Day 05: Heart of Mathematics

Last Time

Next Time

Announcements:
- Your first homework assignment (fun and games) is due today. Please hand those in.
  Now -- do you have questions on any of those?
- Any questions on your assignment for the estimation unit? (Hand this in Friday, 9/12)
  Estimate the following:
  1. How much do you spend in groceries for a year?
  2. How much gasoline do you use in a year? (CNN website) What percentage of your budget is fuel?
  3. How many times will you breathe today?
  4. How tall is the Lucas Administration building?
  5. 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?
What are some of the fundamental mathematical elements of the estimate?
- Collection of (quality) Data
- Approximation using geometrical objects with known characteristics; use of mathematical and statistical models (e.g. a cylinder is a model of a body).
- Units and unit conversion
  - We need useful referents
  - Is one of your fingers three inches long?
  - Is a knuckle an inch long?
  - There are 2.54 centimeters in an inch: how many kilometers are there in a mile? (factors of 10 and English unit conversion)
- Computation (putting it all together)
  - Make sure that you keep units in your calculations;
  - add, subtract, multiply, and divide units, too, so as arrive at an answer that has the correct units.
- Bound where possible (that is, capturing the true value between two values)
- Sometimes it pays to be rough! If you don't need an answer to great accuracy, you can be rough:
  - Where multiplying or adding two numbers, round one up and one down for balance.
  - Where adding many numbers, use rounding (e.g. at the supermarket, just round everything to the dollar -- this should get you in the ballpark of your total).
  - When subtracting or dividing two numbers, you need to either round them both up or both down to stay in the ballpark (as a general rule).
Two things are important in estimation: an estimate and a bound (some idea of how bad your answer can be): e.g.
- A CNN-Opinion Research Corp. poll of 1,041 adults conducted July 27-29 found that 62 percent favor a timetable for withdrawing U.S. troops from Iraq, but 37 percent oppose setting deadlines. The results had a margin of error of plus or minus 3 percentage points.
- We estimated Ed's hair count as about 4-5 million - what can we say about bounds?
Back to the "the pidgeonhole principle"....
- Have you ever played a game, perhaps at a birthday party as a young child, that relies on the pidgeonhole principle?
- So let's review 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 Clock for Population, Illness, etc.
  - Let's generalize: how many people must have the same number of hairs on the bodies, at the very least?
Problems to consider (exercises in estimation):
1. There will be fifty-seven people at your picnic -- how much potato salad do you need?
2. The Republic of Georgia just received $1 billion in US Humanitarian aid in their conflict with Russia. How much are you paying (on average)?
3. In the book "The Three Trillion Dollar War", by Joseph E. Stiglitz (winner of the 2001 Nobel Prize in Economics) and Linda Bilmes, they assess the true cost of the Iraq war at -- you guessed it -- Three Trillion Dollars. About how much is that for each American?
4. How many ordinary dice would it take to fill this classroom?
5. 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?

Website maintained by Andy Long. Comments appreciated.