Day 29, MAT115

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Announcements:
- Your revised exams are due, now! Hopefully you've got yours....
- The spiral gallery votes are also due. We'll now have our
Today's Question of the Day:

Why does the box used in proving the Pythagorean theorem keep showing up?
We'll finish off Chapter 7 (The Joy of x) today.
- Strogatz focuses on x: the variable, that turns arithmetic into algebra. He says that algebra comes in two flavors:
  - "...formulas that ... express elegant patterns about numbers for their own sake." I think of the Fibonacci numbers, ${F(n)=F(n-1)+F(n-2)}$
    for example. (although we've seen that Fibonacci numbers have a lot to do with nature, too); and
  - "...formulas that ... express elegant relationships between numbers in the real world...." I think of the relationship between Fahrenheit and Celsius,
    
    ${F=1.8C+32}$
- Next up is the story of two great physicists, Richard Feynman and Hans Bethe. You owe these two a lot -- do you know why?
  Let's take a look at the figure on p. 48 which is the subject of the discussion: how to square numbers near 50.
  Note: "near 50" means that x is thought of as small.
- Tricks like this are useful in real life, as Richard found out. Here's another one: conversion to Fahrenheit from Celsius: as already said,
  
  ${F=1.8C+32}$
  But we can use an identity to make that calculation easier, as my brother discovered on our recent trip to Canada (where they, like everyone else in the world, use Celsius for temperature -- the US is alone in our love of the ridiculous "British units" -- even the Brits gave up on them!):
  
  ${F=(2-0.2)C+32}$
  I hope that you'll agree that ${1.8=2-0.2}$ .
  The reason why this formula is easier is because of the operations required to compute. Think of it this way: given C, double it, subtract 10 percent of that, and add 32:
  
  ${F=2.0C-0.2C+32}$
  So
  - What's C=100 in Fahrenheit?
  - What's C=0 in Fahrenheit?
  - What's C=50 in Fahrenheit?
  - What's C=-40 in Fahrenheit?
- Strogatz gives another example of a useful algebraic formula:
  
  ${(1-x)(1+x)=1-x^2}$
  presenting it in the context of the stock market; and he ends with the "icky relationship" formula:
  Your partner in a relationship, where your age is x, should be at least ${\frac{x}{2}+7}$ : so your partner's age y should satisfy
  
  ${y\ge\frac{x}{2}+7}$
  in age. What do you think? Let's look at some example, and compute the range of relationship ages you can consider to be "non-icky".
  You must not be too young for your partner, either: hence
  
  ${x\ge\frac{y}{2}+7}$
  which we can solve for y, to yield
  
  ${y\le{2(x-7)}}$
  Putting them together, we get the "appropriate partner age relationship:"
  
  ${\frac{x}{2}+7\le{y}\le{2(x-7)}}$
  I'm x=52, so my range is
  
  ${33\le{y}\le90}$
  Fortunately my wife falls somewhere in that range....
  1. What's your non-icky relationship range?
  2. At what age do the two ages x and y equal each other?
  3. What range do you get for a 12-year old? In what sense does that make sense?

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