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Algebra

Algebra

:This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation). Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world.

History

The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
- Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
- Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
- Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
- Circa 200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
- 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to modern one. Bhaskara II (1114 AD), who wrote the text Bijaganita (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
- 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion".
- 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided into the following categories:
- elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
- abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
- linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- universal algebra, in which properties common to all algebraic structures are studied. In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups

Algebras

The word algebra is also used for various algebraic structures:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory

References

- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).

External links

- [http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics]
- [http://www.sparknotes.com/math/#algebra1 Sparknotes' Review of Algebra I and II]
- [http://www.jamesbrennan.org/algebra/ Understanding Algebra.] An online algebra text by James W. Brennan. Category:Algebra Category:Arabic words ko:대수학 ms:Algebra ja:代数学 simple:Algebra

Algebra (disambiguation)

Algebra is a branch of mathematics. Algebra may also mean:
- elementary algebra
- abstract algebra
- linear algebra
- universal algebra
- computer algebra
- Boolean algebra, in formal logic
- algebra over a commutative ring (or R-algebra), in ring theory In set theory:
- algebra over a set
- sigma algebra In linear algebra, and the study of vector spaces:
- algebra over a field
- associative algebra In category theory:
- F-algebra
- F-coalgebra

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

Structure

:For specific meanings of structure in specific fields, see Structure (disambiguation). The structure of a thing is how the parts of it relate to each other, how it is "put together". This contrast with process, which is how the thing works; but process requires a viable structure. Both reality and language have structure. One of the goals of general semantics, and of science, is to create and use language the structure of which accurately parallels the structure of reality.

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.

Symbol

:For the Romanian choir, see Symbol (choir) A symbol, in its basic sense, is a conventional representation of a concept or quantity; i.e., an idea, object, concept, quality, etc. In more psychological and philosophical terms, all concepts are symbolic in nature, and representations for these concepts are simply token artifacts that are allegorical to (but do not directly codify) a symbolic meaning, or symbolism. Spoken language, for example, consists of distinct auditory tokens for representing symbolic concepts (words), arranged in an order which further suggests their meaning.

Nature of symbols

word]A symbol can be a material object whose shape or origin is related, by nature or convention, to the thing it represents: for instance, the cross is the main symbol of Christianity, and the scepter is a traditional symbol of royal power. A symbol can also be a more or less conventional image (i.e. an icon), or a detail of an image, or even a pattern or color: for example, the olive branch in heraldry represents peace, the halo is a conventional symbol of sainthood in Christian imagery, tartans are symbols of Scottish clans, and the color red is often used as a symbol for socialist movements, especially communism. More often, a symbol is a conventional written or printed sign (specifically, a glyph), usually standing for anything other than a sound (symbols for sounds are usually called graphemes, letters, logograms, diacritics, etc.). Thus mathematical symbols such as π and + represent quantities and operations, currency symbols represent monetary units, chemical symbols represent elements, and so forth. Symbols can also be immaterial entities like sounds, words and gestures. The ringing of gongs and bells, and the banging of a judge's gavel, often have conventional meanings in certain contexts; and bowing is a common way to indicate respect. In fact, every word in a natural language is a symbol for some concept or relationship between concepts. A symbol is usually recognized only within some specific culture, religion, or discipline, but a few hundred symbols are now recognized internationally. See list of common symbols and List of symbols.

Use of symbols

Human beings' ability to manipulate symbols allows them to explore the relationships between ideas, things, concepts, and qualities - far beyond the explorations of which any other species on earth is capable. The discipline of semiotics studies symbols and symbol systems in general; semantics is specifically concerned with the main meaning of words or other linguistic units. Literary works are often admired for their artful use of symbolism, i.e. the use of words, phrases and situations to evoke ideas and feelings beyond their plain interpretations; these uses are the subject of literary semiotics. Religious and metaphysical writings are also known for their use of esoteric symbolism. Alchemical writings made extensive use of symbols for spiritual and chemical processes (which they also saw as symbols of each other). The interpretation of dreams as symbols of one's experiences is a main feature of Freudian psychoanalysis and Jungian analytical psychology.

Etymology

The word "symbol" came to the English language, by way of Middle English, Old French, and Latin, from the Greek σύμβολον súmbolon from the root words σύμ- (sym-) meaning "together" and βολή bolḗ "a throw", having the approximate meaning of "to throw together", so "sign, ticket, or contract".

