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17th century history of mathematics

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  • 1. 17th Century (Fermat & Pascal)
  • 2.  Another Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Stimulated and inspired by the “Arithmetica” of the Hellenistic mathematician Diophantus, he went on to discover several new patterns in numbers which had defeated mathematicians for centuries, and throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory. Pierre De Fermat (1601-1665)
  • 3.  Fermat's mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat.  One example of his many theorems is the Two Square Theorem, which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4n + 1), can always be rewritten as the sum of two square numbers (see image at right for examples). Pierre De Fermat (1601-1665)
  • 4.  His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.  Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 22n + 1. The first five such numbers are: 21 + 3 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to show the value of inductive proof in mathematics. Pierre De Fermat (1601-1665)
  • 5.  Fermat’s Last Theorem Fermat's pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years. The theorem, originally described in a scribbled note in the margin of his copy of Diophantus' “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world’s hardest mathematical problems to prove. Pierre De Fermat (1601-1665)
  • 6.  There are clearly many solutions - indeed, an infinite number - when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent). Pierre De Fermat (1601-1665)
  • 7.  Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it singlehandedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years). The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet’s epsilon theorem, all of which were unavailable in Fermat’s time, so it seems clear that Fermat's claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding). Pierre De Fermat (1601-1665)
  • 8.  In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centers of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series.  Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values. Pierre De Fermat (1601-1665)
  • 9.  The Frenchman Blaise Pascal was a prominent 17th Century scientist, philosopher and mathematician. Like so many great mathematicians, he was a child prodigy and pursued many different avenues of intellectual endeavor throughout his life. Much of his early work was in the area of natural and applied sciences, and he has a physical law named after him (that “pressure exerted anywhere in a confined liquid is transmitted equally and undiminished in all directions throughout the liquid”), as well as the international unit for the measurement of pressure. In philosophy, Pascals’ Wager is his pragmatic approach to believing in God on the grounds that is it is a better “bet” than not to. Blaise Pascal (1623-1662)
  • 10. The table of binomial coefficients known as Pascal’s Triangle  But Pascal was also a mathematician of the first order. 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. Blaise Pascal (1623-1662)
  • 11.  He is best known, however, for Pascal’s Triangle, a convenient tabular presentation of binomial coefficients, where each number is the sum of the two numbers directly above it. A binomial is a simple type of algebraic expression which has just two terms operated on only by addition, subtraction, multiplication and positive whole-number exponents, such as (x + y)2. The coefficients produced when a binomial is expanded form a symmetrical triangle (see image at right).  Pascal was far from the first to study this triangle. The Persian mathematician Al-Karaji had produced something very similar as early as the 10th Century, and the Triangle is called Yang Hui's Triangle in China after the 13th Century Chinese mathematician, and Tartaglia’s Triangle in Italy after the eponymous 16th Century Italian. 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,...) is the natural numbers in order. The next diagonal within that (1, 3, 6, 10, 15,...) is the triangular numbers in order. The next (1, 4, 10, 20, 35,...) is the pyramidal triangular numbers, etc, etc. It is also possible to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series, and even to find fractal patterns within it. Blaise Pascal (1623-1662)
  • 12.  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 Christian Huygens on the subject that the mathematical theory of probability was born. Before Pascal, there was no actual theory of probability - notwithstanding Gerolamo Cardano’s early exposition in the 16th Century - merely an understanding (of sorts) of how to compute “chances” in dice and card games by counting equally probable outcomes. Some apparently quite elementary problems in probability had eluded some of the best mathematicians, or given rise to incorrect solutions. Blaise Pascal (1623-1662)
  • 13.  It fell to Pascal (with Fermat's help) to bring together the separate threads of prior knowledge (including Cardano's early work) and to introduce entirely new mathematical techniques for the solution of problems that had hitherto resisted solution. Two such intransigent problems which Pascal and Fermat applied themselves to were the Gambler’s Ruin (determining the chances of winning for each of two men playing a particular dice game with very specific rules) and the Problem of Points (determining how a game's winnings should be divided between two equally skilled players if the game was ended prematurely). His work on the Problem of Points in particular, although unpublished at the time, was highly influential in the unfolding new field. Blaise Pascal (1623-1662)
  • 14.  The Problem of Points at its simplest can be illustrated by a simple game of “winner take all” involving the tossing of a coin. The first of the two players (say, Fermat and Pascal) to achieve ten points or wins is to receive a pot of 100 francs. But, if the game is interrupted at the point where Fermat, say, is winning 8 points to 7, how is the 100 franc pot to divide? Fermat claimed that, as he needed only two more points to win the game, and Pascal needed three, the game would have been over after four more tosses of the coin (because, if Pascal did not get the necessary 3 points for your victory over the four tosses, then Fermat must have gained the necessary 2 points for his victory, and vice versa. Fermat then exhaustively listed the possible outcomes of the four tosses, and concluded that he would win in 11 out of the 16 possible outcomes, so he suggested that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16 (0.3125) to Pascal. Blaise Pascal (1623-1662)
  • 15.  Pascal then looked for a way of generalizing the problem that would avoid the tedious listing of possibilities, and realized that he could use rows from his triangle of coefficients to generate the numbers, no matter how many tosses of the coin remained. As Fermat needed 2 more points to win the game and Pascal needed 3, he went to the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The first 3 terms added together (1 + 4 + 6 = 11) represented the outcomes where Fermat would win, and the last two terms (4 + 1 = 5) the outcomes where Pascal would win, out of the total number of outcomes represented by the sum of the whole row (1 + 4 + 6 +4 +1 = 16). Blaise Pascal (1623-1662)
  • 16.  Pascal and Fermat had grasped through their correspondence a very important concept that, though perhaps intuitive to us today, was all but revolutionary in 1654. This was the idea of equally probable outcomes, that the probability of something occurring could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This allowed the use of fractions and ratios in the calculation of the likelihood of events, and the operation of multiplication and addition on these fractional probabilities. For example, the probability of throwing a 6 on a die twice is 1⁄6 x 1⁄6 = 1⁄36 ("and" works like multiplication); the probability of throwing either a 3 or a 6 is 1⁄6 + 1⁄6 = 1⁄ ("or" works like addition). 3 Blaise Pascal (1623-1662)
  • 17.  Later in life, Pascal and his sister Jacqueline strongly identified with the extreme Catholic religious movement of Jansenism. Following the death of his father and a "mystical experience" in late 1654, he had his "second conversion" and abandoned his scientific work completely, devoting himself to philosophy and theology. His two most famous works, the "Lettres provinciales" and the "Pensées", date from this period, the latter left incomplete at his death in 1662. They remain Pascal’s best known legacy, and he is usually remembered today as one of the most important authors of the French Classical Period and one of the greatest masters of French prose, much more than for his contributions to mathematics. Blaise Pascal (1623-1662)
  • 18. Thank You!!  Lyka Cabello BSE-II