1. HANNAH WIKUM & BRIAN LAUSCHER Pascal’s, Fibonacci’s Numbers, Algebraic Expansions & Combinations

2. Pascal “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).  June 19, 1623-August 19, 1662  Born in Clermont-Ferrand, France  Mathematician, physicist, religious philosopher  Instrumental in development of economics and social science

3. Pascal’s Triangle 1 1 2 1 01234…01234…

4. Combinations Used to count groupings without regard to order n=total number of objects r=amount taken at a time n C r = n! (n-r)!r!

5. 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

6. 9 C 3 = (9)! (9-3)! (3)! 9 C 3 = (9)! (6)! (3)! 9 C 3 = 9.8.7.6! (6)! (3)! 9 C 3 = 9.8.7 (3)! 9 C 3 = 504 6 9 C 3 = 84

7. Combinations in Pascal’s Triangle Go down n rows Go over r rows Resulting square is the amount of combinations n=9 r=3 9 3

8. 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:

9. Try it! Solution: 35

10. Patterns in Pascal’s Triangle Counting Numbers

11. Patterns in Pascal’s Triangle Triangular Numbers

12. Patterns in Pascal’s Triangle Hexagonal Numbers

13. Patterns in Pascal’s Triangle Tetrahedral Numbers

14. Patterns in Pascal’s Triangle Fibonacci Numbers

15. 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)

16. 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?

17. Fibonacci’s Question (Beginning of) Month Productive *Nonproductive *Total * 1 st 011 2 nd 101 3 rd 112 4 th 213 5 th 325 6 th 538 7 th 8513 8 th 13821 9 th 211334 10 th 342155 * Indicates pairs (2 rabbits)

18. recursive: pertaining to or using a rule or procedure that can be applied repeatedly 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 How does it work?

19. Ratios of Fibonacci Numbers Ratio of Adjacent Fibonacci NumbersDecimal Equivalent 1/11.0 2/12.0 3/21.5 5/31.666… 8/51.6 13/81.625 21/131.6153… 34/211.6190 55/341.6176

20. Golden Ratio Ratios approach 1.6803something 1.6803…=(1+√5)/2 Fibonacci Sequence Graph Golden Ratio

21. Golden Ratio in Nature

22. Review Pascal’s Triangle has many patterns Use Pascal’s Triangle to solve Combinations and Algebraic expansion Example: (x+y) 4 = x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4 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

23. 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://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

24. Bibliography Anderson, Matt, Jeffrey Frazier, and Kris Popendorf. "Nature." The Fibonacci Series. Think Quest. 21 Mar. 2009. "Blaise Pascal." Wikipedia. 2009. MediaWiki. 1 June 2009. Burger, Edward B, and Michael Starbird. Coincedences, Chaos, and All That Math Jazz. New York, NY 10110: W.W. Norton & Company, Inc., 2005. Department of Mathematics. Dept. home page. 13 June 2008. Surrey University. 17 Mar. 2009. Dunham, William. The Mathematical Universe. New York, NY: John Wiley & Sons, Inc., 1994. Gedney, Larry. "Nature's Golden Ratio." Alaska Science Forum. 20 May 1985. University of Alaska Fairbanks. 21 Mar. 2009. Gullberg, Jan. Mathematics: From the Birth of Numbers. New York, NY: W.W. Norton & Company, Inc., 1997. Horadam, A. F. "Eight Hundred Years Young." Fibonacci Numer - Theorists. 21 Mar. 2009. Katsiavriades, Kryss. "Pascal's Triangle." The KryssTal Website. 2004. 1 June 2009. Seward, Kim. "College Algebra Tutorial 57: Combinations." Vitrual Math Lab. 23 June 2003. West Texas A&M University. 1 June 2009. Smith, Harry J. What is a Fibonacci Number? 23 May 2008. 21 Mar. 2009. Weisstein, Eric W, and Pravin Chandra. "Fibonacci Number." Wolfram Math World. 20 Mar. 2009. 21 Mar. 2009.