Algebra

: ''This article is about the branch of mathematics; for other uses of the term see Algebra (disambiguation) .''
Algebra (from s and find their roots.

Algebra is much broader than elementary algebra and can be generalized. Rather than working directly with numbers, one can work with Symbols or Elements of some Set . Addition and multiplication are viewed as general Operations , and their precise definitions lead to structures such as Groups , Rings and Fields .

Together with Geometry and Analysis , algebra is one of the three main branches of mathematics.

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 Variable s, and the rules governing Mathematical Expression s and Equation s 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 Structure s such as Groups , Rings and Field s are Axiomatically defined and investigated;

Linear Algebra , in which the specific properties of Vector Space s are studied (including Matrices );

Universal Algebra , in which properties common to all algebraic structures are studied.

In advanced studies, axiomatic algebraic systems such as 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 Space s

Banach Space s

Hilbert Space s

Banach Algebra s

Normed Algebra s

Topological Algebra s

Topological Group s

ELEMENTARY ALGEBRA

: ''Main article: 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 . Although in arithmetic, only Number s and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra, numbers are often denoted by symbols (such as ''a'', ''x'', ''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 Equation s and the study of how to solve these (for instance, "Find a number ''x'' such that $3x + 1 = 10$ ").

It allows the formulation of Function al 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 the function is performed on.").

ABSTRACT ALGEBRA

: ''Main article: Abstract Algebra ; see also Algebraic Structures ''.

Abstract algebra extends the familiar concepts found in elementary algebra and Arithmetic of Numbers to more general concepts.

'''. All 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 (''ax''²+''bx''+''c''), the set of all two dimensional Vectors in the plane, and the various Finite Groups such as the Cyclic Group s which are the group of integers Modulo ''n''. Set Theory is a branch of Logic and not technically a branch of algebra.

say. For two elements ''a'' and ''b'' in a set ''S'' ''a''---''b'' gives another element in the set, (technically this condition is called Closure ). Addition (+), Subtraction (-), Multiplication (×), and Division (÷) are all binary operations as in addition and multiplication of matrices, vectors, and polynomials.

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'', 1×''a''=''a''. However, if we take the positive natural numbers and addition, there is no identity element.

''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.

'''.

Groups

: ''Main article: Group ; see also Group Theory , Examples Of Groups ''

Combining the above concepts gives one of the most important structures in mathematics: a Group . A group consists of:

A set ''S'' of elements,

A (closed) binary operation (---)

An identity element exists,

Every element has an inverse,

The operation is associative.

If commutativity is included as well, then we get an Abelian Group .

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 Number s form a group under multiplication. Here, the identity element is 1, since

1 \cdot a = a \cdot 1 = a

for any rational number ''a''. The inverse of ''a'' is

rac{1}{a}

, since

a \cdot {1 \over 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 Group s into roughly 30 basic types.

Semigroup s, Quasigroup , and Monoid s are structures similar to groups but not all the conditions hold, for example a semigroup is a set with an associative binary operation on it, but does not have an identity element.

Two operators: Rings and Fields

: ''Main article: Rings , Fields ; see also Ring Theory , Glossary Of Ring Theory , Field Theory , Glossary Of Field Theory ''

Groups just have one operator, to fully explain the 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 Rings has two operators (+) 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 Fields 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, real number and complex numbers are all examples of fields.

ALGEBRAS

The word ''algebra'' is also used for various Algebraic Structures :

Algebra Over A Field

Algebra Over A Set

Boolean Algebra

F-algebra and F-coalgebra in Category Theory

Sigma-algebra

HISTORY

details Geometric al algebra in '' Elements ''.]]

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 formulae and calculate solutions for unknown values for a class of problems typically solved today by using Linear Equation s, Quadratic Equation s, and Indeterminate Linear Equation s. 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 '' Moscow And Rhind Mathematical Papyri '', '' Sulba Sutras '', Euclid's ''Elements'' , and '' The Nine Chapters On The Mathematical Art ''. This 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.

India n mathematicians proceeded to write treatises on algebraic means of solving equations from the end of the first millennium BC, followed by Hellenistic mathematicians from the early First Millennium AD . Important algebraic works from this general era include the '' Bakhshali Manuscript '', the works of Hero Of Alexandria , the '' Arithmetica '' of Diophantus , the ''Aryabhatiya'' of Aryabhata , and the ''Brahma Sputa Siddhanta'' of Brahmagupta .

