The Heart of Mathematics

An invitation to effective thinkingEdward B. Burger and Michael Starbird

Chapter 1Fun 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

• 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 2Number Contemplation

Arithmetic has a very great and

elevating effect, compelling the soul

to reason about abstract number…

PLATO

Section 2.1: CountingHow 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 NatureDiscovering 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 DaisyCount 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 FibonacciNumbers. What do you notice?

Compare 1 to 1Compare 1 to 2Compare 2 to 3Compare 3 to 5Compare 5 to 8… and so on.

Fibonacci QuotientsFind each quotient. What do you notice?

Fraction of adjacent Fibonacci Numbers

Decimal Equivalent

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 Number2 = Fibonacci Number3 = Fibonacci Number4 = 1 + 35 = Fibonacci Number6 = 1 + 57 = 2 + 59 = 1 + 8

Express each non-Fibonacci Number as a sum of non-adjacent Fibonacci NumbersNatural Numbers Sum of Fibonacci

Numbers

10

11

12

13 Fibonacci Number

14

15

16

17

18

19

Unending 1’s

The Golden Ratio

The Golden Ratio

The Golden Ratio

Solve this equation for phi!

Fibonacci Nim

Rules:1) Start with a pile of sticks.

2) Person one removes any number of sticks (at least one but not all) away from the pile.

3) 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.

4) The player who takes the last stick wins.

Section 2.3: Prime Cuts of NumbersHow 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 2, .n n nn x y z

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 BarsCyclical 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 2.5Secret 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 2.6The 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 2.7Get 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 ofRational 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.