# Dictionary Definition

algebra n : the mathematics of generalized
arithmetical operations

# User Contributed Dictionary

## English

### Etymology

From , or mediaeval , from (al-jabr) “reunion”, “resetting of broken parts”, used in the title of al-Khwarizmi’s influential work (ilm al-jabr wa’l-muqābala), “the science of restoration and equating like with like”### Pronunciation

### Noun

- uncountable math A system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols.
- countable math A structure consisting of a set of elements together with one or more operations and rules specifying what expressions are equivalent.
- uncountable mathematics The study of algebras.

#### Derived terms

- abstract algebra
- elementary algebra
- finite algebra
- free algebra
- Lie algebra
- linear algebra
- universal algebra

#### Related terms

#### Translations

system for computation using letters or other
symbols

- Bosnian: algebra
- Catalan: àlgebra
- Chinese: 代数学 (dài shù xué)
- Croatian: algebra
- Czech: algebra
- Dutch: algebra, stelkunde
- Finnish: algebra
- French: algèbre
- German: Algebra
- Greek: άλγεβρα (álgevra)
- Hungarian: algebra
- Ido: algebro
- Italian: algebra
- Japanese: 代数 (だいすう, daisū)
- Maltese: alġebra
- Norwegian: algebra
- Polish: algebra
- Portuguese: álgebra
- Russian: алгебра
- Slovak: algebra
- Spanish: álgebra
- Swedish: algebra
- Turkish: cebir
- Urdu: الجبرا (aljabraa)

structure

study

## Croatian

### Noun

hr-noun f## Czech

### Noun

#### Derived terms

## Italian

### Noun

algebra f#### Related terms

## Spanish

### Noun

# Extensive Definition

Algebra is a branch of mathematics concerning the
study of structure,
relation
and quantity. The name
is derived from the treatise written by the Persian
mathematician,
astronomer,
astrologer
and geographer,
titled Kitab al-Jabr wa-l-Muqabala (meaning "The
Compendious Book on Calculation by Completion and Balancing"),
which provided symbolic operations for the systematic solution of
linear
and quadratic
equations. Al-Khwarizimi's book made its way to Europe and was
translated into Latin as Liber algebrae et almucabala.

Together with geometry, analysis,
combinatorics, and
number
theory, algebra is one of the main branches of mathematics. Elementary
algebra is often part of the curriculum in secondary
education and provides an introduction to the basic ideas of
algebra, including effects of adding and multiplying numbers, the concept of variables, definition of
polynomials, along
with factorization
and determining their roots.

Algebra is much broader than elementary algebra
and can be generalized. In addition to working directly with
numbers, algebra covers working with symbols, variables, and set elements.
Addition and multiplication are viewed as general operations, and their precise
definitions lead to structures such as groups,
rings
and fields.

## Classification

Algebra may be divided roughly 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), also called second year and third year algebra;
- Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
- 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.
- Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
- Algebraic geometry in its algebraic aspect.
- Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.

## Elementary algebra

Elementary algebra is the most basic form of
algebra. It is taught to students who are presumed to have no
knowledge of mathematics beyond the basic
principles of arithmetic. In arithmetic,
only numbers and their
arithmetical operations (such as +, −, ×, ÷) occur. In algebra,
numbers are often denoted by symbols (such as a, x, or y). 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, or f(x) = 3x - 10, where f is the function, and x is the number to which the function is applied.").

### Polynomials

A polynomial is an expression
that is constructed from one or more variables and constants, using
only the operations of addition, subtraction, and multiplication
(where repeated multiplication of the same variable is standardly
denoted as exponentiation with a constant whole number exponent).
For example, x^2 + 2x -3\, is a polynomial in the single variable
x.

An important class of problems in algebra is
factorization of
polynomials, that is, expressing a given polynomial as a product of
other polynomials. The example polynomial above can be factored as
(x-1)(x+3)\,\!. A related class of problems is finding algebraic
expressions for the roots
of a polynomial in a single variable.

## Abstract algebra

see also Algebraic structureAbstract algebra extends the familiar concepts
found in elementary algebra and arithmetic of numbers to more general
concepts.

Sets: Rather than just
considering the different types of numbers, abstract algebra deals
with the more general concept of sets: a collection of all objects
(called elements)
selected by property, specific for the set. All collections of the
familiar types of numbers are sets. Other examples of sets include
the set of all two-by-two matrices,
the set of all second-degree polynomials (ax2 + bx + c),
the set of all two dimensional vectors
in the plane, and the various finite
groups such as the cyclic
groups which are the group of integers modulo
n. Set
theory is a branch of logic and not technically a branch
of algebra.

Binary
operations: The notion of addition (+) is abstracted to
give a binary operation, * say. The notion of binary operation is
meaningless without the set on which the operation is defined. For
two elements a and b in a set S a*b gives another element in the
set; this condition is called closure.
Addition
(+), subtraction
(-), multiplication (×), and
division
(÷) can be binary operations when defined on different sets, as is
addition and multiplication of matrices, vectors, and
polynomials.

