1. The proofs of the Early Greeks 2800 B.C. – 450 B.C.

2. Pythagorean Theorem  Write an equation illustrating the relationship between the area of the rectangles, the area of the white smaller square, and the area of the large square  Write an equation illustrating the relationship between the smaller squares, the rectangles and the larger square.  Use transitivity and the subtraction property of equality.

3. Thales – 624 B.C. – 546 B.C.  Started as a Merchant  Well-traveled and curious  Spent time in Egypt  Why instead of How?  First known Greek Philosopher  Credited with first 5 theorems of Geometry

4. Mathematics of Thlaes  Vertical angle Theorem  Proof  Angle a plus angle c make a straight line  Angle c plus angle b make a straight line  All straight lines are equal  Angle a equals angle b  Applications

5. Mathematics of Thlaes  A circle is bisected by its diameter  The bases of an isosceles triangle are equal  Two triangles are congruent if the have two angles and one sides which are respectively equal

6. Mathematics of Thlaes  An angle inscribed in a semicircle is a right angle.  Thales is credited with the proof.  Babylonia recognized this 1400 years earlier

7. A Greek Tragedy

8. The Sources of Greek Mathematics  Euclid’s Elements in 300 B.C.  Trumped all math before it  No Primary Resources  Eudemian Summary by Proclus around 500 A.D  Had access to works lost to us  Byzantine Greek Codics : 500 – 1500 yrs after Greek works Proclus Euclid

9. Pythagorus: 572 B.C. – 500 B.C  Possible student of Thales  Why do they think this?  Founded Pythagorean School  Mathematics  Natural Sciences  Philosophy  Strange cultish rules  Abstain from Beans  Do not touch a white rooster  Do not pick up what has fallen  Not to stir the fire with iron  Do not look in a mirror beside a light  Do not eat meat

10. Arithmetica vs Logistic  Arithmetica – Study of relationships between numbers  Logistic – Computation  Pythagoras started modern Number Theory Rafael: School of Athens

11. Divisors Definition: Examples:

12. Sum of Divisors Definition: Example: Compute s(284).

13. Perfect, Deficient and Abundant

14. Perfect, Deficient, or Abundant Exercise 1: Is 28 perfect, deficient or abundant? Exercise 2: Is 24 perfect, deficient or abundant? Exercise 3: Is 44 perfect, deficient, or abundant?

15. Euclid and Perfect Numbers Art of Problems Solving II