Algebra

Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.

Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory.

The word ''algebra'' is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an ''algebra''. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

Etymology

The word ''algebra'' comes from the ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr from the title of the early 9th century book '' ^cIlm al-jabr wa l-muqābala'' "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term ''al-jabr'' referred to the operation of moving a term from one side of an equation to the other, المقابلة ''al-muqābala'' "balancing" referred to adding equal terms to both sides. Shortened to just ''algeber'' or ''algebra'' in Latin, the word eventually entered the English language during the 15th 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 English) in the 16th century.

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.
* As a single word without an article, "algebra" names a broad part of mathematics.
* As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
* With a qualifier, there is the same distinction:
** Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
** With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra.
** Sometimes both meanings exist for the same qualifier, as in the sentence: ''Commutative algebra is the study of commutative rings, which are commutative algebras over the integers''.

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 formalized 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 the 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 considerably and includes many branches of mathematics, as can be seen in the Mathematics Subject Classification
where none of the first level areas (two digit entries) are 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

The use of the word "algebra" for denoting a part of mathematics dates probably from the 16th century. The word is derived from the Arabic word ''al-jabr'' that appears in the title of the treatise '' Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala'' (''The Compendious Book on Calculation by Completion and Balancing''), written circa 820 by Al-Kwarizmi.

''Al-jabr'' referred to a method for transforming equations by subtracting like terms from both sides, or passing one term from one side to the other, after changing its sign.

Therefore, ''algebra'' referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by ''algebra''.

In mathematics, the meaning of ''algebra'' has evolved after the introduction by François Viète of symbols ( variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas. So, algebra became essentially the study of the action of operations on expressions involving variables. This includes but is not limited to the theory of equations.

At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces. This new algebra was called '' modern algebra'' by van der Waerden in his eponymous treatise, whose name has been changed to ''Algebra'' in later editions.

Early history

The roots 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 algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the ''Rhind Mathematical Papyrus'', 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, although this would not be realized until mathematics developed in medieval Islam.

By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.See , ''Europe in the Middle Ages'', p. 258: "In the arithmetical theorems in Euclid's ''Elements'' VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's ''Algebra'' made use of lettered diagrams; but all coefficients in the equations used in the ''Algebra'' are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry." Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called ''Arithmetica''. These texts deal with solving algebraic equations, and have led, in number theory, to the modern notion of Diophantine equation.

Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (–850). He later wrote ''The Compendious Book on Calculation by Completion and Balancing'', which established algebra as a mathematical discipline that is independent of geometry and arithmetic.

The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta, continued the traditions of Egypt and Babylon, though Diophantus' ''Arithmetica'' and Brahmagupta's '' Brāhmasphuṭasiddhānta'' are on a higher level. For example, the first complete arithmetic solution written in words instead of symbols, including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book ''Brahmasphutasiddhanta,'' published in 628 AD.Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006). Later, Persian and Arab mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ''ad hoc'' methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".See , page 263–277: "In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers". It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in ''Al-Jabr'' is slightly more elementary than the algebra found in ''Arithmetica'' and that ''Arithmetica'' is syncopated while ''Al-Jabr'' is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of " reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term ''al-jabr'' originally referred to,See , ''The Arabic Hegemony'', p. 229: "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation". and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs while treating algebra as an independent discipline in its own right. His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".

Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book ''Treatise on Demonstrations of Problems of Algebra'' (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,See , ''The Arabic Hegemony'', p. 239: "Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax²ⁿ + bxⁿ = c (only equations with positive roots were considered)," and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σ''n''², Σ''n''³ and used the method of successive approximation to determine square roots.

Modern history

François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published ''La Géométrie'', 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 Kōwa 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

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