Day 18, MAT115

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Announcements:
- Quiz returned: Key
Today's Question of the Day:

What's the hypotenus of an right isosceles triangle with side length 1?
On to the Greeks:
- Reminder: we're talking about the Pythagoras and his 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).
- Pythagoras and his Hypotenuse:
  - The Wizard of Oz: The Scarecrow still might not have a brain.
    Why?: When the Scarecrow receives his diploma he supposedly proves the existence of his brain by spewing out the Pythagorean theorem but what he actually says is
    
    'The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.'
    He's close but incorrect. The Pythagorean Theorem states this:
    
    The sum of the SQUARES of the LEGS of a RIGHT triangle is equal to the SQUARE of the HYPOTENUSE.
    
    From the script:
    - Scarecrow: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh joy, rapture! I've got a brain!"
    - The clip
  - The theorem is often expressed (correctly) this way:
    
    ${a^2+b^2=c^2}$
    Many civilizations knew of special integer examples, called Pythagorean triples which make the theorem true, e.g. (3,4,5) and (5,12,13).
  - Last time we proved the theorem in two ways: geometrically and algebraically, using a sheet of graph paper.
  - For your homework, I've asked you to show that some triangles are not right. If the precarious balance of ${a^2+b^2=c^2}$ is not met, then the triangle is either obtuse (angle greater than 90), or acute (all angles less than 90) -- and your building will not be square!
- Today we look at irrational numbers: The "Crisis of Incommensurable Quantities"
  - The Egyptians were familiar with the rational numbers, which are ratios -- quotients -- of whole numbers (even if they could only express them as sums of unit fractions). All unit fractions are rational numbers, ratios of 1 and some other natural number.
  - Examples of rational numbers:
    - Find a rational number between 1 and 2.
    - Find a rational number between 1001/1003 and 1002/1003.
    - Why is there always a rational number between any two rational numbers?
  - An irrational number is a real number that is not rational. It can be used to measure things, but it can not be represented as the ratio of two integers.
  - Now we'll prove that is irrational. (Proof by contradiction -- famous and easy!) Careful: this one can get you killed....
    - What's the hypotenus of an right isosceles triangle with side length 1? ${\sqrt{2}}$ ....
      So it's a perfectly reasonable quantity.
    - We proceed by contradiction: that is, we assume that it IS rational, then show that that is impossible.
    - Assume ${\sqrt{2}}=\frac{c}{d}$ , where c and d have no common factors. (Use the prime factorization theorem, and cancel any common factors.)
    - Square both sides.
    - Use the prime factorization theorem!

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