Identity
elements: The numbers zero and one are abstracted to give the
notion of an identity element for an operation. Zero is the
identity element for addition and one is the identity element for
multiplication. For a general binary operator * the identity
element e must satisfy a * e = a and e * a = a. This holds for
addition as a + 0 = a and 0 + a = a and multiplication a
× 1 = a and 1 × a = a. However, if we take the
positive natural numbers and addition, there is no identity
element.

Inverse
elements: The negative numbers give rise to the concept of
inverse elements. For addition, the inverse of a is -a, and for
multiplication the inverse is 1/a. A general inverse element a-1
must satisfy the property that a * a-1 = e and a-1 * a = e.

Associativity:
Addition of integers has a property called associativity. That is,
the grouping of the numbers to be added does not affect the sum.
For example: (2+3)+4=2+(3+4). In general, this becomes (a * b) * c
= a * (b * c). This property is shared by most binary operations,
but not subtraction or division or octonion
multiplication.

Commutativity:
Addition of integers also has a property called commutativity. That
is, the order of the numbers to be added does not affect the sum.
For example: 2+3=3+2. In general, this becomes a * b = b * a. Only
some binary operations have this property. It holds for the
integers with addition and multiplication, but it does not hold for
matrix
multiplication or
quaternion multiplication .

### Groups—structures of a set with a single binary operation

Combining the above concepts gives one of the
most important structures in mathematics: a group.
A group is a combination of a set S and a single binary
operation '*', defined in any way you choose, but with the
following properties:

- An identity element e exists, such that for every member a of S, e * a and a * e are both identical to a.
- Every element has an inverse: for every member a of S, there exists a member a-1 such that a * a-1 and a-1 * a are both identical to the identity element.
- The operation is associative: if a, b and c are members of S, then (a * b) * c is identical to a * (b * c).

If a group is also commutative - that is, for
any two members a and b of S, a * b is identical to b * a – then
the group is said to be Abelian.

For example, the set of integers under the
operation of addition is a group. In this group, the identity
element is 0 and the inverse of any element a is its negation, -a.
The associativity requirement is met, because for any integers a, b
and c, (a + b) + c = a + (b + c)

The nonzero rational
numbers form a group under multiplication. Here, the identity
element is 1, since 1 × a = a × 1 = a for any
rational number a. The inverse of a is 1/a, since a × 1/a
= 1.

The integers under the multiplication operation,
however, do not form a group. This is because, in general, the
multiplicative inverse of an integer is not an integer. For
example, 4 is an integer, but its multiplicative inverse is 1/4,
which is not an integer.

The theory of groups is studied in group
theory. A major result in this theory is the
classification of finite simple groups, mostly published
between about 1955 and 1983, which is thought to classify all of
the finite
simple
groups into roughly 30 basic types.

Semigroups,
quasigroups, and
monoids are structures
similar to groups, but more general. They comprise a set and a
closed binary operation, but do not necessarily satisfy the other
conditions. A semigroup has an associative
binary operation, but might not have an identity element. A
monoid is a semigroup
which does have an identity but might not have an inverse for every
element. A quasigroup
satisfies a requirement that any element can be turned into any
other by a unique pre- or post-operation; however the binary
operation might not be associative. All are instance of groupoids, structures with a
binary operation upon which no further conditions are
imposed.

All groups are monoids, and all monoids are
semigroups.

### Rings and fields—structures of a set with two particular binary operations, (+) and (×)

Groups just have one binary operation. To fully
explain the behaviour of the different types of numbers, structures
with two operators need to be studied. The most important of these
are rings,
and fields.

Distributivity
generalised the distributive law for numbers, and specifies the
order in which the operators should be applied, (called the
precedence). For the integers (a + b) × c =
a×c+ b×c and c × (a + b) =
c×a + c×b, and × is said to be
distributive over +.

A ring
has two binary operations (+) and (×), with × distributive over +.
Under the first operator (+) it forms an Abelian group. Under the
second operator (×) it is associative, but it does not need to have
identity, or inverse, so division is not allowed. The additive (+)
identity element is written as 0 and the additive inverse of a is
written as −a.

The integers are an example of a ring. The
integers have additional properties which make it an integral
domain.

A field
is a ring with the additional property that all the elements
excluding 0 form an Abelian group under ×. The multiplicative (×)
identity is written as 1 and the multiplicative inverse of a is
written as a-1.

The rational numbers, the real numbers and the
complex numbers are all examples of fields.

## Objects called algebras

The word algebra is also used for various
algebraic
structures:

- Algebra over a field or more generally Algebra over a ring
- Algebra over a set
- Boolean algebra
- F-algebra and F-coalgebra in category theory
- Sigma-algebra
- T-algebras of monads.

## History

The origins of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.Later, the Indian
mathematicians developed algebraic methods to a high degree of
sophistication. Although Diophantus and the Babylonians used mostly
special ad hoc methods to solve equations, Brahmagupta was
the first to solve equations using general methods. He solved the
linear indeterminate equations, quadratic equations, second order
indeterminate equations and equations with multiple variable.

