Tamara Davis MTH 4960 June 1, 2006

1. The PythagoreansandFigurate Numbers Pythagoreans carried number worship to its extreme, basing their philosophy and their way of life upon it The combination of mathematics and theology began with Pythagoras (Next Slide) There were four parts of mathematics, music, arithmetic (meant number theory), astronomy and geometry known as the quadrivium (next slide) Pythagoreans carried number worship to its extreme, basing their philosophy and their way of life upon it The combination of mathematics and theology began with Pythagoras (Next Slide) There were four parts of mathematics, music, arithmetic (meant number theory), astronomy and geometry known as the quadrivium (next slide)

2. Pythagoreans Four parts of Mathematics: Music Arithmetic (Arithmetica) Number Theory according to the Ancient Greeks Astronomy Geometry Known as the quadrivium In the united states, Arithmetic refers to algorithms and computational procedures Whereas in Ancient Greece, it is known as number theory and was studied by philosophers and gentlemen of leisureIn the united states, Arithmetic refers to algorithms and computational procedures Whereas in Ancient Greece, it is known as number theory and was studied by philosophers and gentlemen of leisure

3. Pythagoreans Natural numbers was the key to the universe. Everything was composed of numbers. The explanation of objects existence could only be found in numbers. Their view of the universe rested on the belief that Natural numbers was the key to the various qualities of matter and mankind They were responsible for number theory and introduction and development of number mysticism in Western SocietyTheir view of the universe rested on the belief that Natural numbers was the key to the various qualities of matter and mankind They were responsible for number theory and introduction and development of number mysticism in Western Society

4. Pythagoreans Numerical Attributes: 1 � number of reason, generator of numbers 2 � first true female number (because it is even) 3 � first true male number (because it is odd) 4 � squaring of accounts 5 � first female + first male 6 � first female + first male + 1 10 � the number of the Universe 1+2+3+41 � number of reason, generator of numbers 2 � first true female number (because it is even) 3 � first true male number (because it is odd) 4 � squaring of accounts 5 � first female + first male 6 � first female + first male + 1 10 � the number of the Universe 1+2+3+4

5. Pythagoreans Tetracyts (10) was the holiest number. 1+2+3+4 = 10 Some argue that it was the Pythagoreans and the properties of the tetracyts that was responsible for our use of Base 10 They also developed other concepts of fours, in nature, for example the four elements: earth, air, fire and water1+2+3+4 = 10 Some argue that it was the Pythagoreans and the properties of the tetracyts that was responsible for our use of Base 10 They also developed other concepts of fours, in nature, for example the four elements: earth, air, fire and water

7. Pythagoreans Do these look familiar? Triangular Numbers n(n+1)/2 and Square Numbers n2 These are what the Pythagoreans called figurate numbersTriangular Numbers n(n+1)/2 and Square Numbers n2 These are what the Pythagoreans called figurate numbers

8. Pythagoreans Are a sequence of numbers that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, they are called Polygonal Numbers.

9. Pythagoreans

10. Pythagoreans

11. Pythagoreans 1, 5, 11, 221, 5, 11, 22

12. Pythagoreans

13. Pythagoreans You can also form other shapes, such as three-dimensional solids.

14. Pythagoreans First to� to bring mysticism to numbers numerology. to study and analyze number theory / figurate numbers. understand the universe through mathematics

17. The Pentagram was the mystical symbol for the Pythagorean order. They saw in the Pentagram a mathematical perfection�phi also known as the Golden Ratio (1.61803399 )

18. f = 1.61803399 When drawn with perfect angles each line is divided into several smaller segments, and if you divide the length of the longer segment with the shorter segment of any pair of segments you will get f When drawn with perfect angles each line is divided into several smaller segments, and if you divide the length of the longer segment with the shorter segment of any pair of segments you will get f

19. Pythagoreans �Were it not for number and its nature, nothing that exists would be clear to anybody either in itself or in its relation to other things�You can observe the power of number exercising itself�in all acts and the thoughts of men, in all handicrafts and music.� ~ Pythagorean Philolaus (425 B.C.E.)

20. Work Cited Allen, Don; �Pythagoras and the Pythagoreans�; Feb. 6, 1997; <http://www.math.tamu.edu/~don.allen/history/pythag/pythag.html>. archytech.org; �The History of Pythagoras and his Theorem�; Jun. 9, 2003; <http://www.arcytech.org/java/pythagoras/history.html>. Bunt, Jones, and Bedient; The Historical Roots of Elementary Mathematics; Dover Publications, Inc.; New York; 1988; pp. 71-83. Math Gym; �Pythagoras � Number�; Mathgym.com; May 2006; <http://www.mathgym.com.au/history/pythagoras/pythnum.htm>. Weisstein, Eric W.; �Figurate Numbers�; MathWorld-A Wolfram Web Resource; 1999; <http://mathworld.wolfram.com/FigurateNumber.html>.