Blaise Pascal

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2 Blaise Pascal

3 Blaise Pascal background and early life Born 1623, Clermont-Ferrand, France Mother died when he was 3 Father was a senior government administrator, which meant he was a minor member of the nobility Two sisters, one older and one younger Father interested in maths and science, and in education

4 Blaise Pascal background and early life Family moved to Paris in 1631 Father educated Blaise and his sisters at home intention was to teach Blaise languages first, then move on to maths and science from age 14 Blaise soon showed amazing abilities especially in maths and science family legend is that he had derived most of Euclid from scratch by age 12

5 Blaise Pascal background and early life Family lost its money in 1631 when Cardinal Richelieu defaulted on the government s bonds to pay for the 30 years war Father forced to flee Paris, having fallen out with Richelieu left his children in the care of a aristocratic society lady Father eventually pardoned and made King s commissioner of taxes in Rouen

6 Blaise Pascal background and early life First serious work on mathematics when he was 16 essay on conics, sent to Pere Mersenne Pascal s theorem in projective geometry Descartes didn t believe a 16 year old boy could have done it

7 Pascal s Theorem If six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

8 Pascal s Theorem

9 The Pascaline A calculating machine designed and built by Pascal between 1642 and 1644 (when Pascal was between 19 and 21years old!) His motivation for building it was to try to help his father do his tax calculations more efficiently. The Pascal computer language is named after him

10 The Pascaline Besides being the first calculating machine made public during its time, the Pascaline is also: the only operational mechanical calculator in the 17th century the first calculator to have a controlled carry mechanism that allowed for an effective propagation of multiple carries

11 The Pascaline the first calculator to be used in an office (his father's to compute taxes) the first calculator sold commercially (around twenty machines were built) the first calculator to be patented (royal privilege of 1649 [Louise XIV])

12 Scientific work Pascal invented the hydraulic press and the syringe He proved the existence of the vacuum (disagreeing with Descartes) Pascal s law: any change in pressure applied at any given point of a fluid is transmitted undiminished throughout the fluid This is the basis of hydraulics

13 Scientific work The SI unit of pressure is called the Pascal (1 Pascal = 1Nm -2 ) Pascal was a pioneer of the scientific method: In order to show that a hypothesis is evident, it does not suffice that all the phenomena follow from it; instead, if it leads to something contrary to a single one of the phenomena, that suffices to establish its falsity.

14 Theological work In 1646 he became interested in Jansenism a type of Catholicism He then lost interest in religion until 1654 when he had a near death experience in a carriage accident that was followed by a religious vision

15 Theological work His religious vision prompted him to give up maths and science and concentrate on theology instead The Pensees, a theological work published posthumously, is considered one of the greatest works of French literature it contains Pascal s wager

16 Blaise Pascal Pascal died in 1662, aged just 39 If he hadn t died so young, or got distracted from maths and science by religion, how much more might he have achieved?

17 The problem of points The problem: Two people each put 10 into a pot. They then choose who will play heads and who will play tails when tossing a coin. They get a point each time the coin shows their choice. The first one to get 10 points wins. If the game is interrupted before the end, depending on the number of points each has at that time, how should the pot be divided?

18 The problem of points Pascal s friend, the Chevalier de Méré, asked him to consider this problem. Pascal wrote to Pierre de Fermat about it Each produced the same answer, but using different methods Pascal s method introduced the notion of the expected value and Pascal s and Fermat s work on this problem became the foundation of probability theory

19 The problem of points Imagine the game is between Jeremy and Theresa An unexpected interruption means the game stops when Theresa (playing heads) is leading Jeremy (playing tails) by 8 to 7

20 The problem of points Theresa says she is ahead and so should get all the money Jeremy says the game is incomplete so all bets are off and they should each get back their stake Tim suggests that since Theresa has won 8/15 of the games, she should get 8/15 X 20 = What do you think?

21 The problem of points When the score is 8 to 7, what is the maximum number of coin tosses need to complete the game?

22 To solve the problem: Fermat used and approach based on listing all possible outcomes to find the probability of each winning Pascal used an approach based on the probability of each outcome and the expected return Try it! The problem of points

23 Pascal s wager You are a believer God exists + Infinity (eternal bliss in heaven) God does not exist Something finite You are a nonbeliever - Infinity (eternal misery in hell) Something finite Pascal is suggesting a way to make a rational decision, based upon expectation

24 Pascal s triangle Using Pascal s triangle to determine binomial coefficients features in AS Maths, both to expand brackets and to calculate binomial probabilities

25 Pascal s triangle Why do the numbers in Pascal s triangle give the binomial coefficients?

26 Pascal s triangle If the initial 1 in the triangle forms row 0, investigate and, where possible, prove these statements (at least convince your neighbour!): a.the sum of the numbers in the n th row is 2 n b.the number formed by the digits in the n th row of Pascal s triangle is 11 n c.the numbers in the 3 rd diagonal are the triangular numbers

27 Pascal s triangle Discuss how the binomial coefficients relate to combinations: Explain why the r th entry in the n th row is equal to n C r n! r!( n r)!

28 Why does the way that Pascal s triangle is constructed suggest Prove it! Pascal s triangle C C C n r n r 1 n 1 r 1?