- Quiz today, over Egyptian division).
- 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).
- Students divided into
- In particular, we're investigating the Pythagorean Theorem
- Is often expressed this way:
And many civilizations knew of special integer examples, e.g. (3,4,5), which make the theorem true. These special sets of natural numbers are called Pythagorean triples. I've given you a worksheet that features some Pythagorean triples to solve. Let's look at a few.
- Now we'll go through the proof of the theorem, using a standard sheet of graph paper.
- Is often expressed this way:
- That
is irrational. (Proof by contradiction -- famous and easy!) Careful: this one can get you killed....