical notation in use today was not invented until the 16th century. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering ^[10] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the 18th century, Euler was responsible for many of the notations in use today. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system 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, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.

Mathematical language also is hard for beginners. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them Words such as or and only have more precise meanings than in everyday speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Mathematical jargon includes technical terms such as homeomorphism and integrable. The Language of mathematics has a vast Vocabulary of specialist and technical terms Topological equivalence redirects here see also Topological equivalence (dynamical systems. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Rigor is fundamentally a matter of mathematical proof. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements ^[11] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. A computer-assisted proof is a Mathematical proof that has been at least partially generated by computer Since large computations are hard to verify, such proofs may not be sufficiently rigorous. ^[12] Axioms in traditional thought were "self-evident truths", but that conception is problematic. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject 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. Symbolic logic is the area of Mathematics which studies the purely formal properties of strings of symbols 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 It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Hilbert's program, formulated by German mathematician David Hilbert in the 1920s was to formalize all existing theories to a finite complete set of axioms and provide In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor 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 Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. ^[13]

Mathematics as science

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German ^[14] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application Albert Einstein has 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. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical "^[15]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. Falsifiability (or "refutability" is the logical possibility that an assertion can be shown false by an observation or a physical experiment Sir Karl Raimund Popper ( July 28 1902  &ndash September 17 1994) was an Austrian and British Philosopher and a professor However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently. "^[16] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself. Imre Lakatos ( November 9, 1922 – February 2, 1974) was a Philosopher of mathematics and science,

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. John Michael Ziman ( May 16, 1925 - January 2, 2005) was a Physicist and a humanist who worked in the area of Condensed ^[17] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Experimental mathematics 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. For the mathematical journal of the same name see Experimental Mathematics (journal Experimental mathematics is an approach to mathematics in which Scientific method refers to bodies of Techniques for investigating phenomena In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right. A New Kind of Science is a Controversial book by Stephen Wolfram, published in 2002 Stephen Wolfram (born August 29, 1959 in London) is a British Physicist, Mathematician and Businessman known for his

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; 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 has driven much development in mathematics. The term liberal arts refers to a particular type of educational Curriculum broadly defined as a Classical education. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and 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 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. A university is an institution of Higher education and Research, which grants Academic degrees in a variety of subjects 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. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics.

Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,^[18][19] established in 1936 and now awarded every 4 years. The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical It is often considered, misleadingly, the equivalent of science's Nobel Prizes. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature The Wolf Prize in Mathematics, instituted in 1979, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. 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 These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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", was published in 2000. The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved

Fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i. e. , arithmetic, algebra, geometry, and analysis). Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Analysis has its beginnings in the rigorous formulation of Calculus. In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Uncertainty is a term used in subtly different ways in a number of fields including Philosophy, Statistics, Economics, Finance, Insurance

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object These, in turn, are contained within the real numbers, which are used to represent continuous quantities. In Mathematics, the real numbers may be described informally in several different ways Real numbers are generalized to complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division This is the field of abstract algebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Linear algebra is the branch of Mathematics concerned with The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

Number theory	Abstract algebra	Group theory	Order theory

Space

The study of space originates with geometry - in particular, Euclidean geometry. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Within differential geometry are the concepts of fiber bundles and calculus on manifolds. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Lie groups are used to study space, structure, and change. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country

Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Functions arise here, as a central concept describing a changing quantity. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. For functional analysis as used in psychology see the Functional analysis (psychology article Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another One of many applications of functional analysis is quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that In Mathematics, a deterministic system is a system in which no Randomness is involved in the development of future states of the system

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any formal system 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). In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science. Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

Mathematical logic	Set theory	Category theory

Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such This includes computability theory, computational complexity theory, and information theory. In Computer science, computability theory is the branch of the Theory of computation that studies which problems are computationally solvable using different Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm 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. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy. There are close parallels between the mathematical expressions for the thermodynamic Entropy, usually denoted by S, of a physical system in the Statistical thermodynamics

As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems. The relationship between the Complexity classes P and NP is an unsolved question in Theoretical computer science. The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 ^[20]

Combinatorics	Theory of computation	Cryptography	Graph theory

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects The theory of computation is the branch of Computer science that deals with whether and how efficiently problems can be solved on a Model of computation, using an Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding A business (also called firm or an enterprise) is a legally recognized organizational entity designed to provide goods and/or services to An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group. ) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the

Mathematical physics	Mathematical fluid dynamics	Numerical analysis	Optimization
Probability	Statistics	Financial mathematics	Game theory

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Fluid mechanics is the study of how Fluids move and the Forces on them Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function Probability is the likelihood or chance that something is the case or will happen Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Mathematical finance is the branch of Applied mathematics concerned with the Financial markets. Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.

Mathematics is not numerology, nor is it accountancy; nor is it restricted to arithmetic. Numerology is any of many Systems Traditions or Beliefs in a mystical or Esoteric relationship between Numbers and physical Accountancy or accounting is the measurement statement or provision of assurance about financial information primarily used by Lenders managers, Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. Pseudomathematics is a form of Mathematics -like activity that does not work within the framework definitions rules or rigor of formal mathematical models It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. Pseudoscience is defined as a body of knowledge methodology belief or practice that is claimed to be Scientific or made to appear scientific but does not adhere to the The misconceptions involved are normally based on:

• misunderstanding of the implications of mathematical rigor;
• attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
• lack of familiarity with, and therefore underestimation of, the existing literature. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse A mathematical journal is a Scientific journal which publishes exclusively (or almost exclusively mathematical papers. An academic journal is a peer-reviewed Periodical in which scholarship relating to a particular Academic discipline is published Peer review (also known as refereeing) is the process of subjecting an author's scholarly work research or Ideas to the scrutiny of others who are

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. Kurt Heegner (1893–1965 was a German private scholar from Berlin, who specialized in Radio Engineering and Mathematics. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Marin Mersenne, Marin Mersennus or le Père Mersenne ( September 8, 1588 &ndash September 1, 1648) was

Mathematics and physical reality

Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many axiom systems are derived from our perceptions and experiments, they are not dependent on them. 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

For example, we could say that the physical concept of two apples may be accurately modeled by the natural number 2. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an On the other hand, we could also say that the natural numbers are not an accurate model because there is no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of fractional or partial apples. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from any actual physical entities.

Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led physicist Eugene Wigner to write an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a In 1960 the Physicist Eugene Wigner published an article titled " The Unreasonable Effectiveness of Mathematics in the Natural Sciences " arguing that the

See also

Notes

References

External links

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