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IB Math Exploration

"Should Blaise Pascal be famous for what he is famous for?"
by

Mio Okayanagi

on 24 April 2013

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Transcript of IB Math Exploration

Should Blaise Pascal be famous for what he is famous for? Pascal's triangle Triangular array of binomial coefficients Each number is the sum of the two numbers above. Diagonals Triangular numbers sequence: (1,3,6,10,15,21,28,36,45....) Rule : Xn = n(n+1)/2 Tetrahedral numbers Rule : Xn = n(n+1)(n+2)/6 Patterns Horizontal sums The horizontal sums double each time. Exponents of 11 Why shouldn't Blaise Pascal be famous for the Pascal's triangle? History Odds and Evens Odds and evens colored Sierpinski triangle The Sierpinski Triangle An infinite pattern of triangles 1050 1523 Italy China Nicolo Tartaglia first published the generalization of figurate numbers Figurate numbers:
Figurate numbers are sequences generated by figures made up of evenly spaced dots. Figurate numbers include the triangular numbers, square numbers, pentagonal numbers, etc. Sometimes they are called polygonal numbers since they form polygons. source: http://math.youngzones.org/FigurateNumbers.html 1539 Gerolamo Cardano Determined that the number of ways to take 2 or more things from a set of n things is 2n-n-1. 1544 Michael Stifel published the extended figurate triangle Germany Italy Stifel gives credit to Cardano's work published five years earlier. 1653 France Bliase Pascal publishes the Pascal's triangle. Said to have originated with Pythagoras, possibly based on Babylonian or Egyptian precursors source: Wikipedia Why is Blaise Pascal famous for the Pascal's triangle? Pascal was the first one to publish a treatise on the Arithmetical triangle describing it. Established a fomula :

C(n, k) = n(n - 1)(n - 2)·...·(n - k + 1)
k(k - 1)(k - 2)·...·1 was using the triangle to extract square and cube roots of numbers Chia Hsien Omar Khayyam around 1048 A Persian mathematician, seemed to have had knowledge of the so-called Pascal’s Triangle. source: http://pages.csam.montclair.edu/~kazimir/history.html http://pages.csam.montclair.edu/~kazimir/history.html Pascal developed many of the triangle’s properties and applications using the already known array of binomial coefficients. Combination formula find out how many combinations there are of k things from a set of n Pierre Raymond de Montmort (1708) and
Abraham de Moivre (1730) names the triangle after Pascal. Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21 Fibonacci sequence: 1,1,2,3,5,8,13,21.. PINEAPLES
There are two spirals; one going in a clockwise direction, and one the opposite. The number of these spirals is a number in the Fibonacci sequence.
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