Day 04: Heart of Mathematics

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Announcements:
- Your first homework assignment (from fun and games) is due. Please hand that in.
- Your next homework assignment is due next Monday, 2/2 (estimation)
  Reminder: your answers should be typed (although figures and diagrams may be hand-drawn).
Today: more Estimation (Section 2.1: Counting)
- Last time we discussed how the authors went about getting an estimate of hairs on a human body (among other things).
  Their objective was not exactly an estimate of hairs, but rather to show that there are two people on Earth with exactly the same number of hairs. Their demonstration rests on something called "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?
    - Something useful, and something to notice: World Clock for Population, Illness, etc.
- What are some of the fundamental mathematical elements of "the estimate"?
  - Collection of (quality) Data
  - Approximation of the body by geometrical objects with known characteristics.
  - Units and unit conversion
    - We need useful referents.
    - Is one of your fingers three inches long?
    - Is a knuckle an inch long?
    - How many centimeters in an inch? (2.54cm/in)
    - How many kilometers in a mile? (factors of 10 and English unit conversion)
  - Computation (putting it all together)
  - Bounding (that is, capturing the true value between two values)
  - Sometimes it pays to be rough! (that is, we don't need a fine estimate)
  - Can we find another estimate?
  - Can we do a better job of estimating?
- Two things are important: an estimate and a bound (some idea of how bad your answer can be). Here are some examples of bad estimates without any bounds:
  - 42 (my favorite estimate, no matter what the question!)
  - Let's look at some very topical examples of really bad estimates without any error bounds: these relate to the housing bubble.
- Problems to consider (exercises in estimation):
  1. Every time our government spends a billion dollars, how much do you owe?
  2. How much water do you require to take a shower? (we'll spend five minutes working on this one, and then discuss)
  3. How many ordinary dice would it take to fill this classroom? (10 minutes on this one)
  4. Do you have a temporal twin? That is, is there someone who was born on the same date as you (i.e. within 24 hours of you), and will die on the same date (within 24 hours of you)?
    - Is there anyone who obviously does?
    - Is there anyone who obviously doesn't?

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