Day 12: Heart of Mathematics

I'll expect a full, one-page description of what you plan to do by then.

• investigate the history of the various kinds of numbers: zero, natural, rational, irrational, imaginary(!), quaternions(!!), etc.
• Reminder: this is as good time as any to go through the book, looking for subjects of interest. As mentioned last time, "missed sections" are prime material!
• An on-line game, another form of Nim. There are four rows, of 6/5/4/3 pearls. One chooses any number of pearls from a single row, and the last one to take a pearl is the loser. You might study the strategy of Fibonacci Nim, and games like Nim.

A final word on Primes:

Generalize "the gap" problem from last time (II.10)

• First of all, what is a rational number? And what are these so called lowest terms?

• Find a rational number between 1001/1003 and 1002/1003

• What makes these numbers "irrational"? (The story of the Pythagoreans - a local copy)

• The proof that the square root of 2 is irrational (here's an alternative proof).

• Problems
  • I.1 - what are the rules?
  • I.5 - "We mustn't waste a good proof!" - generalize?
  • II.5 - Alter slightly a good proof....