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Arithmetic

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.

Greek language

Greek (Greek Ελληνικά, IPA – "Hellenic") is an Indo-European language with a documented history of 3,500 years. Today, it is spoken by 15 million people in Greece, Cyprus, the former Yugoslavia, particularly The Former Yugoslav Republic of Macedonia, Bulgaria, Albania and Turkey. There are also many Greek emigrant communities around the world, such as those in Melbourne, Australia which is the third-largest Greek-populated city in the world, after Athens and Thessaloniki. Greek has been written in the Greek alphabet, the first true alphabet, since the 9th century B.C. and before that, in Linear B and the Cypriot syllabaries. Greek literature has a long and rich tradition.

History

This article does not cover the reconstructed history of Greek prior to the use of writing. For more information, see main article on Proto-Greek language. Greek has been spoken in the Balkan Peninsula since the 2nd millennium BC. The earliest evidence of this is found in the Linear B tablets dating from 1500 BC. The later Greek alphabet (q.v.) is unrelated to Linear B, and was derived from the Phoenician alphabet (abjad); with minor modifications, it is still used today. Greek is conventionally divided into the following periods:
- Mycenean Greek: the language of the Mycenean civilisation. It is recorded in the Linear B script on tablets dating from the 16th century BC onwards.
- Classical Greek (also known as Ancient Greek): In its various dialects was the language of the Archaic and Classical periods of Greek civilisation. It was widely known throughout the Roman empire. Classical Greek fell into disuse in western Europe in the Middle Ages, but remained known in the Byzantine world, and was reintroduced to the rest of Europe with the Fall of Constantinople and Greek migration to Italy.
- Hellenistic Greek (also known as Koine Greek): The fusion of various ancient Greek dialects with Attic (the dialect of Athens) resulted in the creation of the first common Greek dialect, which gradually turned into one of the world's first international languages. Koine Greek can be initially traced within the armies and conquered territories of Alexander the Great, but after the Hellenistic colonisation of the known world, it was spoken from Egypt to the fringes of India. After the Roman conquest of Greece, an unofficial diglossy of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. Through Koine Greek it is also traced the origin of Christianity, as the Apostles used it to preach in Greece and the Greek-speaking world. It is also known as the Alexandrian dialect, Post-Classical Greek or even New Testament Greek (after its most famous work of literature).
- Medieval Greek: The continuation of Hellenistic Greek during medieval Greek history as the official and vernacular (if not the literary nor the ecclesiastic) language of the Byzantine Empire, and continued to be used until, and after the fall of that Empire in the 15th century. Also known as Byzantine Greek.
- Modern Greek: Stemming independently from Koine Greek, Modern Greek usages can be traced in the late Byzantine period (as early as 11th century). Two main forms of the language have been in use since the end of the medieval Greek period: Dhimotikí (Δημοτική), the Demotic (vernacular) language, and Katharévousa (Καθαρεύουσα), an imitation of classical Greek, which was used for literary, juridic, and scientific purposes during the 19th and early 20th centuries. Demotic Greek is now the official language of the modern Greek state, and the most widely spoken by Greeks today. It has been claimed that an "educated" speaker of the modern language can understand an ancient text, but this is surely as much a function of education as of the similarity of the languages. Still, Koinē , the version of Greek used to write the New Testament and the Septuagint, is relatively easy to understand for modern speakers. Greek words have been widely borrowed into the European languages: astronomy, democracy, philosophy, thespian, etc. Moreover, Greek words and word elements continue to be productive as a basis for coinages: anthropology, photography, isomer, biomechanics etc. and form, with Latin words, the foundation of international scientific and technical vocabulary. See English words of Greek origin, and List of Greek words with English derivatives.

Classification

Greek is an independent branch of the Indo-European language family. The ancient languages which were probably most closely related to it, Ancient Macedonian language (which may be regarded as a dialect of Greek) and Phrygian, are not well enough documented to permit detailed comparison. Among living languages, Armenian seems to be the most closely related to it.

