Pascal’s Triangle — Blaise- ing a Trail of Mathematics

1. Pascal’s Triangle — Blaise-ing a Trail of Mathematics Eddie Tchertchian Los Angeles Pierce College CMC3-S 2019 – Pomona, CA

2. Eddie TchertchianLos Angeles Pierce College • This presentation is available on SLIDESHARE: http://www.slideshare.net/EddieMath tchertea@piercecollege.edu

3. “All of men’s miseries derive from not being able to sit quietly in a room alone.” • Born June 19, 1623 in Clermont-Ferrand, Auvergne, France to father Etienne Pascal & mother Antoinette Begon • Lost mother at the age of 3 – family relocated to Paris five years later

4. “Do you wish people to think well of you? Don’t speak well of yourself.” • At age 16 – Pascal’s theorem: If six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) 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 at three points which lie on a straight line, called the Pascal line of the hexagon.

5. “I would prefer an intelligent hell to a stupid paradise.” • When Etienne became the king’s commissioner of taxes in the city of Rouen, Pascal tried to aid his father in doing many computations by constructing the world’s first mechanical calculator, the “Pascaline.”

6. “To make light of philosophy is to be a true philosopher.” • Pascal’s contributions to mathematics were numerous, as were his contributions outside of mathematics: • Philosophy (of mathematics – axiomatic method; formalism based on Descarte’s work) • Literature & religion (The Pensées – “Thoughts”) • Physical sciences (pressure – Pascal’s principle) • Probability theory/gambling (primitive form of roulette wheel)

7. “You always admire what you really don’t understand.” • Pascal’s triangle & binomial coefficients were studied by Pascal in 1653, but had been described and well-known centuries before that around the world: • Indian studies of combinatorics & the numbers of the triangle date back to Pingala (2nd century BC) • Iran: Al-Karaji wrote a now lost book which contained the first description of Pascal’s triangle; repeated later by Omar Khayyam (1048-1131) – Khayyam’s triangle • China: Jia Xian (1010-1070); Yang Hui (1238-1298) presented the triangle – Yang Hui’s triangle

8. “It is man's natural sickness to believe that he possesses the truth.” • Germany: Petrus Apianus – full triangle published (1527) • Italy: Tartaglia’s triangle (1556 – first six rows); Cardano published the triangle & additive and multiplicative rules for constructing it. (1570)

9. The entry in the n-th row, r-th column is simply the binomial coefficient “n choose r”

23. If the 1st element in a row is a prime number (remember, the 0th element of every row is 1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.

24. The sum of the squares of the elements of row n equals the middle element of row 2n. For example, 12 + 42 + 62 + 42 + 12 = 70. In general form: Row 4 Row 8 = 2 x 4

25. Sum of first k number of rows is the Mersenne prime number

27. There are infinitely many numbers that occur at least six times in Pascal’s (whole) triangle, namely the solutions to: given by where is the -th Fibonacci number. • The numbers that occur five or more times in Pascal’s triangle are 1; 120; 210; 1540; 3003; 7140; 11628; 24310; and the number 61,218,182,743,304,701,891,431,482,520 with no others up to

30. Using Nilakantha’s infinite series for

34. “The last thing one discovers in composing a work is what to put first.” • Thanks to: • A Piece of the Mountain: The Story of Blaise Pascal (Joyce McPherson; Greenleaf press; 1995) • Pascal’s Triangle: A Study in Combinations (Jason VanBilliard; CreateSpace Publishing; 2014) • Blaise Pascal: Reasons of the Heart (Marvin O’Connell; Wm. B. Eerdmans Publishing Co; 1997) • http://mathworld.wolfram.com/PascalsTriangle.html and associated links within • http://www.cut-the-knot.org/arithmetic/algebra/PiInPascal.shtml • https://en.wikipedia.org/wiki/Blaise_Pascal • https://en.wikipedia.org/wiki/Pascal%27s_triangle • https://www.goodreads.com/author/quotes/10994.Blaise_Pascal • https://www.mathsisfun.com/pascals-triangle.html

35. Thank you!Eddie TchertchianLos Angeles Pierce College • This presentation will be available on SLIDESHARE: http://www.slideshare.net/EddieMath tchertea@piercecollege.edu