External links

- [http://www.symbols.com Symbol search engine]
- [http://altreligion.about.com/library/glossary/blsymbols.htm Religious and Cultural Symbols]
- ja:シンボル simple:Symbol

Elementary algebra

:This article is about basic algebra in mathematics. For other uses of the term "algebra" see algebra (disambiguation). Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as a, x, y) to denote numbers. This is useful because:
- It allows the general formulation of arithmetical laws (such as

a + b = b + a

for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
- It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number x such that

3x + 1 = 10

).
- It allows the formulation of functional relationships (such as "if you sell x tickets, then your profit will be

3x - 10

dollars"). These three are the main strands of elementary algebra, which should be distinguished from abstract algebra, a much more advanced topic generally taught to college seniors. In algebra, an "expression" may contain numbers, variables and arithmetical operations; a few examples are: :

x + 3\,

y^ - 3\,

z^ + a(b + x^) + 42/y - \pi.\,

An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as

a + (b + c) = (a + b) + c

); these are also known as "identities". Other equations contain symbols for unknown values and we are then interested in finding those values for which the equation becomes true:

x^ - 1 = 4.

These are the "solutions" of the equation.

Laws of elementary algebra

- The order of operations in a mathematical expression are as follows:
- groupings -> exponents -> multiplication -> addition
- Addition is a commutative operation.
- Subtraction is the reverse of addition.
- To subtract is the same as to add a negative number: :::

a - b = a + (-b). \

:: Example: if

5 + x = 3

then

x = -2.

- Multiplication is a commutative operation.
- Division is the reverse of multiplication.
- To divide is the same as to multiply by a reciprocal: :::

= a \left(  \right).

- Exponentiation is not a commutative operation.
- Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with fractional exponents (e.g. square roots).
- Examples: if

3^x = 10

then

x = \log_3 10 .

x^ = 10

then

x = 10^.

- The square roots of negative numbers do not exist in the real number system. (See: complex number system)
- Associative property of addition:

(a + b) + c = a + (b + c).

- Associative property of multiplication:

(ab)c = a(bc).

- Distributive property of multiplication with respect to addition:

c(a + b) = ca + cb.

- Distributive property of exponentiation with respect to multiplication:

(a b)^c = a^c b^c .

- How to combine exponents:

a^b a^c = a^ .

- Power to a power property of exponents:

(a^b)^c = a^ .

- If

a = b

and

b = c

, then

a = c

(transitivity of equality).
-

a = a

(reflexivity of equality).
- If

a = b

then

b = a

(symmetry of equality).
- If

a = b

and

c = d

then

a + c = b + d.

- If

a = b

then

a + c = b + c

for any c (addition property of equality).
- If

a = b

and

c = d

then

ac

bd.

- If

a = b

then

ac = bc

for any c (multiplication property of equality).
- If two symbols are equal, then one can be substituted for the other at will (substitution principle).
- If

a > b

and

b > c

then

a > c

(transitivity of inequality).
- If

a > b

then

a + c > b + c

for any c.
- If

a > b

and

c > 0

then

ac > bc.

- If

a > b

and

c < 0

then

ac < bc.

Examples

Linear equations

The simplest equations to solve are linear equations. They contain only constant numbers and a single variable without an exponent. For example: :

2x + 4 = 12. \,

The central technique is add, subtract, multiply, or divide both sides of the equation by the same thing in such a way to eventually arrive at the value of the unknown variable. If we subtract 4 from both sides in the equation above we get: :

2x = 8 \,

and if we then divide both sides by 2, we get our solution :

x = \frac = 4.

Quadratic equations

Quadratic equations contain variables raised to the first and second (square) power, and can be solved using factorization or the quadratic formula. As an example of factoring: :

x^ + 3x = 0. \,

This is the same thing as :

x(x + 3) = 0. \,

Setting x to 0 or -3 will make this true. All quadratic equations will either have one or two solutions.

System of linear equations

If we have a system of linear equations, for example, two equations in two variables, it is often possible to find two answers that satisfy both. :

4x + 2y = 14 \,

2x - y = 1. \,

Now, multiply the second equation by 2 on both sides, and you have the following equations: :

4x + 2y = 14 \,

4x - 2y = 2. \,

Now we add the two equations together to get: :

8x = 16 \,

x = 2. \,

You can see that since we multiplied the second equation by 2, we can combine the equations and cancel out y, and then we can solve for x. Note that you can multiply by any numbers (positive or negative, but not zero) to both sides of any to get to a point where a variable cancels out when you combine them. To find y, choose either one of the equations from the beginning. :

4x + 2y = 14. \,

Substitute in 2 for x. :

4(2) + 2y = 14. \,

Simplify using the rules of algebra. :