Geographic distribution

Modern Greek is spoken by about 15 million people mainly in Greece and Cyprus. There are also Greek-speaking populations in Georgia, Ukraine, Egypt, Turkey, Albania, Former Yugoslav Republic of Macedonia and Southern Italy. The language is spoken also in many other countries where Greeks have settled, including Armenia, Australia, Austria, Belgium, Bulgaria, Canada, Denmark, France, Germany, Netherlands, Sweden, United Kingdom, and the United States.

Official status

Greek is the official language of Greece where it is spoken by about 99.5% of the population. It is also, alongside Turkish, the official language of Cyprus. Due to the membership of Greece and Cyprus, Greek is one of the 20 official languages of the European Union.

Phonology

This section generally describes the post-Classic phonology of the Greek language. :All phonetic transcriptions in this section use the International Phonetic Alphabet

Vowel sounds

Greek has 5 vowel sounds, all phonemic:

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much 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.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. 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. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires 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. 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. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. 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. 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 (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. 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.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) 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. [http://info.med.yale.edu/therarad/summers/ziman.htm] 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.

Overview of 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. 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). 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) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. 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. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. 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. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. 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 part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. 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. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Mathematician

A mathematician is a person whose area of study and research is mathematics. Today, most mathematicians are professors at a university or other research institution; however, a minority have a non-academic career and are often known as amateur mathematicians. While a number of misinformed people may believe mathematics is fully understood (as it is often presented this way in elementary textbooks), in fact, there is ongoing research into many areas of mathematics. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics). Unlike the other sciences, research in mathematics generally does not consist of performing experiments. Rather, mathematics is about problem-solving, where truths are deduced from other known truths. Computer experiments and other numerical evidence might be a part of this process, but in the end, mathematics research is about constructing proofs of theorems. In particular, calculation is not a big part of mathematics research, and mathematicians need not have any extraordinary ability in adding or multiplying numbers. See mental calculators to read about prodigies at performing such calculations.

Motivation

Mathematicians are typically interested in finding and describing patterns that may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. Problems have come from physics, economics, games, generalizations of earlier mathematics, and some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown the eventually applications are found. For example, number theory originally seemed to be without purpose, but after the invention of computers it gained countless applications to algorithms and cryptography.

Differences

Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question. Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal, a theorem that is proven true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematical theories are static. Once a statement is considered a theorem, it remains true forever.

Demographics

As is the case in many scientific disciplines, the field of mathematics has been disproportionately dominated by men. Among the minority of prominent female mathematicians are Emmy Noether (1882 - 1935), Sophie Germain (1776 - 1831), Sofia Kovalevskaya (1850 - 1891), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Mary Ellen Rudin, Eva Tardos, Émilie du Châtelet, Mary Cartwright and Marianna Csörnyei.

Quotes

...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. :-Saint Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such) A mathematician is a machine for turning coffee into theorems. :-Paul Erdős Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. (Mathematicians are [like] a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.) :-Johann Wolfgang von Goethe Some humans are mathematicians; others aren't. :-Jane Goodall (1971) In the Shadow of Man

Jokes

Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows: :An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep. :The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep. :Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs - and returns to his slumbers. Another joke goes thus: :Three men are flying in a hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing a field and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in a hot air balloon". One of the men in the air then says to his friends, "He was a mathematician - he thought before answering, his answer was totally right and totally useless" And another: :An astrologer, a chemist, and a mathematician are on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by. The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!" The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black." The mathematician then says, "Actually, we can only be sure there is at least one Scottish sheep of which at least one side is black" And finally: : An experiment is being made. A physicist (or an engineer) and a mathematician are asked to boil hot water, but the kettle is in the living room. The physicist goes to the living room, takes the kettle, returns to the kitchen and puts it on the stove and boils the water. The mathematician does the same. In the second stage, the kettle is in the kitchen and the two are again asked to boil hot water. The physicist simply puts the kettle on the stove and boils the water. However, the mathematician takes the kettle, puts it in the living room and declares: "We have already solved this problem!"

Links and references

References

- A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
- Reprint edition, Cambridge University Press, 1992; ISBN 0521427061
- First edition, 1940
- Dunham, William. The Mathematical Universe. John Wiley 1994.

