Day 04: Heart of Mathematics

Last Time

Next Time

Announcements:
- Assignment for next time: Read 2.2 for next time, and work on problems I.1; II.6, 9, 18, 21; III.28. Either:
  1. bring an artichoke, pinecone, dandelion (good luck!), daisy, pineapple, or something that illustrates Fibonacci numbers (look around!) to class to share; or
  2. hand in your problems, Wednesday 2/6. III.28 should be typed up, and will be graded for complete sentences, spelling, etc., in addition to the mathematical content.
Any further questions/thoughts/observations from last time? How are the problems coming? Remember, they are to be handed in next time!
Our mission today: Section 2.1, Number Contemplation:
- The first day I shared some "strange mathematical musings" from my youth. Do any of you have any "Number Contemplation" stories from your childhood?
- A reading from Hardy's A Mathematician's Apology, concerning Srinivasa Ramanujan (p. 37)
  "Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, 'rather a dull number,' adding that he hoped that wasn't a bad omen. 'No, Hardy,' said Ramanujan, 'it is a very interesting number. It is the smallest number expressible as the sum of two [positive] cubes in two different ways"'.
- Obviously 1729 is an interesting number:
  - 1729=12³+1³
  - 1729=10³+9³
  and it's the smallest that we can write in this way, as the sum of two cubes in two different ways.
  Are all counting numbers
  (We'll call them natural numbers: 1, 2, 3, ....)
  interesting?
- What is "the pidgeonhole principle"?
  - Have you ever played a game, perhaps at a birthday party as a young child, that relies on the pidgeonhole principle?
  - How do we know that there are two people on earth who have exactly the same number of hairs on their bodies?
    - Why do the authors exaggerate?
    - World Population Clock
- Problems to consider (exercises in estimation):
  1. Do you have a temporal twin (someone who was born on the same day as you, and will die on the same day)?
    - Is there anyone who obviously does?
    - Is there anyone who obviously doesn't?
  2. How many ordinary dice would it take to fill this classroom?
  3. The southbound lanes of I-471 are backed up from the Big Mac bridge at the Ohio river, to Nunn drive. How many cars are stuck in traffic on that section?

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