8 + 2y = 14 \,

2y = 6 \,

y = 3. \,

The full solution to this problem is then :

\begin x = 2 \\ y = 3. \end\,

External links

Charles Smith, [http://mathbooks.library.cornell.edu:8085/Dienst/UIMATH/1.0/Display/cul.math/Smit025 A Treatise on Algebra], in [http://historical.library.cornell.edu/math Cornell University Library Historical Math Monographs]. Other example problems can be found at [http://www.exampleproblems.com www.exampleproblems.com].
- Category:School subjects

Group (mathematics)

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Évariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms. A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right. Groups underlie many other algebraic structures such as fields and vector spaces. They are also important tools for studying symmetry in all its forms; the principle that the symmetries of any object form a group is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to mathematical physics (for example, in particle physics).

History

See Group theory.

Basic definitions

A group (G,
- ) is a nonempty set G together with a binary operation
- : G × G → G, satisfying the group axioms below. "a
- b" represents the result of applying the operation
- to the ordered pair (a, b) of elements of G. The group axioms are the following:
- Associativity: For all a, b and c in G, (a
- b)
- c = a
- (b
- c).
- Identity element: There is an element e in G such that for all a in G, e
- a = a
- e = a.
- Inverse element: For all a in G, there is an element b in G such that a
- b = b
- a = e, where e is the identity element from the previous axiom. You will often also see the axiom
- Closure: For all a and b in G, a
- b belongs to G. The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure. When determining if
- is a group operation, however, it is nonetheless necessary to verify that
- satisfies closure; this is part of verifying that it is in fact a binary operation. The above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms. It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a
- b ≠ b
- a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a
- b = b
- a. Groups lacking this property are called non-abelian. The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set. Note that we often refer to the group (G,
- ) as simply "G", leaving the operation
- unmentioned. But to be perfectly precise, different operations on the same set define different groups.

Notation for groups

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:
- We write "a · b" or even "ab" for a
- b and call it the product of a and b;
- We write "1" for the identity element and call it the unit element;
- We write "a⁻¹" for the inverse of a and call it the reciprocal of a. However, sometimes the group operation is thought of as analogous to addition and written additively:
- We write "a + b" for a
- b and call it the sum of a and b;
- We write "0" for the identity element and call it the zero element;
- We write "−a" for the inverse of a and call it the opposite of a. Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "
- ") and terminology that was introduced in the definition, using the notation a⁻¹ for the inverse of a. If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products ; similarly the notation Sx = ; and for two subsets S and T of G, we write ST for . In additive notation, we write x + S, S + x, and S + T for the respective sets.

Some elementary examples and nonexamples

An abelian group: the integers under addition

A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, , and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively). Proof:
- If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)
- If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
- 0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element)
- If a is an integer, then there is an integer b := −a, such that a + b = b + a = 0. (Inverse element) This group is also abelian: a + b = b + a. The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

Not a group: the integers under multiplication

On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:
- If a and b are integers then a · b is an integer. (Closure)
- If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)
- 1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element)
- However, it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is a integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails) Since not every element of (Z,·) has an inverse, (Z,·) is not a group. The most we can say is that it is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, that is the set of numbers a/b such that a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group. However, if we instead use the set Q \ instead of Q, that is include every rational number except zero, then (Q \ ,·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, the rational numbers form the algebraic structure of a field. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block". field In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
- e : RGB → RGB
- a : RGB → GRB
- b : RGB → RBG
- ab : RGB → BRG
- ba : RGB → GBR
- aba : RGB → BGR Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,
- bb = e,
- (aba)(aba) = e, and
- (ab)(ba) = (ba)(ab) = e; so each of the above actions has an inverse. By inspection, we can also determine associativity and closure; note for example that
- (ab)a = a(ba) = aba, and
- (ba)b = b(ab) = bab. This group is called the symmetric group on 3 letters, or S₃. It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ ba). Since S₃ is built up from the basic actions a and b, we say that the set generates it. Every group can be expressed in terms of permutation groups like S₃; this result is Cayley's theorem and is studied as part of the subject of group actions.

Further examples

For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.

Simple theorems

- A group has exactly one identity element.
- Every element has exactly one inverse.
- You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x
- a = b and exactly one solution y in G to the equation a
- y = b.
- The expression "a₁
- a₂
- ···
- a_n" is unambiguous, because the result will be the same no matter where we place parentheses.
- (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a
- b)⁻¹ = b⁻¹
- a⁻¹. These and other basic facts that hold for all individual groups form the field of elementary group theory.