External links

- [http://www-history.mcs.st-and.ac.uk/history/index0.html The MacTutor History of Mathematics archive], a very complete list of detailed biographies.
- [http://genealogy.math.ndsu.nodak.edu/ The Mathematics Genealogy Project], which allows to follow the succession of thesis advisors for most mathematicians, living or dead. Category:Mathematical science occupations
- ja:数学者 ko:수학자 th:นักคณิตศาสตร์ __NOTOC__

Number theory

Traditionally, number theory is the branch of pure mathematics concerned with the properties of integers. It contains many results and open problems that are easily understood, even by non-mathematicians. More generally, the field has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. See for example the list of number theory topics. Mathematicians working in the field of number theory are called number theorists. The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, fundamental theorem of arithmetic). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system.

Fields

Elementary number theory

In elementary number theory, the integers are studied without use of techniques from other mathematical fields. Questions of divisibility, the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, investigation of perfect numbers and congruences belong here. Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity. The properties of multiplicative functions such as the Möbius function and Euler's φ function are investigated; so are integer sequences such as factorials and Fibonacci numbers. Many questions in elementary number theory appear simple but may require very deep consideration and new approaches. Examples are
- The Goldbach conjecture concerning the expression of even numbers as sums of two primes,
- Catalan's conjecture regarding successive integer powers,
- The twin prime conjecture about the infinitude of prime pairs, and
- The Collatz conjecture concerning a simple iteration. The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem).

Analytic number theory

Analytic number theory employs the machinery of calculus and complex analysis to tackle questions about integers. The prime number theorem and the related Riemann hypothesis are examples. Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well. Proofs of the transcendence of mathematical constants, such as π or e, are also classified as analytical number theory. While statements about transcendental numbers may seem to be removed from the study of integers, they really study the possible values of polynomials with integer coefficients evaluated at, say, e; they are also closely linked to the field of Diophantine approximation, where one investigates "how well" a given real number may be approximated by a rational one.

Algebraic number theory

In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. These domains contain elements analogous to the integers, the so-called algebraic integers. In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed -- Galois theory, group cohomology, class field theory, group representations and L-functions -- is that it allows to recover that order partly for this new class of numbers. Many number theoretical questions are best attacked by studying them modulo p for all primes p (see finite fields). This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.

Geometric number theory

Geometric number theory (traditionally called geometry of numbers) incorporates all forms of geometry. It starts with Minkowski's theorem about lattice points in convex sets and investigations of sphere packings.

Combinatorial number theory

Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.

Computational number theory

Computational number theory studies algorithms relevant in number theory. Fast algorithms for prime testing and integer factorization have important applications in cryptography.

History

Early history

Number theory was a favorite study among the Ancient Greeks, who were aware of the Diophantine equation concept in numerous special cases. It revived in the sixteenth and seventeenth centuries, in Europe, with François Viète, Bachet de Meziriac, and especially Fermat, whose infinite descent method was the first general idea for dealing with diophantine questions. In the eighteenth century Euler and Lagrange made major contributions.

Beginnings of a systematic theory

Around the beginning of the nineteenth century books of Legendre (1798), and Gauss put together the first systematic theories. Gauss's Disquisitiones Arithmeticae (1801) may be said to begin the modern theory of numbers. The formulation of the theory of congruences starts with Gauss's Disquisitiones. He introduced the symbolism :

a \equiv b \pmod c,

and explored most of the field. Chebyshev published in 1847 a work in Russian on the subject, and in France Serret popularised it. Besides summarizing previous work, Legendre stated the law of quadratic reciprocity. This law, discovered by induction and enunciated by Euler, was first proved by Legendre in his Théorie des Nombres (1798) for special cases. Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. To the subject have also contributed: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker. The theory extends to include cubic and biquadratic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer). To Gauss is also due the representation of numbers by binary quadratic forms.

