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{{short description|Study of mathematical symbols and the rules for manipulating them}}{{for|the kind of algebraic structure|Algebra over a field}}{{pp-move-indef}}{{pp-semi-indef}}File:Quadratic formula.svg|thumb|The 1=ax2 + bx + c = 0}}, where {{mvar|a}} is not zero, in terms of its coefficients {{math|a, b}} and {{mvar|c}}.Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts"WEB, algebra,weblink Oxford English Dictionary, Oxford University Press, ) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;See {{harvnb|Herstein|1964}}, page 1: "An algebraic system can be described as a set of objects together with some operations for combining them". it is a unifying thread of almost all of mathematics.See {{harvnb|Herstein|1964}}, page 1: "...it also serves as the unifying thread which interlaces almost all of mathematics". It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5 the letter x is unknown, but applying additive inverses can reveal its value: x=3. In {{math|1=E = mc{{smallsup|2}}}}, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.A mathematician who does research in algebra is called an algebraist.

## Etymology

File:Muá¸¥ammad ibn MÅ«sÄ al-KhwÄrizmÄ«.png|thumb|180px|The name of algebra comes from the title of a book by 978-0-19-988041-6}}.The word algebra comes from the Arabic ({{transl|ar|al-jabr}} lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muá¸³Äbala by the Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century.ENCYCLOPEDIA, Algebra, T. F. Hoad, The Concise Oxford Dictionary of English Etymology, Oxford University Press, Oxford, 2003,weblink subscription, 10.1093/acref/9780192830982.001.0001, 978-0-19-283098-2,

## Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

## Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers. This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation
ax^2+bx+c=0,
a, b, c can be any numbers whatsoever (except that a cannot be 0), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity x which satisfy the equation. That is to say, to find all the solutions of the equation.Historically, and in current teaching, the study of algebra starts with the solving of equations such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials. The structural properties of these non-numerical objects were then abstracted into algebraic structures such as groups, rings, and fields.Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject ClassificationWEB,weblink 2010 Mathematics Subject Classification, 2014-10-05, where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

## History

### Modern history of algebra

File:Gerolamo Cardano (colour).jpg|thumb|200px|Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.]]FranÃ§ois ViÃ¨te's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, RenÃ© Descartes published La GÃ©omÃ©trie, inventing analytic geometry and introducing modern algebraic notation. 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 Seki KÅwa in the 17th century, followed independently 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. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper "RÃ©flexions sur la rÃ©solution algÃ©brique des Ã©quations{{-"}} devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues."The Origins of Abstract Algebra". University of Hawaii Mathematics Department. George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra)."The Collected Mathematical Papers". Cambridge University Press.

## Areas of mathematics with the word algebra in their name

Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.
Many mathematical structures are called algebras:

## Elementary algebra

(File:algebraic equation notation.svg|thumb|right|Algebraic expression notation:  1 â€“ power (exponent)  2 â€“ coefficient  3 â€“ term  4 â€“ operator  5 â€“ constant term  x y c â€“ variables/constants)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 +, âˆ’, Ã—, Ã·) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). 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" or going a bit further "Find a number x such that ax + b = c". This step leads to the conclusion that it is not the nature of the specific numbers that allows us to solve it, but that of the operations involved.)
• It allows the formulation of functional relationships. (For instance, "If you sell x tickets, then your profit will be 3x âˆ’ 10 dollars, or f(x) = 3x âˆ’ 10, where f is the function, and x is the number to which the function is applied".)

### Polynomials

File:Polynomialdeg3.svg|The graphgraphA polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x2 + 2x âˆ’ 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (x âˆ’ 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x âˆ’ 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

### Education

{{see also|Mathematics education}}It has been suggested that elementary algebra should be taught to students as young as eleven years old,WEB, Hull's Algebra, New York Times, July 16, 1904,weblink pdf, 2012-09-21, though in recent years it is more common for public lessons to begin at the eighth grade level (â‰ˆ 13 y.o. Â±) in the United States.WEB, Quaid, Libby, Kids misplaced in algebra, Associated Press, 2008-09-22,weblink Report, 2012-09-23, However, in some US schools, algebra is started in ninth grade.

## Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are listed fundamental concepts in abstract algebra.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 groups 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, âˆ— 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 âˆ— b is another element in the set; this condition is called closure. Addition (+), subtraction (âˆ’), multiplication (Ã—), and division (Ã·) can be binary operations when defined on different sets, as are 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 âˆ— the identity element e must satisfy a âˆ— e = a and e âˆ— a = a, and is necessarily unique, if it exists. This holds for addition as a + 0 = a and 0 + a = a and multiplication a Ã— 1 = a and 1 Ã— a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written âˆ’a, and for multiplication the inverse is written aâˆ’1. A general two-sided inverse element aâˆ’1 satisfies the property that a âˆ— aâˆ’1 = e and aâˆ’1 âˆ— a = e, where e is the identity element.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: {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4)}}. In general, this becomes (a âˆ— b) âˆ— c = a âˆ— (b âˆ— c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a âˆ— b = b âˆ— a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

### Groups

{{see also|Group theory|Examples of groups}}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 âˆ—, 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 âˆ— a and a âˆ— e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member aâˆ’1 such that a âˆ— aâˆ’1 and aâˆ’1 âˆ— a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a âˆ— b) âˆ— c is identical to a âˆ— (b âˆ— c).
If a group is also commutative â€“ that is, for any two members a and b of S, a âˆ— b is identical to b âˆ— a â€“ 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, âˆ’a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)The non-zero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 Ã— a = a Ã— 1 = a for any rational number a. The inverse of a is 1/a, since a Ã— 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 Â¼, 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 separates the finite simple groups into roughly 30 basic types.Semi-groups, quasi-groups, 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 semi-group has an associative binary operation, but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however the binary operation might not be associative.All groups are monoids, and all monoids are semi-groups.{| class="wikitable"|+Examples!Set
Natural numbers NIntegers ZRational numbers Q (also Real numbers R and Complex numbers>complex C numbers)Integers modulo 3: Z3 = {0, 1, 2}
!Operation| +| Ã— (w/o zero)| +| Ã— (w/o zero)| +| âˆ’| Ã— (w/o zero)| Ã· (w/o zero)| +| Ã— (w/o zero)
!Closed| Yes| Yes| Yes| Yes| Yes| Yes| Yes| Yes| Yes| Yes
| Identity| 0| 1| 0| 1| 0| N/A| 1| N/A| 0| 1
| Inverse| N/A| N/A| âˆ’a| N/A| âˆ’a| N/A| 1/a| N/A| 0, 2, 1, respectively| N/A, 1, 2, respectively
| Associative| Yes| Yes| Yes| Yes| Yes| No| Yes| No| Yes| Yes
| Commutative| Yes| Yes| Yes| Yes| Yes| No| Yes| No| Yes| Yes
| Structure| monoid| monoid| abelian group| monoid| abelian group| quasi-group| abelian group| quasi-group| abelian group| abelian group (Z2)

### Rings and fields

{{see also|Ring theory|Glossary of ring theory|Field theory (mathematics)|Glossary of field theory}}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.A ring has two binary operations (+) and (Ã—), with Ã— distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (Ã—) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as âˆ’a.Distributivity generalises the distributive law for numbers. For the integers {{nowrap|1=(a + b) Ã— c = a Ã— c + b Ã— c}} and {{nowrap|1=c Ã— (a + b) = c Ã— a + c Ã— b,}} and Ã— is said to be distributive over +.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 Ã—. The multiplicative (Ã—) identity is written as 1 and the multiplicative inverse of a is written as aâˆ’1.The rational numbers, the real numbers and the complex numbers are all examples of fields.

## References

{{reflist}}

### Works cited

• BOOK, Boyer, Carl B., Carl Benjamin Boyer, A History of Mathematics, 2nd, John Wiley & Sons, 1991, 978-0-471-54397-8,weblink
• JOURNAL, Gandz, S., The Sources of Al-KhowÄrizmÄ«'s Algebra, Osiris (journal), Osiris, 1, January 1936, 263â€“277, 301610, 10.1086/368426,
• BOOK, Herstein, I. N., 1964, Topics in Algebra, Ginn and Company, 0-471-02371-X,

• BOOK, Allenby, R. B. J. T., Rings, Fields and Groups, 0-340-54440-6,
• BOOK, Asimov, Isaac, Realm of Algebra, 1961, Houghton Mifflin, Isaac Asimov,
• BOOK, Boyer, Carl B., Carl Benjamin Boyer, A History of Mathematics, 2nd, John Wiley & Sons, 1991, 978-0-471-54397-8,weblink
• BOOK, Euler, Leonhard, Leonhard Euler,weblink" title="web.archive.org/web/20110413234352weblink">weblink Elements of Algebra, 978-1-899618-73-6,
• BOOK, Herstein, I. N., Topics in Algebra, 0-471-02371-X,
• BOOK, Hill, Donald R., Islamic Science and Engineering, Edinburgh University Press, 1994,
• BOOK, Joseph, George Gheverghese, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, 2000,
• WEB, O'Connor, John J., Robertson, Edmund F., 2005, History Topics: Algebra Index,weblink MacTutor History of Mathematics archive, University of St Andrews,
• BOOK, Sardar, Ziauddin, Ravetz, Jerry, Loon, Borin Van, 1999, Introducing Mathematics, Totem Books,

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