Constructing new groups from given ones

#If a subset H of a group (G,
- ) together with the operation
- restricted on H is itself a group, then it is called a subgroup of (G,
- ). #The direct product of two groups (G,
- ) and (H,•) is the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁
- g₂,h₁•h₂). The product can also be defined with an infinite number of terms. #The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non zero terms. If the family is finite the direct sum and the product are of course the same. #Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.

Ring (mathematics)

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. The branch of abstract algebra which studies rings is called ring theory. For a history and overview of rings see that article.

Formal definition

A ring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:
- (R, +) is an abelian group with identity element 0:
- (a + b) + c = a + (b + c)
- a + b = b + a
- 0 + a = a + 0 = a
- ∀a ∃(−a) such that a + −a = −a + a = 0
- (R, ·) is a monoid with identity element 1:
- 1·a = a·1 = a
- (a·b)·c = a·(b·c)
- Multiplication distributes over addition:
- a·(b + c) = (a·b) + (a·c)
- (a + b)·c = (a·c) + (b·c) As with groups the symbol · is usually omitted and multiplication is just denoted by juxtaposition. Also the standard order of operation rules are used, so that e.g. a+bc is an abbreviation for a+(b·c). Although ring addition is commutative (i.e. a+b = b+a), note that the commutativity for multiplication (a·b = b·a) is not among the ring axioms listed above. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Not all rings are commutative. For example

M_n(K)

, the ring of

n\times n

matrices over a field 'K', is a non-commutative ring. Also note that an element of a ring need not have a multiplicative inverse. An element a in a ring is called a unit if it is invertible with respect to multiplication, i.e., if there is an element b in the ring such that a·b = b·a = 1. If that is the case, then b is uniquely determined by a and we write a⁻¹ = b. The set of all units in R forms a group under ring multiplication; this group is denoted by U(R).

Alternative definitions

There are some alternative definitions of rings of which the reader should be aware:
- Some authors add the additional requirement that 0 ≠ 1. This omits only one ring: the so called trivial ring, which has only a single element.
- A more significant difference is that some authors, I. N. Herstein for example, omit the requirement that a ring have a multiplicative identity. These authors call rings which do have multiplicative identities unital rings, unitary rings, or simply rings with a 1. Authors such as Bourbaki, who do require rings to have an identity, call algebraic objects which meet all the requirements of a ring except the identity requirement pseudo-rings.
- Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are called associative rings.
- In category theory, a ring is defined to be an additive category with a single object. The categorical dual is also then a ring, called the opposite ring. In this article all rings are assumed to be associative and unital unless otherwise stated.

Examples

- The trivial ring has only one element which serves both as additive and multiplicative identity. Note that some authors define the ring as to specifically exclude this from being considered a ring.
- The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
- The rational, real and complex numbers form rings (in fact, they are even fields). These are likewise commutative rings.
- Generally, every field is by definition a commutative ring in which one can also divide by any nonzero element.
- The Gaussian integers form a ring.
- And so do the Eisenstein integers.
- The set R[X] of all polynomials over some coefficient ring R forms a ring.
- The set of formal power series RX₁,...,X_n over a commutative ring R is a ring.
- Noncommutative ring: For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n=1, this matrix ring is just (isomorphic to) R itself. For n>2, this matrix ring is an example of a noncommutative ring (unless R is the trivial ring).
- Finite ring: If n is a positive integer, then the set Z_n = Z/nZ of integers modulo n (as an additive group the cyclic group of order n ) forms a ring with n elements (see modular arithmetic).
- If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.
- The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). The operations are addition and multiplication of functions.
- If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
- If G is a group and R is a ring the group ring of G over R is a free module over having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
- Non-example: The set of natural numbers is not a ring as (N,+) is not a group (the elements are not invertible with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. To make it a ring one needs to add negative numbers to the set, thus obtaining the ring of integers.

Basic theorems

From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have
- 0a = a0 = 0
- (−1)a = −a
- (−a)b = a(−b) = −(ab)
- (ab)⁻¹ = b⁻¹ a⁻¹ if both a and b are invertible Other basic theorems
- The identity element 1 is unique
- If the ring has at least two elements then 0 ≠ 1
- The binomial theorem

(x+y)^n=\sum_^nx^ky^

works in a ring if xy=yx

Constructing new rings from given ones

- If a subset S of a ring R is closed under multiplication and subtraction and contains the multiplicative identity element, then S is called a subring of R.
- The center of a ring R is the set of elements of R that commute with every element of R; that is, c lies in the center if cr=rc for every r in R. The center is a subring of R. We say that a subring S of R is central if it is a subring of the center of R.
- The direct product of two rings R and S is the cartesian product R×S together with the operations :(r₁, s₁) + (r₂, s₂) = (r₁+r₂, s₁+s₂) and :(r₁, s₁)(r₂, s₂) = (r₁r₂, s₁s₂).
- Given a ring R and an ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations :(a+I) + (b+I) = (a+b) + I and :(a+I)(b+I) = (ab) + I.
- Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.

