Erdös: The Man Who Loved Only Numbers
Exercise from Class #1
(Reminder: All class-exercise materials are permanently provided in Little Problems in Mathematics.)
Example (blue) and proof (red) that the sum of the first n odd numbers always equals the square of n.
Assignments for Class #2
• In our text, read pages 25 through 57 (Chapter 1, "Straight from the Book")
• Watch these videos:
1) A proof that's "from the book"
2) Sieve of Eratosthenes -- finding primes
3) Goldbach Conjecture -- every even number greater than 2 is the sum of two primes
Questions to Think About
• What is a mathematical proof? (Also, what is a theorem; a conjecture?)
• What did Erdös mean by saying that a proof is "straight from the book"?
• How does mathematical proof of a statement differ from trying to test the truth of a statement with a large number of examples?
• Apply the answers to the two previous questions to the question of whether there are an infinite number of primes.
• What is a perfect number? Why are perfect numbers hard to find?
• What is a Mersenne number? A Mersenne prime?
• How would you describe the map-coloring problem to a friend who is not a scientist or mathematician?
• The second law of thermodynamics means that hot things always cool unless you do something to stop them. It expresses a fundamental and simple truth about the universe: that disorder, characterised as a quantity known as entropy, always increases. Do you think this law can be proven analytically, in the manner of the proof above that there are an infinite number of prime numbers? Why or why not?
• In science, is there any kind of knowledge that is equivalent to a proof in mathematics?
Other Resources
• Positive Integers: http://www.positiveintegers.org
This is a handy list of some properties of the positive integers, including whether they are prime or composite, whether they are perfect, abundant, or deficient, and a list of all their factors.
• Preview of Class #3: A Visual Proof of the Pythagorean Theorem
• What kinds of memories do you have about math in school?
You might enjoy this poem:
Trouble with Math in a One-Room Country School, Jane Kenyon