The word "algebra" is named after the Arabic word
"al-jabr" from the title of the book , meaning The book of Summary
Concerning Calculating by Transposition and Reduction, a book
written by the Persian
mathematician in 820. The word Al-Jabr means "reunion". The
Hellenistic mathematician Diophantus has
traditionally been known as "the father of algebra" but debate now
exists as to whether or not Al-Khwarizmi should take that title.
Those who support Al-Khwarizmi point to the fact that much of his
work on reduction
is still in use today and that he gave an exhaustive explanation of
solving quadratic equations. Those who support Diophantus point
to the fact that the algebra found in Al-Jabr is more elementary
than the algebra found in Arithmetica and that Arithmetica is
syncopated while Al-Jabr is fully rhetorical. Another Persian
mathematician, Omar
Khayyam, developed algebraic
geometry and found the general geometric solution of the
cubic
equation. The Indian mathematicians Mahavira
and Bhaskara II,
and the Chinese mathematician Zhu Shijie,
solved various cases of cubic, quartic,
quintic
and higher-order polynomial equations.

Another key event in the further development of
algebra was the general algebraic solution of the cubic and quartic
equations, developed in the mid-16th century. The idea of a
determinant was
developed by Japanese
mathematician Kowa Seki in
the 17th century, followed by Gottfried
Leibniz ten years later, for the purpose of solving systems of
simultaneous linear equations using matrices.
Gabriel
Cramer also did some work on matrices and determinants in the
18th century. Abstract
algebra was developed in the 19th century, initially focusing
on what is now called Galois
theory, and on constructibility
issues.

The stages of the development of symbolic algebra
are roughly as follows:

- Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
- Geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
- Syncopated algebra, as developed by Diophantus, Brahmagupta and the Bakhshali Manuscript; and
- Symbolic algebra, which was initiated by Abū al-Hasan ibn Alī al-Qalasādī
- 1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.
- 1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation.
- 1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
- 1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.
- 1832: Galois theory is developed by Évariste Galois in his work on abstract algebra.

## See also

## References

- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000).
- John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).
- I.N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
- R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
- L. Euler: Elements of Algebra, ISBN 978-1-89961-873-6

## External links

- 4000 Years of Algebra, lecture by Robin Wilson, at Gresham College, 17th October 2007 (available for MP3 and MP4 download, as well as a text file).
- ExampleProblems.com Example problems and solutions from basic and abstract algebra.

algebra in Afrikaans: Algebra

algebra in Arabic: جبر

algebra in Aragonese: Alchebra

algebra in Asturian: Álxebra

algebra in Azerbaijani: Cəbr

algebra in Bengali: বীজগণিত

algebra in Min Nan: Tāi-sò͘

algebra in Bashkir: Алгебра

algebra in Belarusian: Алгебра

algebra in Bulgarian: Алгебра

algebra in Catalan: Àlgebra

algebra in Czech: Algebra

algebra in Corsican: Algebra

algebra in Welsh: Algebra

algebra in Danish: Algebra

algebra in German: Algebra

algebra in Estonian: Algebra

algebra in Modern Greek (1453-): Άλγεβρα

algebra in Emiliano-Romagnolo: Algebra

algebra in Spanish: Álgebra

algebra in Esperanto: Algebro

algebra in Basque: Aljebra

algebra in Persian: جبر

algebra in French: Algèbre

algebra in Western Frisian: Algebra

algebra in Scottish Gaelic: Ailseabra

algebra in Galician: Álxebra

algebra in Classical Chinese: 代數學

algebra in Korean: 대수학

algebra in Hindi: बीजगणित

algebra in Ido: Algebro

algebra in Indonesian: Aljabar

algebra in Interlingua (International Auxiliary
Language Association): Algebra

algebra in Icelandic: Algebra

algebra in Italian: Algebra

algebra in Hebrew: אלגברה

algebra in Javanese: Aljabar

algebra in Georgian: ალგებრა

algebra in Haitian: Aljèb

algebra in Lao: ພຶດຊະຄະນິດ

algebra in Latin: Algebra

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algebra in Ligurian: Algebra

algebra in Lojban: turki'icmaci

algebra in Hungarian: Algebra

algebra in Macedonian: Алгебра

algebra in Marathi: बीजगणित

algebra in Malay (macrolanguage): Algebra

algebra in Dutch: Algebra

algebra in Japanese: 代数学

algebra in Norwegian: Algebra

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algebra in Novial: Algebra

algebra in Piemontese: Àlgebra

algebra in Polish: Algebra

algebra in Portuguese: Álgebra

algebra in Romanian: Algebră

algebra in Russian: Алгебра

algebra in Scots: Algebra

algebra in Albanian: Algjebra

algebra in Sicilian: Àlgibbra

algebra in Simple English: Algebra

algebra in Slovak: Algebra

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algebra in Serbian: Алгебра

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algebra in Tagalog: Aldyebra

algebra in Tamil: இயற்கணிதம்

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algebra in Vietnamese: Đại số

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algebra in Turkish: Cebir

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algebra in Ukrainian: Алгебра

algebra in Võro: Algõbra

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algebra in Yiddish: אלגעברע

algebra in Yoruba: Áljẹ́brà

algebra in Chinese: 代数