Day 16, MAT115

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Announcements:
- Your first test:
  - Results:
    - Median: 75.8
  - The Exam
  - Key
  - Curve
  - Thought #1: didn't do your reading?
  - Thought #2: Attendance matters....
- Quiz Friday, over Egyptian division).
- You had a reading assignment for today: Chapters 4 and 12. Did you read?
Today's Question of the Day:

Who were the Pythagoreans?
We're leaving the Egyptians, and moving on to the Greeks. Do you have any questions about Egyptian division before we go?
Now we're not really leaving the Egyptians, because the things that the Pythagoreans did were known, in many cases, by them.
- The Greeks thought of numbers very geometrically, as indicated in Chapter 4. 7*3 (=3*7) was best thought of as the area of a rectangle with side lengths 3 and 7. A "square" (e.g. 9=3*3) was literally thought of as a square with side lengths 3.
  Before we go on to consider the famous Pythagorean Theorem, however, let's reflect on the idea of "commutativity" that is discussed in Chapter 4.
  Indeed, 7*3=3*7.
  This was a bit mysterious (perhaps even miraculous) to our author as a child, since it says that 3 sevens is the same as 7 threes. Who would have thought it?
  He does a nice example to illustrate that many operations of real life are commutative (although many others aren't). For example, if clothing is discounted by 50%, then by 40%, is that the same as clothing discounted by 40% followed by 50%? And is that the same as clothing discounted by 90%?). Let's do some calculations....
  Here's the kind of question that begs the question whether commutitivity holds:
  
  Coffee+milk+wait-5-minutes ${\ne}$ Coffee+wait-5-minutes+milk
  Which is warmer? Or are they the really equal?
- Pythagoras's School of Crotona
  - Students divided into
    - acoustici -- "listeners" -- novices (three years)
    - mathematici -- inner circle
  - Women were allowed (including Theano -- wife, child, student?)
  - Political, philosophical, religious
  - Secrecy, honor, virtue valued highly
  - Strict taboos, dietary restrictions, etc.
  - Pythagoreans persisted for centuries even after Pythagoras's death.
  - They "worshipped" whole numbers, and believed that all things in the universe could be represented using them (if one also included fractional expressions, ratios of whole numbers).
- The Pythagorean Theorem
  - Is often expressed this way:
    
    ${a^2+b^2=c^2}$
    And many civilizations knew of special integer examples, e.g. (3,4,5) which make the theorem true. These special sets of numbers are called Pythagorean triples.
  - Plimpton 322 (1900-1600 BCE): a tablet containing "Pythagorean triples" (many years before Pythagoras, 6th century BCE)
  - From the Cairo papyrus (circa 300 BCE):
    - Egyptians were aware of several Pythagorean triples.
    - Examples of the ol' ladder problems A ladder of 10 cubits has its foot 6 cubits from a wall; to what height will it reach?
      (Babylonian version: "A beam of length ....")
- Now we'll go through the proof of the theorem, using a standard sheet of graph paper.
- Irrational numbers: The "Crisis of Incommensurable Quantities"
  - That ${\sqrt{2}}$ is irrational. (Proof by contradiction -- famous and easy!) Careful: this one can get you killed....
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