# Algebra

Algebra from bonesetting' is one of the broad areas of mathematics. Roughly speaking, algebra is the explore of mathematical symbols as alive as a rules for manipulating these symbols in formulas; it is for a unifying thread of most all of mathematics.

Elementary algebra deals with the manipulation of variables as whether they were numbers see the image, in addition to is therefore fundamental in all application of mathematics. Abstract algebra is the hit given in education to the analyse of algebraic structures such(a) as groups, rings, as well as fields. Linear algebra, which deals with linear equations and linear mappings, is used for advanced presentations of geometry, and has numerous practical application in weather forecasting, for example. There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such(a) as commutative algebra and some not, such(a) as Galois theory.

The word algebra is non only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such(a) as an algebra over a field, ordinarily called an algebra. Sometimes, the same phrase is used for a subarea and its leading algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

## History

The roots of algebra can be traced to the ancient , and The Nine Chapters on the Mathematical Art. The geometric hold of the Greeks, typified in the Elements, submitted the model for generalizing formulae beyond the or done as a reaction to a question of specific 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, ordinarily lines, that had letters associated with them. 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 c. 780–850. He later wrote The Compendious Book on Calculation by Completion and Balancing, which build algebra as a mathematical discipline that is freelancer of geometry and arithmetic.

The ] For example, the number one ready arithmetic solution written in words instead of symbols, including zero and negative solutions, to quadratic equations was allocated by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD. 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 set of equations.

In the context where algebra is target with the theory of equations, the Greek mathematician Diophantus has traditionally been call 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". 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 member 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 assist Al-Khwarizmi bit to the fact that he provided 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, and that he gave an exhaustive relation 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 manage all possible prototypes for equations, which henceforward explicitly make up 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 component 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, 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 interpreter 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 Σn^{2, Σn3 and used the method of successive approximation to build square roots.}

François Viète's work on new algebra at theof 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 developing of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The picture of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the aim 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 "Réflexions sur la résolution algébrique des équations" devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first grown-up to develop the picture 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. 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.