Blaise pascal and pierre de fermat relationship questions

Pierre de Fermat - Wikipedia

Pierre Fermat's father was a wealthy leather merchant and second consul of more interested in proving geometrical theorems than in their relation to the real world. Frenicle de Bessy became annoyed at Fermat's problems which to him were mathematicians restarted in when Blaise Pascal, Étienne Pascal's son. Pierre de Fermat was an intensely reticent part-time mathematician, a lawyer by Blaise Pascal exchanged letters about questions of probability in relation to. Rene Descartes, Pierre Fermat and Blaise Pascal Pierre de Fermat 17th August or – 12th January Blaise Pascal, - . The notation below describes the relationship of two coordinate systems (x′ and x) in The Poincaré Conjecture is first and only of the Clay Millennium problems to be.

But Pascal was also a mathematician of the first order.

probability theory / Blaise Pascal / Pierre de Fermat

At the age of sixteen, he wrote a significant treatise on the subject of projective geometry, known as Pascal's Theorem, which states that, if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a single line, called the Pascal line. As a young man, he built a functional calculating machine, able to perform additions and subtractions, to help his father with his tax calculations. The co-efficients produced when a binomial is expanded form a symmetrical triangle see image at right.

Pascal was far from the first to study this triangle.

Blaise Pascal - Wikipedia

But Pascal did contribute an elegant proof by defining the numbers by recursion, and he also discovered many useful and interesting patterns among the rows, columns and diagonals of the array of numbers. For instance, looking at the diagonals alone, after the outside "skin" of 1's, the next diagonal 1, 2, 3, 4, 5, The next diagonal within that 1, 3, 6, 10, 15, The next 1, 4, 10, 20, 35, It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it.

Pascal also made the conceptual leap to use the Triangle to help solve problems in probability theory. In fact, it was through his collaboration and correspondence with his French contemporary Pierre de Fermat and the Dutchman Christiaan Huygens on the subject that the mathematical theory of probability was born.

Pierre de Fermat

These curves in turn directed him in the middle s to an algorithmor rule of mathematical procedure, that was equivalent to differentiation. This procedure enabled him to find equations of tangents to curves and to locate maximum, minimum, and inflection points of polynomial curves, which are graphs of linear combinations of powers of the independent variable.

Blaise Pascal

During the same years, he found formulas for areas bounded by these curves through a summation process that is equivalent to the formula now used for the same purpose in the integral calculus.

Such a formula is: It is not known whether or not Fermat noticed that differentiation of xn, leading to nan - 1, is the inverse of integrating xn.

• Pierre de Fermat

Through ingenious transformations he handled problems involving more general algebraic curves, and he applied his analysis of infinitesimal quantities to a variety of other problems, including the calculation of centres of gravity and finding the lengths of curves. He also solved the related problem of finding the surface area of a segment of a paraboloid of revolution.

Descartes had sought to justify the sine law through a premise that light travels more rapidly in the denser of the two media involved in the refraction.

Twenty years later Fermat noted that this appeared to be in conflict with the view espoused by Aristotelians that nature always chooses the shortest path. Through the mathematician and theologian Marin Mersennewho, as a friend of Descartes, often acted as an intermediary with other scholars, Fermat in maintained a controversy with Descartes on the validity of their respective methods for tangents to curves.

In he had enjoyed an exchange of letters with his fellow mathematician Blaise Pascal on problems in probability concerning games of chance, the results of which were extended and published by Huygens in his De Ratiociniis in Ludo Aleae Work on theory of numbers Fermat vainly sought to persuade Pascal to join him in research in number theory. Inspired by an edition in of the Arithmetic of Diophantusthe Greek mathematician of the 3rd century ad, Fermat had discovered new results in the so-called higher arithmetic, many of which concerned properties of prime numbers those positive integers that have no factors other than 1 and themselves.

Fermat seldom gave demonstrations of his results, and in this case proofs were provided by Gottfried Leibniz, the 17th-century German mathematician and philosopher, and Leonhard Eulerthe 18th-century Swiss mathematician.