Day 13: Heart of Mathematics

Last time: Irrational numbers. Rational numbers are ratios of integers. A rational number is in lowest terms when the numerator and denominators have no common factors. Rational numbers have terminating or repeating decimal representations (i.e., rational representations!); Irrational numbers do not - there is no pattern!
Today:
- Reminder:
  - Project Assignment 1: by Friday, 9/20, pick your partner(s) (you can go alone!) for the poster project; it would also be nice if you have a topic, but that's not required!
- Project ideas: A couple of students proposed some project ideas, which I summarize as
  - Mathematics of Fire-Fighting
  - Math and Music
  - The mathematics of contradiction (e.g. 2=1)
  These are great project ideas, because they are relevant to you!
- More on Irrational Numbers:
  - The proof that the square root of 2 is irrational is based on contradiction: we assume that it is rational, and then discover that it leads to an impossibility.
  - Pi is perhaps the most famous irrational number, but historically people have had trouble with that. For example, in the Bible,
    1 Kings 7:23 He [Solomon] made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim [diameter = 10] and five cubits high. It took a line of thirty cubits to measure around it. [circumference = 30]
    Pi = 3!?! Even so, that's better than the Indiana legislature did, a few thousand years later....
    (From http://www.yfiles.com/pi.html)
  - Irrational Problems:
    - I.5 - "We mustn't waste a good proof!" - generalize?
    - II.5 - Alter slightly a good proof....
    - III.5 - a challenge!
Next time: Real numbers (enough of these fake ones!) We're creeping up on infinity (chapter 3)....

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