Transcript Pascal
HANNAH WIKUM & BRIAN LAUSCHER Pascal’s , Fibonacci’s Numbers, Algebraic Expansions & Combinations Pascal June 19, 1623-August 19, 1662 Born in Clermont-Ferrand, France Mathematician, physicist, religious philosopher Instrumental in development of economics and social science “Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” Quoted in W H Auden and L Kronenberger, The Viking Book of Aphorisms (New York 1966). Pascal’s Triangle 1 11 1 2 1 0 1 2 3 4 … Combinations Used to count groupings without regard to order n=total number of objects r=amount taken at a time nCr = n! (n-r)!r! Combinations If there are 9 coins on a table and you are asked to take 3 of the coins how many different combinations are possible? Hint: n=9 r=3 Combinations 9C3 = 9C3 = 9C3 = 9C3= 9C3= 9C3= (9)! (9-3)! (3)! (9)! (6)! (3)! 9.8.7.6! (6)! (3)! 9.8.7 (3)! 504 6 84 Combinations in Pascal’s Triangle n=9 r=3 9 3 •Go down n rows •Go over r rows •Resulting square is the amount of combinations Try it! Bob wants to order an ice cream sundae. Of the seven toppings, he can choose three. Assuming he chooses three different toppings, how many different combinations can he choose from? Remember ~ Pascal’s Triangle can also be used for algebraic expansion. Example: Try it! Solution: 35 Patterns in Pascal’s Triangle Counting Numbers Patterns in Pascal’s Triangle Triangular Numbers Patterns in Pascal’s Triangle Hexagonal Numbers Patterns in Pascal’s Triangle Tetrahedral Numbers Patterns in Pascal’s Triangle Fibonacci Numbers Fibonacci Born 1170 in Pisa, Italy Died 1250 Educated in Northern Africa where he was introduced to the Hindu-Arabic numeral system (0-9 instead of Roman Numerals) Fibonacci’s Question A pair of newly born rabbits, male and female, were placed in a hutch. In two months, these rabbits began their breeding cycle and produced one pair of rabbits, one male and one female. The original rabbits and their offspring continued to breed in this manner, that is the first pair of offspring appearing at the parental age of two months and every new pair every month thereafter-always one male and one female. All rabbits survived their first year. What then is the total number of pairs of rabbits at the beginning of the months during the first year? Fibonacci’s Question (Beginning of) Month Productive * Nonproductive * Total * 1st 0 1 1 2nd 1 0 1 3rd 1 1 2 4th 2 1 3 5th 3 2 5 6th 5 3 8 7th 8 5 13 8th 13 8 21 9th 21 13 34 10th 34 21 55 * Indicates pairs (2 rabbits) Fibonacci Number Sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 0+1=1 1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 8+13=21 13+21=34 recursive: pertaining to or using a rule or procedure that can be applied repeatedly Ratios of Fibonacci Numbers Ratio of Adjacent Fibonacci Numbers Decimal Equivalent 1/1 1.0 2/1 2.0 3/2 1.5 5/3 1.666… 8/5 1.6 13/8 1.625 21/13 1.6153… 34/21 1.6190 55/34 1.6176 Golden Ratio Ratios approach 1.6803something 1.6803…=(1+√5)/2 Golden Ratio Fibonacci Sequence Graph Golden Ratio in Nature Review Pascal’s Triangle has many patterns Use Pascal’s Triangle to solve Combinations and Algebraic expansion Example: (x+y)4 = x4 +4x3 y+6x2 y2+4xy3+y4 Fibonacci Numbers are known as the natural numbers (0, 1, 1, 2, 3, 5, 8, 13, etc.) Ratio of adjacent Fibonacci Numbers equals the Golden Ratio Picture Bibliography http://recycle.lbl.gov/apac2007/Blaise_pascal.jpg http://goitaly.about.com/od/pisa/p/pisa.htm http://en.wikipedia.org/wiki/File:Pascal%27s_Triangle_rows_0-16.svg http://mathforum.org/workshops/usi/pascal/pascal_hexagonal.html http://mathforum.org/workshops/usi/pascal/pascal_triangular.html http://goldennumber.net/pascal.htm http://creativecag.com/art/fibonacci-graph.jpg http://www.nazmath.net/Online_Classes/HTML2/Wk2/parthenon.jpg http://farm1.static.flickr.com/58/182577397_aa27d7830d.jpg http://www.abc.net.au/science/photos/mathsinnature/img/13.jpg http://z.about.com/d/webdesign/1/0/E/K/1/nautilus.jpg http://www.scibuff.com/blog/wp-content/uploads/2009/05/fibonacci-00.jpg http://4.bp.blogspot.com/_V8KsSIiGjBk/SP4fm7IoNBI/AAAAAAAACXQ/OKha9SuU4y g/s400/ice+cream+sundae.jpg http://www.wvi.com/~coinguy/coins.jpg http://www.petsworld.co.uk/images/rabbit.jpg Bibliography Anderson, Matt, Jeffrey Frazier, and Kris Popendorf. 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Fibonacci Numer - Theorists. 21 Mar. 2009 <http://faculty.evansville.edu/ck6/bstud/fibo.html>. Katsiavriades, Kryss. "Pascal's Triangle." The KryssTal Website. 2004. 1 June 2009 <http://www.krysstal.com/binomial.html>. Seward, Kim. "College Algebra Tutorial 57: Combinations." Vitrual Math Lab. 23 June 2003. West Texas A&M University. 1 June 2009 <http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/ col_alg_tut57_comb.htm>. Smith, Harry J. What is a Fibonacci Number? 23 May 2008. 21 Mar. 2009 <http://www.geocities.com/ hjsmithh/Fibonacc/FibWhat.html>. Weisstein, Eric W, and Pravin Chandra. "Fibonacci Number." Wolfram Math World. 20 Mar. 2009. 21 Mar. 2009 <http://mathworld.wolfram.com/FibonacciNumber.html>.