Download 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
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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
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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
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<http://www.krysstal.com/binomial.html>. Seward, Kim. "College Algebra Tutorial 57:
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<http://www.geocities.com/
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