Prime number theory

A recurring and productive theme in number theory is the study of the distribution of prime numbers. Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager. Chebyshev (1850) gave useful bounds for the number of primes between two given limits. Riemann introduced complex analysis into the theory of the Riemann zeta function. This led to a relation between the zeros of the zeta function and the distribution of primes, eventually leading to a proof of prime number theorem independently by Hadamard and de la Vallée Poussin in 1896. However, an elementary proof was given later by Paul Erdős and Atle Selberg in 1949+. Here elementary means that it does not use techniques of complex analysis; however, the proof is still very ingenious and difficult. The Riemann hypothesis, which would give much more accurate information, is still an open question.

Nineteenth-century developments

Cauchy, Poinsot (1845), Lebesgue(?) (1859, 1868), and notably Hermite have added to the subject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy advance in the theory of forms in general. Smith gave a complete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith. Dirichlet was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's theorem on :

x^n+y^n \neq z^n,

which Euler and Legendre had proved for

n = 3, 4

, Dirichlet showing that

x^5+y^5 \neq az^5

. Among the later French writers are Borel; Poincaré, whose memoirs are numerous and valuable; Tannery, and Stieltjes. Among the leading contributors in Germany were Kronecker, Kummer Schering, Bachmann, and Dedekind. In Austria Stolz's Vorlesungen über allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) were scholarly of general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory.

Quotations

- Mathematics is the queen of the sciences and number theory is the queen of mathematics. — Gauss
- God invented the integers; all else is the work of man. — Kronecker
- I know numbers are beautiful. If they aren't beautiful, nothing is. — Erdős

References

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- Smith, David. [http://www.gutenberg.net/etext05/hsmmt10p.pdf History of Modern Mathematics (1906)] (adapted public domain text)
- Important publications in number theory

External links

- [http://www.numbertheory.org Number Theory Web] Category:Discrete mathematics ko:수론 ja:数論 th:ทฤษฎีจำนวน

Business

Business refers to at least three closely related commercial topics. The first is a commercial, professional or industrial organization or enterprise, generally referred to as "a business." The second is commercial, professional, and industrial activity generally, as in "business continues to evolve as markets change." Finally, business can be used to refer to a particular area of economic activity, such as the "record business" or the "computer business" (see Industry). This article is concerned primarily with the first definition of individual businesses, but also contains links to general business and management topics, in the sense of the second definition. Individual businesses are established in order to perform economic activities. With some exceptions (such as cooperatives, non-profit organizations and generally, institutions of government), businesses exist to produce profit. In other words, the owners and operators of a business have as one of their main objectives the receipt or generation of a financial return in exchange for expending time, effort and capital.

Types of Businesses

There are many types of businesses, and, as a result, businesses can be classified in many ways. One of the most common focuses on the primary profit-generating activities of a business, for example:
- Manufacturers produce products, from raw materials or component parts, which they then sell at a profit. Companies that make physical goods, such as cars or pipes, are considered manufacturers.
- Service businesses offer intangible goods or services and typically generate a profit by charging for labor or other services provided to other businesses or consumers. Organizations ranging from house painters to consulting firms to restaurants are types of service businesses.
- Retailers and Distributors act as middle-men in getting goods produced by manufacturers to the intended consumer, generating a profit as a result of providing sales or distribution services. Most consumer-oriented stores and catalogue companies are distributors or retailers.
- Agriculture and mining businesses are concerned with the production of raw material, such as plants or minerals.
- Financial businesses include banks and other companies that generate profit through investment and management of capital.
- Information businesses generate profits primarily from the resale of intellectual property and include movie studios, publishers and packaged software companies.
- Utilities produce public services, such as heat, electricity, or sewage treatment, and are usually government chartered.
- Real estate businesses generate profit from the selling, renting, and development of properties, homes, and buildings.
- Transportation businesses deliver goods and individuals from location to location, generating a profit on the transportation costs. There are many other divisions and subdivisions of businesses. The authoritative list of business types for North America (although it is widely used around the world) is generally considered to be the NAICS, or North American Industry Classification System. The equivalent European Union list is the [http://www.fifoost.org/database/nace/nace-en_2002AB.php NACE].

Business departments

Within businesses one can often find similar departments, named (and not limited to):
- Administration
- Finance & controlling
- Human ressources
- Management
- Marketing & sales
- Production/service
- Purchasing

Business and Government

Most legal jurisdictions specify the forms that a business can take, and a body of commercial law has developed for each type. Some common types include partnerships, corporations (also called limited liability companies), and sole proprietorships.

