HANNAH WIKUM & BRIAN LAUSCHER

Pascal’s ,Fibonacci’s Numbers,

Algebraic Expansions & Combinations

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, 1662Born in Clermont-Ferrand, FranceMathematician, physicist, religious philosopherInstrumental in development of economics and social science

Pascal’s Triangle

1

1 1

1 2 1

01234…

Combinations

Used to count groupings without regard to order

n=total number of objectsr=amount taken at a time

nCr = n!

(n-r)!r!

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

Combinations

9C3 = (9)!

(9-3)! (3)!

9C3 = (9)!

(6)! (3)!

9C3 = 9.8.7.6!

(6)! (3)!

9C3= 9.8.7

(3)!

9C3= 504

6

9C3= 84

Combinations in Pascal’s Triangle

•Go down n rows•Go over r rows•Resulting square is the amount of combinations

n=9r=3

9

3

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)

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=11+1=21+2=32+3=53+5=85+8=138+13=2113+21=34

How does it work?

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 1.6803something1.6803…=(1+√5)/2

Fibonacci

Sequence

GraphGolden Ratio

Golden Ratio in Nature

Review

Pascal’s Triangle has many patternsUse 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 equalsthe 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/

OKha9SuU4yg/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. "Nature." The Fibonacci Series. Think Quest. 21 Mar. 2009 <http://library.thinkquest.org/27890/applications5.html>. "Blaise Pascal." Wikipedia. 2009. MediaWiki. 1 June 2009 <http://en.wikipedia.org/wiki/ Blaise_Pascal>. 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 <http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html>. 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 <http://www.gi.alaska.edu/ScienceForum/ASF7/716.html>. 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 <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>.