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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 12 – MONDAY 18TH OCTOBER Blaise Pascal At first glance one may think that the relatively small mathematical output of Blaise Pascal (1623‐1662) is due to the fact that he died at the age of 39. However a closer look at his life reveals a more complicated explanation, starting with the fact that he was educated by his father, Etienne, who expressly prohibited him from studying mathematics until the age of 15. Etienne was forced to change his mind though when Blaise started showing a great intuitive knowledge of geometry, and within a short time of being given a copy of Euclid’s Elements he had fully understood the initial 32 properties. So talented was the young Pascal that when Etienne was invited to the weekly meetings of Marin Mersenne (see Lecture 11), he took Blaise along. In 1640, still only 16, Blaise presented a paper Essay pour les coniques, to the distinguished members of Mersenne’s circle which stated a number of new results concerning a hexagon inscribed inside a circle. One result states that given a hexagon with vertices A, B, C, D, E, F inscribed inside a circle, then if we extend the lengths of the sides, then the intersection of lines AB and DE, AF and CD, BC and EF lie on the same line. Pascal’s next productive period occurred between 1651 and 1654, culminating with the publication of Traité du Triangle Arithmétique, which contains his famous triangle. Pascal’s Triangle Properties of Pascal’s Triangle There are many interesting patterns in the triangle, including the fact that if the numbers are written so as to form a right triangle rather than an equilateral triangle then the diagonal elements sum to the values of the Fibonacci numbers. However the most important feature is when a binomial term such as x + y is raised to a positive integer power, then by the binomial theorem we have where the values of ai are exactly the numbers in the nth row (starting from n=0) of Pascal’s triangle. The numbers also help us determine the solution to combinatorial problems, where a selection of k unordered objects is selected from a total of n objects. The answer is ak = !
! ! Pascal’s Triangle and Perfect Numbers Pascal and Probability Theory Pascal, along with Fermat, was instrumental in developing probability theory in the 17th century. Their interest was sparked by letters sent by the Flemish gentleman, the Chevalier de Méré, who enjoyed gambling, and wished to know more about the theoretical aspects of his favorite games, in particular the two below. Game 1: Roll a single die 4 times and bet on getting a six. One may naively assume that since the chance of getting a 6 in a single throw is 1 out of 6, it therefore follows that the chance of getting a 6 in four rolls is four times 1 out of 6; i.e. 2 out of 3. Game 2: Roll two dice 24 times and bet on getting a double six. Again, one might initially believe that since the chance of getting a double six in one roll is 1 out of 36, it follows that the chances of getting a double six in 24 rolls is 24 times 1 out of 36; i.e. 2 out of 3. Pascal’s life took a very different course starting in November 1654, when a riding accident almost led to hisdeath. From then on Pascal devoted himself to religion, and he became a popular writer on the issues of the day. He did however turn his attention back to mathematics for a brief period in 1658, during which he believed that by solving various problems involving a cycloid it would distract him from the toothache from which he was suffering. He extended the work of Gilles Personne de Roberval (1602‐1675), who first solved the problem of finding the area generated by a circle of radius r moving along a flat surface in 1634. This prompted Descartes to say of the result that “It is a pretty one, which I had not noticed before, but [one] which would cause no difficulty to any moderately skilled geometer.” Mathematical Induction While the use of induction as a method of proof was first applied by Franceco Maurolico (1494‐1575), it was Pascal who first used the method that it is familiar today. Consequence XII in the Triangle Arithmétique states that 1 1
Pascal’s proof reads as follows: “Although this proposition has an infinite number of cases, I will give a rather short demonstration by supposing two lemmas. The first one, which is self‐evident, is that the proposition occurs in the second base [i.e. when n = 1]. The second one is that if the proposition is true for an arbitrary base, it will necessarily be true in the next base. From which it is clear that it will necessarily be true in all bases, because it is true in the second base by the first lemma; hence by means of the second lemma, it is true in the third base, hence in the fourth base, and so on to infinity. It is therefore only necessary to prove the second lemma.” This is the process that we call mathematical induction today, and amounts to three steps: i) Showing that the result holds for the initial case (usually when n = 0 or n = 1). ii) Assuming that the result holds when n = k, and writing out the consequence. iii) Using the hypothesis in ii) to show that the result holds when n = k + 1. Ex. Ex. Note: Since the time of Pascal a lot of work has been done on induction (sometimes called “weak induction”), and we now use strong induction, the axiom of choice, and the well‐ordering principle as methods of proof, which are all mathematically equivalent to the original form. Strong induction involves replacing step ii) with the assumption that the result holds for all values of n less than or equal to some value k. Ex. Prove that every postage of 12 cents or more can be formed using 4‐cent and 5‐cent stamps. ...
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