Business and Management

The study of the efficient and effective operation of a business is called management. The main branches of management are financial management, marketing management, human resource management, strategic management, production management, service management, information technology management, and business intelligence.

External links

- [http://business-articles.us/ Business Articles]
- [http://www.growfolio.com/ growFolio - Online Business Magazine for Fresh Thinkers]
- [http://finance.yahoo.com/ Yahoo! Finance] Aggregates some really good business articles
- Category:Academic disciplines Category:School subjects ja:ビジネス th:ธุรกิจ

Subtraction

Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. Subtraction is denoted by an minus sign in infix notation. The traditional names for the parts of the formula :c − b = a are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage, while "difference" is very common. Subtraction is used to model several closely related processes: #From a given collection, take away (subtract) a given number of objects. #Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal. #Compare two objects to find their difference. For example, the difference between $800 and $600 is $800 − $600 = $200. In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative— in fact, it is anticommutative— but addition of signed numbers is both.

Basic subtraction: integers

anticommutative Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition: :a + b = c. From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction: :c − b = a. addition Now, imagine a line segment labelled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction. The solution is to consider the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …). From 3, it takes 4 steps to the left to get to −1, so :3 − 4 = −1.

External links

Printable Worksheets: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1214&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3 One Digit Subtraction], [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1202&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3 Two Digit Subtraction], and [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1273&CurriculumID=3&Method=Worksheet&NQ=24&NQ4P=3 Four Digit Subtraction]
- [http://www.cut-the-knot.org/Curriculum/Arithmetic/SubtractionGame.shtml Subtraction Game] at cut-the-knot
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Subtraction1 Subtraction on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead] Category:Arithmetic ko:뺄셈 ja:減法 simple:Subtraction th:การลบ

Multiplication

:This article is about multiplication in mathematics. For multiplication in music, see multiplication (music). In its simplest form, multiplication is the sum of a list of identical numbers. For example, the product 7 × 4 is 7 + 7 + 7 + 7. The numbers being multiplied are called the multiplicand and multiplier or the factors.

Notation

Multiplication can be denoted in several equivalent ways. All of the following mean, "5 times 2": :5×2 :5·2 :(5)2, 5(2), (5)(2), 5[2], [5]2, [5][2] :5
- 2 The asterisk (
- ) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by Juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like :5x and xy This is potentially confusing if variables are permitted to have names longer than one letter. The notation is not used with numbers alone: 52 never means 5 × 2. If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written

1 \cdot 2 \cdot \ldots \cdot 99 \cdot 100

. This can also be written with the ellipsis vertically placed in the middle of the line, as

1 \cdot 2 \cdot \cdots \cdot 99 \cdot 100

. Alternatively, the product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. This is defined as: :

\prod_^ x_ := x_ \cdot x_ \cdot x_ \cdot \cdots \cdot x_ \cdot x_.

The subscript gives the symbol for a dummy variable (

i

in our case) and its lower value (

m

); the superscript gives its upper value. So for example: :

\prod_^ \left(1 + \right) = \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) = .

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). The product of such a series is defined as the limit of the product of the first

n

terms, as

n

grows without bound. That is: :

\prod_^ x_ := \lim_ \prod_^ x_.

One can similarly replace

m

with negative infinity, and :

\prod_^\infty x_i := \left(\lim_\prod_^m x_i\right) \cdot \left(\lim_\prod_^n x_i\right),

for some integer

m

, provided both limits exist.

