Transcript Chapter 2: Number Contemplation

Chapter 2: Number
Contemplation
2.1 Counting: The Pigeonhole Principle
Monday, January 19, 2009
Numbers
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Natural or counting numbers
Integers
Rational Numbers
Real Numbers
Estimation
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How many blades of grass on a
football field?
How many cars would be needed to
line them up, bumper to bumper
from New York City to San
Francisco?
How many golf balls could fit in this
room?
Estimation
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How many blades of grass on a football
field? (Kentucky Bluegrass is 1/8” wide)
How many cars would be needed to line
them up, bumper to bumper from New
York City to San Francisco? (Total Est.
Distance: 2906.10 miles)
How many golf balls could fit in this
room? (Diameter = 1.86”)
Example
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Order the following quantities from
smallest to largest:
1.
2.
3.
4.
5.
6.
Number
Number
Number
Number
Number
Number
of
of
of
of
of
of
telephones in the US
US Congressmen
people in the US
grains of sand
states in the US
cars in the US
Quantification
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Consider your everyday activities
Can you view them quantitatively rather
than qualitatively?
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Skipping class
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$39,434 for tuition =$19,717 per semester
4 courses per semester = $4929.25 per course
42 days of class per semester = $117.36 per
day of class
Pigeonhole Principle
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If we have an antique desk with
slots for envelopes (known as
pigeonholes), and we have more
envelopes than slots, the certainly
some slot must contain at least two
envelopes.
Examples: Same SAT scores, same
zip codes, leaves on trees, temporal
twins, etc.
Birthday Problem
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Suppose we had a room filled with
400 people. Will there be at least 2
people who celebrate their birthday
on the same day?
Birthday Problem
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In a group of at least 23 randomly chosen
people, there is more than 50%
probability that some pair of them will
both have been born on the same day.
For 57 or more people, the probability is
more than 99%, and it reaches 100%
when the number of people reaches 367
(there are a maximum of 366 possible
birthdays).
Birthday Problem
A Historical Note
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Ramanujan & Hardy
The Intrigue of Numbers
“Every natural number is interesting”
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http://numeropedia.googlepages.com/
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Some Problems
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In a standard deck of 52 cards, what is
the smallest number of cards you must
draw in order to guarantee that you have
at least one pair?
What is the smallest number of cards you
must draw in order to guarantee that you
have five cards of one suit?
Some Problems
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Someone offers to give you a
million dollars in $1 bills. To
receive the money, you must lie
down. The bills will be placed on
your stomach. If you keep the
money on your stomach for 10
minutes, the money is yours. Do
you accept the offer?
Answer
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Each note no matter what the
denomination weighs one gram.
There are 454 grams to a pound.
One pound would have 454 notes.
Take 1,000,000 dollar notes divided
by 454 and come out with
approximately 2202.6 pound for
one million dollars in one dollar bills
More Dollar Facts
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How high will the stack be?
The Dollar Bill is 6.6294 cm wide by
15.5956 cm long. One bill is 0.010922 cm
thick
Approximately 232 Will stack up in an
inch if new.
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1,000,000 ∕232 = 4310.34in = 359.195ft
Animal Crackers
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Each box of animal crackers
contains exactly 24 crackers. There
are exactly 18 different shapes
made. Are there always two
crackers of the same shape in each
box? Explain why or why not.
Problem of the Day
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What is the next term in the
following sequence?
1
11
21
1211
111221
312211
13112221