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The Heart of Mathematics
An invitation to effective thinking
Edward B. Burger and Michael Starbird
Chapter 1
Fun and Games
An introduction to rigorous thought
1. Make an earnest attempt to solve each puzzle.
2. Be creative.
3. Don’t give up: If you get stuck, look at the story
in a different way.
4. If you become frustrated, stop working, move
on, and then return to the story later.
5. Share these stories with your family and
friends.
6. HAVE FUN!
Lessons for Life
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Just do it.
Make mistakes and fail, but never give up.
Keep an open mind.
Explore the consequences of new ideas.
Seek the essential.
Understand the issue.
Understand simple things deeply.
Break a difficult problem into easier ones.
Examine issues from several points of view.
Look for patterns and similarities.
Story 1.
That’s a Meanie Genie
Story 2. Damsel in Distress
Story 3.
The Fountain of Knowledge
Story 4. Dropping Trou
Story 5. Dodgeball
Story 6. A Tight Weave
Story 7. Let’s Make A Deal
Story 8.
Rolling Around in Vegas
Story 9. Watsamattawith U?
Chapter 2
Number Contemplation
Arithmetic has a very great and
elevating effect, compelling the soul
to reason about abstract number…
PLATO
Section 2.1: Counting
How the Pigeonhole Principle Leads to
Precision Through Estimation
Understand simple thing deeply.
Question of the day
How many Ping-Pong balls are needed to fill
up the classroom?
The Hairy Body Question
Are there two non-bald people on the Earth
who have the exact same number of hairs
on their bodies?
Johnny Carson
Johnny Carson was the most watched
person in human history. Estimate the total
number of viewers who watched Carson
over his 30 year reign on the Tonight Show.
Pigeonhole Principle
Why are there two trees with leaves on the
earth with the exact same number of
leaves?
Why does every person have many
temporal twins on earth, that is, people who
were born on the same day and will die on
the same day?
Pigeonhole Principle
State the Pigeonhole Principle in your own words.
Section 2.2: Numerical Patterns in Nature
Discovering the Beauty and Nature
of Fibonacci Numbers
There can be great value in looking
at simple things deeply, finding a pattern,
and using the pattern to gain new insights.
Question of the day
What is the next number in the sequence?
1, 1, 2, 3, 5, 8, 13, 21, ___
Pineapples
List as many observations about the
pineapple as you can.
The Daisy
Count the spirals in a daisy.
Comparing Numbers
The pineapple has two sets of spirals: 8, 13
The daisy has two sets of spirals: 21, 34
Compare these numbers: 8, 13, 21, 34
Do you notice a pattern?
Noticing a pattern
Find the next two numbers in the sequence:
8, 13, 21, 34, ___, ___
More of the pattern…
What numbers must have come before 8,
and how many numbers before 8 exist?
__?__, 8, 13, 21, 34, 55, 89, …
Fibonacci Numbers
The following sequence of numbers are
called the Fibonacci Numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Comparing Fibonacci Numbers
Compare the size of adjacent Fibonacci
Numbers. What do you notice?
Compare 1 to 1
Compare 1 to 2
Compare 2 to 3
Compare 3 to 5
Compare 5 to 8… and so on.
Fibonacci Quotients
Find each quotient. What do you notice?
Fraction of
Decimal
adjacent Fibonacci Equivalent
Numbers
1/1
2/1
3/2
5/3
8/5
13/8
21/13
34/21
55/34
89/55
What number do we get?
As the Fibonacci Numbers in the previous
quotients get larger and larger, what number
are we approaching?
Express each non-Fibonacci Number as a sum of
non-adjacent Fibonacci Numbers
1 = Fibonacci Number
2 = Fibonacci Number
3 = Fibonacci Number
4=1+3
5 = Fibonacci Number
6=1+5
7=2+5
9=1+8
Express each non-Fibonacci Number as a sum of
non-adjacent Fibonacci Numbers
Natural Numbers
Sum of Fibonacci
Numbers
10
11
12
13
14
15
16
17
18
19
Fibonacci Number
Unending 1’s
The Golden Ratio
The Golden Ratio
The Golden Ratio
Solve this equation for phi!
Fibonacci Nim
Rules:
1)
2)
3)
4)
Start with a pile of sticks.
Person one removes any number of sticks (at least
one but not all) away from the pile.
Person two removes as many as they wish with the
restriction that they must take at least one stick but no
more than two times the number of sticks the previous
person took.
The player who takes the last stick wins.
Section 2.3: Prime Cuts of Numbers
How the Prime Numbers are the
Building Blocks of All Natural Numbers
Are there infinitely many primes,
why or why not?
Question of the day
Can you write 71 as a product
of two smaller numbers?
Write the following numbers as products of smaller
numbers other than one.
12
21
36
108
Prime Numbers
A natural number greater than 1 is a prime
number if it cannot be expressed as a
product of two smaller natural numbers.
The Prime Factorization of Natural Numbers
Every natural number greater than 1 is either a
prime number or it can be expressed as a product
of prime numbers.
The Infinitude of Primes
There are infinitely many prime numbers.
Fermat’s Last Theorem
It is impossible to write a cube as a sum of
two cubes, a fourth power as a sum of two
fourth powers, and, in general, any power
beyond the second as a sum of two similar
powers.