Definition

As for what multiplication means, the product of two whole numbers n and m is: :

mn := \sum_^n m

This is just a shorthand for saying, "Add m to itself n times." Expanding the above to make its meaning more clear: :m × n = m + m + m + ... + m such that there are n m's added together. So for instance:

5 × 2 = 5 + 5 = 10
2 × 5 = 2 + 2 + 2 + 2 + 2 = 10
4 × 3 = 4 + 4 + 4 = 12
m × 6 = m + m + m + m + m + m

Using this definition, it is easy to prove some interesting properties of multiplication. As the first two examples above hint at, the order in which two numbers are multiplied does not matter. This is called the commutative property and it turns out to be true in general that for any two numbers x and y, :x · y = y · x. Multiplication also has what is called the associative property. The associative property means that for any three numbers x, y, and z, :(x · y)z = x(y · z). Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done. Multiplication also has what is called a distributive property with respect to the addition, because :x(y + z) = xy + xz. Also of interest is that any number times 1 is equal to itself, thus, :1 · x = x. and this is called the identity property What about zero? Well, we have: :m · 0 = m + m + m +...+ m where there are zero m's added together. The sum of zero m's is zero, so :m · 0 = 0 no matter what m is (as long as it is finite). Multiplication with negative numbers also requires a little thought. First consider negative 1. For any positive integer m: :(−1)m = (−1) + (−1) +...+ (−1) = −m This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s. All that remains is to explicitly define (−1)(−1): :(−1)(−1) = −(−1) = 1 In this way, the multiplication of any two integers is defined. The definitions can be extended to larger and larger sets of numbers: first to vulgar fractions called the rational numbers, then to infinitely long decimals called real numbers, and then to the complex numbers. Students are sometimes mystified when told that the result of multiplying no numbers is 1. A formal recursive definition of multiplication can be given by the rules: : x · 0 = 0 : x · y = x + x·(y − 1) where x is a real number, and y is a natural number. Once multiplication has been defined for natural numbers, it can be extended to include integers, and then to real and complex numbers.

Computation

For fast ways to compute products of large numbers, see multiplication algorithms. Some algorithms are suitable for multiplying numbers using pencil and paper. Most, such as lattice multiplication, require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9); the peasant multiplication algorithm does not.

External links

- [http://www.cut-the-knot.org/do_you_know/multiplication.shtml Multiplication] at cut-the-knot
- [http://www.mathsisfun.com/multiplying-negatives.html Multiplying Negative Numbers]
- [http://www.cut-the-knot.org/blue/SysTable.shtml Arithmetic Operations In Various Number Systems] at cut-the-knot
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Multiplication1 Multiplication on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead]
- [http://webhome.idirect.com/~totton/suanpan/mod_mult/ Modern Chinese Multiplication Techniques on an Abacus] Category:Elementary arithmetic ko:곱셈 ja:乗法 simple:Multiplication th:การคูณ

Division (mathematics)

:This article is about the arithmetic operation. For other uses, see Division (disambiguation). In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. Specifically, if :

c \times b = a

where b is not zero, then :

\frac ab = c

that is, a divided by b equals c. For instance,

\frac 63 = 2

since

2 \times 3 = 6\,

. In the above expression, a is called the dividend, b the divisor and c the quotient. Division by zero (i.e. where the divisor is zero) is usually not defined.

Notation

Division is most often shown by placing the dividend over the divisor with a horizontal line between them. For example, a divided by b is written

\frac ab

. This can be read out loud as "a divided by b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this:

a/b\,

. This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters. A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor: ^a⁄_b Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further. A less common way to show division is to use the obelus (or division sign) in this manner:

a \div b

. This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. In some non-English-speaking cultures, "a divided by b" has sometimes been written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios.

Computing division

With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. Division can be calculated with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset. In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. In such a case, division can be calculated by multiplication. This approach is useful in computers that do not have a fast division instruction.

Division of integers

Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches. # Say that 26 cannot be divided by 10. # Give the answer as a decimal fraction or a mixed number, so

\frac = 2.6

26/10 = 2 \frac 35

. This is the approach usually taken in mathematics. # Give the answer as a quotient and a remainder, so

\frac = 2

remainder 6. # Give the quotient as the answer, so

\frac = 2

. This is sometimes called integer division. One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.

Division of rational numbers

The result of dividing two r

Arithmetic

Arithmetic operations

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Number theory

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Elementary number theory

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Algebraic number theory

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Computational number theory

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Prime number theory

Nineteenth-century developments

Quotations

References

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Subtraction

Basic subtraction: integers

See also

Algorithms

External links

Multiplication

Notation

Definition

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See also

External links

Division (mathematics)

Notation

Computing division