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 Persian Mathematician in 820 . The word ''al-jabr'' means ''"reunion"''. Al-Khwarizmi is often considered the "father of algebra" (though that title is also given to Diophantus), as much of his works on Reduction are still in use today. Another Persian mathematician Omar Khayyam developed Algebraic Geometry and found the general geometric solution of the Cubic Equation . Indian mathematician Bhaskara and Chinese mathematician Zhu Shijie solved cubic, Quartic (biquadratic) and higher-order Polynomial equations.

Another key event in the further development of algebra was the general algebraic solution of the Cubic and Quartic Equation s, 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 and in the ''Bakhshali Manuscript''; and

Symbolic algebra, which sees its culmination in the work of Leibniz .

by Claude Gaspard Bachet De Méziriac .]]A timeline of key algebraic developments are as follows:

Circa Strassburg tablet seeks the solution of a quadratic elliptic equation.

Circa '' tablet gives a table of Pythagorean Triples in Babylonia n Cuneiform Script .

Circa , in his ''Baudhayana Sulba Sutra '', discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax² = c and ax² + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine Equation s.

Circa , in his ''Apastamba Sulba Sutra'', solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.

Circa gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.

Circa .

Circa '' (''The Nine Chapters on the Mathematical Art''), which contains solutions of linear equations solved using the Rule Of Double False Position , geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.

Circa '' written in Ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of Linear Equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.

Circa Egyptian mathematician Hero Of Alexandria , treats algebraic equations in three volumes of mathematics.

Circa mathematician Diophantus , who lived in Egypt and is often considered the "father of algebra", writes his famous '' Arithmetica '', a work featuring solutions of algebraic equations and on the theory of numbers.

, in his treatise ''Aryabhatiya'', obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a Differential Equation .

Circa finds numerical solutions of cubic equations.

, in his treatise ''Brahma Sputa Siddhanta'', invents the ''chakravala'' Method of solving indeterminate quadratic equations, including Pell's Equation , and gives rules for solving linear and quadratic equations. He discovers that Quadratic Equation s have two Root s, including both Negative as well as Irrational roots.

mathematician titled ''Al-Kitab al-Jabr wa-l-Muqabala'' (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of Linear and Quadratic Equation s. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.

Circa to problems in algebra.

Circa , in his treatise ''al-Fakhri'', further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the Monomial s x, x², x³, ... and 1/x, 1/x², 1/x³, ... and gives rules for the products of any two of these.

Circa finds numerical solutions of polynomial equations.

develops algebraic geometry and, in the ''Treatise on Demonstration of Problems of Algebra'', gives a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.

, in his ''Bijaganita'' (''Algebra''), recognizes that a positive number has both a positive and negative Square Root , and solves quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's Equation , the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations.

1150 : Bhaskara, in his ''Siddhanta Shiromani'', solves differential equations.

largely through the work of Leonardo Fibonacci of Pisa in his work '' Liber Abaci ''.

Circa deals with Polynomial Algebra , solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, Quintic and higher-order polynomial equations.

Circa finds the solution of Transcendental Equations by Iteration , Iterative Method s for the solution of non-linear equations, and solutions of differential equations.

1515 : Scipione del Ferro solves a cubic such that the quadratic term is missing.

solves a cubic such that the linear term is missing.

publishes ''Ars magna'' -''The great art'' which gives solutions for a variety of cubics as well as Ludovico Ferrari's solution of a special quartic equation.

recognizes the complex roots of the cubic and improves current notation.

develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in ''In artem analyticam isagoge''.

1631 : Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".

develops his notion of symbolic manipulation with formal rules which he calls ''characteristica generalis''.

, in his ''Method of solving the dissimulated problems'', discovers the Determinant , Discriminant , and Bernoulli Number s.

1685 : Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations.

1693 : Leibniz solves systems of simultaneous linear equations using matrices and determinants.

, in his treatise ''Introduction to the analysis of algebraic curves'', states Cramer's Rule and studies Algebraic Curves , matrices and determinants.

in his work on abstract algebra.

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).

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