Algebra - SpeedyLook encyclopedia

History

Antiquity

The Babylonians could already solve the equation of the 2nd degree (or quadratic equation).

The first known document stating a algebraic problem such as we know it is the Papyrus Rhind. This papyrus, currently (2007) with the British Museum of London, date of -1650, Christian era. It comprises the following statement:

One must divide 100 bread round loaves between ten men including/understanding a navigator, a foreman and a guard, all three receiving double share. What is necessary to give to each one?

Diophante, in IVe century, develops the method of resolution in rational numbers and discovers that the Discriminant must be the square of a rational number.

Persia and Muslim world

The word algebra from the Arab Al-jabr comes (rear RTL الجبر), which became will algebra in Latin and which means “the meeting” (pieces), “the rebuilding” or “connection” (in Spanish the word algebrista indicates that which practices the calculus but also the Rebouteux, that which can reduce the osseous fractures).

It is one of the first words of Arab title in of a work of the mathematician Perse Al-Khawarizmi who begins again, in the first part of the 9th century, work of Diophante of Alexandria (4th century). This last had imagined to represent an unknown factor by a symbol named arithme . The title of this work (Al-jabr wa' l-muqabalah) which fell under the time of rise of the Sciences and Islamic technology (the culture of the time wanted that all to know either translated into Arabic and disseminated in all the Empire), gave the modern word algebra .

After a voyage in the north of Africa, Léonard of Pisa known as Fibonacci was allured by this new way of writing the figures (different from the Roman numerals) and by the decimal system. As of its return to the country, it is among the first to popularize the Arab numerals and the Decimal system in Europe and works on its famous continuation.

XVIe century: Europe

The pope Gerbert d' Aurillac had brought back from Spain about the year the 1000 Zero, invention Indian that the mathematicians Al-Khawarizmi and Abu Kamil had made known themselves in all the Empire, and also with Cordoue.

This numeration of position launches one era of calculus, initially by means of the algorisms named thus in homage to Al-Kawarizmi, which replace little by little the use of the abacus. The Italian mathematicians of XVIe century (LED Ferro, Tartaglia and Cardan joint) solve the equation of the 3rd degree (or cubic equation). Ferrari, raises Cardan joint, solves the equation of the 4th degree (or equation quartic), and the method is improved by Bombelli. At the end of the century, the French Viète discovers that the symmetrical functions of the roots are related to the coefficients of the polynomial equation.

Until the 17th century, the algebra can be characterized overall like the continuation or the beginning of the equations and like an extension of the Arithmétique; it consists mainly of the study of the resolution of the algebraic equations, and the progressive coding of the operations symbolic systems allowing this resolution. To note that it is with French François Viète (1540-1603) that one must the idea note the unknown factors using letters.

At the XVIIe century, the mathematicians use “fictitious” numbers gradually, such as one of the square roots of -1 , to manage to calculate the nonreal roots of their equations. This “extension” of the real numbers (which will take the name of complex numbers) brings of Alembert (in 1746) and Gauss (in 1799) to state and show the fundamental theorem of the algebra (or Théorème of Alembert-Gauss): any equation polynomial of degree N of complex numbers has N roots exactly (by counting each one with its possible multiplicity). Or, in its modern form: the body $\ _ \ mathbb C$ of the complex numbers provided with the addition and the multiplication is algebraically closed .

The XIXe century is interested from now on in the calculability of the roots, and in particular in the possibility of expressing them by general formulas containing radicals. The failures concerning the equations of degree 5 lead the mathematician Abel (after Vandermonde, Lagrange and Gauss) to look further into the transformations on the whole of the roots of an equation. Welsh Évariste (1811 - 1832), in a fulgurating report, introduced for the first time the concept of group (by studying the group of the permutations of the roots of a polynomial equation) and leads to the impossibility of the resolution by radicals for the equations of degree equal to or higher than 5.

A decisive stage was reached with the writing of the fractional exhibitors . This one will make it possible Euler to state its famous formula $e^ {I \ pi} = -1$ .

Modern algebra

Consequently, the modern algebra starts a fertile course: Boole creates the algebra which bears its name, Hamilton invents the Quaternion S, and the English mathematicians Cayley, Hamilton and Sylvester study the structures of matrices. The Linear algebra, a long time restricted with the resolution of linear systems of equations to 2 or 3 unknown factors, takes its rise with the Théorème of Cayley-Hamilton (“Any square matrix with coefficients in $\ _ \ mathbb R$ or $\ _ \ mathbb C$ divides its characteristic Polynôme”). Follow the transformations by basic change, the Diagonalisation and the Trigonalisation of the matrices, and the methods of calculating which will nourish, at the XXe century, the programming of the computers. In parallel, Kummer generalizes the structures galoisiennes and studies the structures of body and ring. Dedekind defines the ideals (already foreseen by Gauss) which will make it possible to generalize and reformulate the great theorems of Arithmétique. The linear algebra spreads in Multilinear algebra and tensorial Algèbre.

At the beginning of the XXe century, under the impulse of German Hilbert and French Poincaré, the mathematicians wonder about the bases of mathematics: Logical and axiomatization occupy the front of the scene. Peano axiomatizes the arithmetic one, then the vector spaces. The structure of vector space and the structure of algebra are deepened by Artin in 1925, with basic bodies others that $\ _ \ mathbb R$ or $\ _ \ mathbb C$ and operators increasingly more abstract. One also must with Artin, regarded as the father of the contemporary algebra, of the fundamental results on the bodies of algebraic numbers. The noncommutative bodies bring to define the structure of Module on a ring and the generalization of the traditional results on the vector spaces.

The French school “Nicolas Bourbaki”, taken along by Weil, Cartan and Dieudonné, undertakes to rewrite the whole of mathematical knowledge on an axiomatic basis: this gigantic work starts with the set theory and the algebra in the middle of the century, and confirms the algebra like universal language of mathematics. Paradoxically, whereas the number of publications follows a Exponential growth throughout the world, whereas no mathematician can claim to dominate that an any small portion of knowledge, mathematics never appeared unified as much than today.

Modern European notations

the symbols " +" and " - " appear in 1489 in the work Arithmétique of John Widmann (Leipzig)
the sign " =" in 1557 at Robert Recorde " appears; because two things could be more equal only two parallel lines ".
the signs " <" and " >" appear in 1610 at Thomas Harriot (1560-1621).
William Oughtred (1574-1660) introduced the sign of the multiplication " x" in its Clavis Mathematica (1631). It introduces also the terms of sine , cosine and tangent .
the sign of division " /" is used by J.H. Rahn in 1659 and is introduced in England by John Pell in 1668.

See too

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