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Transcript
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 • • • • • • • • • • 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.