Field (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

Introduction

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except for commutativity, is today called a division ring or sometimes a skew field, but also non-commutative field is still widely used. Other languages have retained the old usage: for example, in Italian and French, division rings are called corpo and corps, both literally meaning 'body'. Instead, in German and Spanish, Körper (whence the blackboard bold K used to denote a field) and cuerpo mean 'field'. Notice that French language has no single word for field, they are simply called corps commutatif. Italian for field is campo, with the same literal meaning as English. The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. See field theory for more information.

Definition

A field is a commutative ring (F, +,
- ) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelled out, this means that the following hold: ; Closure of F under + and
- : For all a, b belonging to F, both a + b and a
- b belong to F (or more formally, + and
- are binary operations on F). ; Both + and
- are associative : For all a, b, c in F, a + (b + c) = (a + b) + c and a
- (b
- c) = (a
- b)
- c. ; Both + and
- are commutative : For all a, b belonging to F, a + b = b + a and a
- b = b
- a. ; The operation
- is distributive over the operation + : For all a, b, c, belonging to F, a
- (b + c) = (a
- b) + (a
- c). ; Existence of an additive identity : There exists an element 0 in F, such that for all a belonging to F, a + 0 = a. ; Existence of a multiplicative identity : There exists an element 1 in F different from 0, such that for all a belonging to F, a
- 1 = a. ; Existence of additive inverses : For every a belonging to F, there exists an element −a in F, such that a + (−a) = 0. ; Existence of multiplicative inverses : For every a ≠ 0 belonging to F, there exists an element a⁻¹ in F, such that a
- a⁻¹ = 1. The requirement 0 ≠ 1 ensures that the set which only contains a single element is not a field. Directly from the axioms, one may show that (F, +) and (F − ,
- ) are commutative groups (abelian groups) and that therefore (see elementary group theory) the additive inverse −a and the multiplicative inverse a⁻¹ are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses: :(a
- b)⁻¹ = b⁻¹
- a⁻¹ = a⁻¹
- b⁻¹ provided both a and b are non-zero. Other useful rules include :−a = (−1)
- a and more generally :−(a
- b) = (−a)
- b = a
- (−b) as well as :a
- 0 = 0, all rules familiar from elementary arithmetic. If the requirement of commutativity of the operation
- is dropped, one distinguishes the above commutative fields from non-commutative fields, usually called division rings or skew fields).

Examples of fields

- The complex numbers C, under the usual operations of addition and multiplication. The field of complex numbers contains the following subfields (a subfield of a field F is a set containing 0 and 1, closed under the operations + and
- of F and with its own operations defined by restriction):
- The rational numbers Q = where Z is the set of integers. The rational number field contains no proper subfields.
- An algebraic number field is a finite field extension of the rational numbers Q, that is, a field containing Q which has finite dimension as a vector space over Q. Such fields are very important in number theory.
- The field of algebraic numbers, the algebraic closure of Q.
- The real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field which is categorical — it is this structure that provides the foundation for most formal treatments of calculus.
- The real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers, and the definable numbers.
- If q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements, usually denoted F_q, Z/qZ, or GF(q). Every other finite field is isomorphic to one of these fields. Such fields are often called a Galois field, whence the notation GF(q).
- In particular, for a given prime number p, the set of integers modulo p is a finite field with p elements: F_p = where the operations are defined by performing the operation in Z, dividing by p and taking the remainder; see modular arithmetic.
- Taking p = 2, we obtain the smallest field, F₂, which has only two elements: 0 and 1. It can be defined by the two Cayley tables + 0 1
- 0 1 0 0 1 0 0 0 1 1 0 1 0 1 ::::This field has important uses in computer science, especially in cryptography and coding theory.
- The rational numbers can be extended to the fields of p-adic numbers for every prime number p. These fields are very important in both

Algebra

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Group (mathematics)

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Further examples

Simple theorems

Constructing new groups from given ones

See also

Ring (mathematics)

Formal definition

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Examples

Basic theorems

Constructing new rings from given ones

See also

Field (mathematics)

Introduction

Definition

Examples of fields