Algebra

Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b – 2 = 5 are algebraic equations, but 2 + 3 = 5 and 73 * 46 = 3,358 are not.

Classical Algebra
The work of François Viète at the end of the 16th century, described in the section Viète and formal equations, marks the beginning of the classical discipline of algebra. Further developments include a number of related trends, of which the following deserve particular mention: the pursuit of systematic solutions to higher-order equations, including approximation techniques; the rise of polynomials and their study as independent mathematical entities; and the increasing adoption of algebraic approaches in other areas of mathematics, such as geometry, analysis, and logic. During the same period, new mathematical objects arose that eventually replaced polynomials as the main focus of algebraic study.

Analytical Geometry
The creation of what is known as analytic geometry can be attributed to two great French thinkers of the 17th century: Pierre de Fermat and René Descartes. Using algebraic techniques developed by Viète and Girolamo Cardano, as described earlier in this article, Fermat and Descartes tackled geometric problems that had not been solved since the time of the classical Greeks. The new kind of organic connection he established between algebra and geometry was an important breakthrough, without which the development of mathematics in general and geometry and calculus in particular would have been inconceivable.

French mathematician René Descartes (1596–1650) showed that the square root of any line segment can be formed by the simple, but ingenious, addition of line segments of unit length.
French mathematician René Descartes (1596–1650) showed that the square root of any line segment can be formed by the simple, but ingenious, addition of line segments of unit length.
In his famous book La Géométrie (1637), Descartes established equivalence between algebraic operations and geometric constructions. To do this, he introduced a unit length that served as a reference for all other lengths and all operations between them. For example, suppose Descartes was given a segment AB and asked to find its square root. He would draw the straight line DB (see figure), where DA was defined as unit length. Then he will bisect DB at C, draw a semicircle on the diameter of DB with center C, and finally draw a perpendicular from A to E on the semicircle. The elementary properties of a circle imply that ∠DEB = 90 °, which in turn implies that ∠ADE = ∠AEB and ∠DEA = ∠EBA. Thus, △DEA is similar to △EBA, or in other words, the ratio of the corresponding sides is equal. Substituting x, 1, and y for AB, DA, and AE, respectively, one obtains x/y = y/1. Simplifying, x = y2, or y is the square root of x. Thus, in what may be a general application of classical Greek techniques, Descartes showed that he could find the square root of any number, as represented by a line segment. The main step in its construction was the introduction of the unit length DA. This seemingly trivial move, or anything like it, had never been done before, and it had huge implications for what could then be done by applying algebraic reasoning to geometry.

Descartes also introduced a notation that allowed greater flexibility in symbolic manipulation. For example, he would write the Descartes solution to express the cube root of this algebraic expression. It was a direct continuation (with some improvements) of the techniques and gestures introduced by Viète. Descartes also introduced a new idea with truly far-reaching consequences when he explicitly dropped the demand for uniformity between terms in an equation—although for convenience he tried to stick to uniformity wherever possible. of

Descartes’ program was based on the idea that certain geometric loci (straight lines, circles, and conic sections) could be specified in terms of certain types of equations that included dimensions that represented line segments. were taken for However, he did not conceive of the equally important, reciprocal idea of ​​finding curves that correspond to an arbitrary algebraic expression. Descartes knew that much information about the properties of a curve – such as its tangents and enclosed areas – could be derived from its equation, but he did not explain it.

Descartes, on the other hand, was the first to discuss polygebraic equations separately and systematically. These include his observations on the correspondence between the degree of an equation and the number of its roots, the factorization of polynomials with known roots into linear factors, the principle of counting the number of positive and negative roots of an equation, and the method of a new To obtain an equation whose roots were equal to the given equation, though increased or decreased by a given amount.