If n  2, x  y  z .
n
n
n
The Twin Prime Question
Are there infinitely many pairs of prime
numbers that differ from one another by
two?
Examples:
11 and 13, 29 and 31, 41 and 43 are twin primes.
The Goldbach Question
Can every positive, even number greater
than 2 be written as the sum of two primes?
Examples:
4=2+2
6=3+3
8=3+5
10 = 5 + 5
12 = 5 + 7
14 = ?
16 = ?
Section 2.4: Crazy Clocks and Checking Out Bars
Cyclical Clock Arithmetic and Bar codes
Identifying similarities among different objects is
often the key to understanding a deeper idea.
Question of the day
Today is, Monday, March 10.
On what day of the week will the Fouth of
July fall this year?
Mod Clock Arithmetic
Devise a method for figuring out the day of
the week for any day next year.
How many years pass before the days of the
week are back to the same cycle?
More Mod Clock Arithmetic…
Formulate a numerical statement about
when x = y mod 12.
Check Digits
Devise a check digit scheme where there
are two check digits, perhaps combining
two fot he schemes.
How accurate would this system be?
Section 2.5
Secret Codes and How to Become a Spy
Encrypting Information Using Modular Arithmetic and Primes
Attractive ideas in one realm often have
unexpected uses elsewhere.
Question of the Day
Which is easier, multiplying or factoring?
ATM’s
Have you ever taken cash out of an ATM or
used a credit card to buy something
online?
Do you feel confident that the bank records
are accurate and safe? Why or why not?
Break the code!
What does this say?
ZKK RXRSDLR ZQD ETMBSHNMHMF
MNQLZKKX, CZUD
The code is broken!
ZKK RXRSDLR ZQD ETMBSHNMHMF
MNQLZKKX, CZUD.
All systems are functioning normally, Dave.
Is it possible?
Is it possible to create a code with which
anyone can send encrypted messages to
the owner, but no one other than the
owner can decode the messages?
Product of Primes
The number 6 is the product of two prime
numbers. What are the two numbers?
Product of Primes
Now try it with the following numbers:
77
187
851
19,549
802,027,811
Section 2.6
The Irrational Side of Numbers
Are There Numbers Beyond Fractions?
Explore the consequences of assumptions.
Question of the Day
How can we prove that all numbers are
rational (all numbers are fractions)?
Rational Numbers
What are rational numbers?
Are all numbers rational?
Claim: 2 is a rational number.
Prove:
3 is irrational.
Think about it…
You are thinking of a number B and it has
the property that 3^B = 10. Could B be
rational?
Section 2.7
Get Real
The Point of Decimals and
Pinpointing Numbers on the Real Line
Look for new ways of expressing an idea.
Rational and Irrational Numbers
What are Rational Numbers? Give five
examples.
What are Irrational Numbers? Give five
exampes.
Real Numbers
What are Real Numbers?
Rationals everywhere…
Why does every interval on the line contain
infinitely many rational numbers?
Irrationals everywhere…
Why does every interval on the line contain
infinitely many irrational numbers?
Irrational Numbers
and the Real Number Line
Draw a real number line and locate an
irrational number such as the square root
of 2.
Decimal Expansions
How do the decimal expansions of the
rational numbers differ from those of
irrational numbers?
Find the decimal expansion of the following
numbers: 11/4, 1/3, 22/7
Reversing a Decimal Expansion
Transform the following repeating decimals
to fraction form:
7.63636363…
12.34567567567…
Decimals
Draw a real line labeling the integers.
Suppose a decimal number has been
smudged, so all you can read is the tenths
digit, which is 3:
XXX.3XXXXXX…
Shade in all possible locations for this
number on the real line.
Decimals
Draw a real line labeling the integers.
Suppose a decimal number has been
smudged, so all you can read is the
hundredths digit, which is 7:
XXX.X7XXXXX…
Shade in all possible locations for this
number on the real line.
Decimal Representation of
Rational Numbers
Neatly write out the long division 7 into 45
doing at least 14 places after the decimal
point.
Why is it quick to see what the decimal
answer is forever?
Explain why any rational number must have
a repeating decimal representation.
Shuffling Rationals
Suppose you take two rationals represented
as decimals, say 0.1234 and 0.5678, and
you shuffle their digits to get 0.15263748.
Is the shuffled number rational?
Is this true for all such numbers represented
as decimals?
Unshuffling Rationals
Take a rational number in its decimal form.
Why is the decimal number constructed by
just using the digits in the odd positions
still rational?
Bag of 0’s and 1’s
Suppose you have a bad of infinitely many
0’s and 1’s. How can you use them to
wreck the rationality of any decimal
number? That is, how can you insert 0’s
and 1’s, never putting in consecutive
inserts, into the decimal expansion of a
number to make certain that the result is
not rational?
Bag of 0’s and 1’s
Example:
Given the decimal 0.XXXXXXX…, add 0’s
and 1’ to create a number like
0.1X0X1X0X0X1X0X0X1X1X…
so that you can be certain the number is not
rational.
Rational or Irrational?
Which numbers are rational and why?
1.25
0.333…
17.3965
4.121212…
Think about it…
If a number is irrational, what must its
decimal expansion look like?
Create other examples of irrational numbers
in decimal forms.