EVOLUTION OF MATHEMATICS We will deal with the evolution of mathematical concepts and ideas following their historical development. In fact, mathematics is as old as mankind itself: in prehistoric designs of pottery, textiles and rock paintings one can find evidence of geometric sense and interest in geometric figures. Primitive calculus systems were probably based on the use of one- or two-handed fingers, which is evident from the great abundance of numerical systems in which the bases are numbers 5 and 10. The earliest references to advanced and organized mathematics date back to the third millennium BC in Babylon and Egypt. EGYPT The first Egyptian books, written about 1800 BC, show a system of decimal numeration with different symbols for successive powers of 10 (1, 10, 100 ...), similar to the system used by the Romans. The numbers were represented by writing the symbol of 1 as many times as units had the given number, the symbol of 10 as many times as there were tens in the number, and so on. To add numbers, the units, tens, hundreds ... of each number were added together. The multiplication was based on successive duplications and division was the reverse process. BABYLON The Babylonian numbering system was quite different from the Egyptian. In the Babylonian one used tablets with several notches or marks in the form of wedge (cuneiform); A simple wedge represented 1 and an arrow-shaped mark represented 10. Numerals smaller than 59 were formed by these symbols using an additive process, as in Egyptian mathematics. Over time, the Babylonians developed more sophisticated mathematics that allowed them to find the positive roots of any equation of the second degree. They were even able to find the roots of some third-degree equations, and solved more complicated problems using the Pythagorean Theorem. MATHEMATICS IN GREECE The Greeks took elements of the mathematics of the Babylonians and the Egyptians. The most important innovation was the invention of abstract mathematics based on a logical structure of definitions, axioms and demonstrations. According to Greek chroniclers, this advance began in the sixth century BC. With Thales of Miletus and Pythagoras of Samos. The latter taught the importance of the study of numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, which are attributed to Pythagoras himself. THE MAYA AND MATHEMATICS The Maya were partially advanced in mathematics and astronomy. Although the first documented use of zero is by the Maya (in 36 BC), they remained stagnant since they knew of no other developments such as decimals, complex numbers, infinitesimal calculus, etc. In mathematics they developed a system of numeration using three symbols and of base 20. In astronomy they made calculations of cycles with great precision considering that they were realized in simple sight, without using instruments like telescopes. However, they were inferior compared to the advances that can be made thanks to these instruments. Neither did they know the sphericity of the Earth, the heliocentric model, etc. MATHEMATICS IN THE MIDDLE AGES (5TH AND 15TH CENTURIES) In Greece, after Ptolemy, established the tradition of studying the works of these mathematicians of previous centuries in the schools. The fact that these works have been preserved to this day is mainly due to this tradition. Nevertheless, the first mathematical advances consequent of the study of these works appeared in the Arab world. The Arabs provided European culture with their numbering system, which replaced Roman numeration. This system was practically unknown in Europe before the mathematician Leonardo Fibonacci introduced it in 1202 in his work Liber abbaci (Book of the abacus). MATHEMATICS IN THE ISLAMIC WORLD By the year 900, the incorporation period had been completed and Muslim scholars began to build on the acquired knowledge. Among other advances, Arabic mathematicians extended the Indian system of decimal places into integer arithmetic, extending it to decimal fractions. In the twelfth century Persian mathematician Omar Jayyam generalized Indian methods of extracting square and cubic roots to calculate fourth, fifth and fifth degree roots. The Arab mathematician Al-JwDrizm¬; (From its name comes the word algorithm, and the title of one of its books is the origin of the word algebra) developed the algebra of the polynomials; Al-Karayi completed it for polynomials even with infinite number of terms. MATHEMATICS DURING THE RENAISSANCE Although the end of the medieval period witnessed important mathematical studies on problems of the infinite by authors like Nicole Oresme, it was not until the early sixteenth century when a mathematical discovery of transcendence was made in the West. It was an algebraic formula for the resolution of the third and fourth degree equations, and was published in 1545 by the Italian mathematician Gerolamo Cardano in his Ars magna. This finding led mathematicians to become interested in complex numbers and stimulated the search for similar solutions for fifth-degree and higher-order equations. It was this search that in turn generated the first works on the theory of groups in the late eighteenth century and the theory of equations of the French mathematician Évariste Galois in the early nineteenth century. ADVANCES IN THE SEVENTEENTH CENTURY Europeans dominated the development of mathematics after the Renaissance. During the seventeenth century took place the most important advances in mathematics since the era of Archimedes and Apollonius. The century began with the discovery of logarithms by the Scottish mathematician John Napier (Neper); Its great utility led the French astronomer Pierre Simon Laplace to say, two centuries later, that Neper, by reducing the work of astronomers by half, had doubled his life. The science of number theory, which had remained dormant since medieval times, is a good example of the advances made in the seventeenth century on the basis of studies of classical antiquity. Diophantan's The Arithmetic helped Fermat make important discoveries in number theory. SITUATION IN THE 18TH CENTURY During the rest of the seventeenth century and much of the eighteenth century, Newton and Leibniz's disciples relied on their work to solve various problems of physics, astronomy and engineering, which allowed them, at the same time, to create new fields within mathematics . Thus the brothers Jean and Jacques Bernoulli invented the calculus of variations and the French mathematician Gaspard Monge the descriptive geometry. Joseph Lagrange, also French, gave a completely analytical treatment of mechanics in his great work Analytical Mechanics (1788), where you can find the famous Lagrange equations for dynamical systems. In addition, Lagrange made contributions to the study of differential equations and number theory, and developed group theory. MATHEMATICS IN THE 19TH CENTURY In 1821, a French mathematician, Augustin Louis Cauchy, got a logical and appropriate approach to calculus. Cauchy based his view of calculus only on finite quantities and the concept of limit. However, this solution posed a new problem, that of the logical definition of real number. Although Cauchy's definition of calculation was based on this concept, it was not he but the German mathematician Julius W. R. Dedekind who found a suitable definition for real numbers, from the rational numbers, which is still taught today; The German mathematicians Georg Cantor and Karl T. W. Weierstrass also gave other definitions almost at the same time. MATHEMATICS AT THE END OF THE 20TH CENTURY At the International Mathematical Conference held in Paris in 1900, the German mathematician David Hilbert expounded his theories. Hilbert was a professor at Göttingen, the academic home of Gauss and Riemann, and had contributed substantially in almost all branches of mathematics, from his classical Foundations of Geometry (1899) to his Foundations of Mathematics in collaboration with other authors . Hilbert's lecture in Paris consisted of a review of 23 mathematical problems that he believed might be the goals of mathematical research of the beginning of the century. These problems, in fact, have stimulated much of the mathematical work of the twentieth century, and every time news comes that another "Hilbert problem" has been solved, the international mathematical community awaits details impatiently. MATHEMATICS TODAY Today, mathematics is used throughout the world as an essential tool in many fields, including natural sciences, engineering, medicine and social sciences, and even disciplines which are apparently not linked to She, like music (for example, in matters of harmonic resonance). Applied mathematics, the branch of mathematics for the application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of new disciplines. Mathematicians also participate in pure mathematics, without regard to the application of this science, although the practical applications of pure mathematics are usually discovered over time.