Pythagoras: 2500 Years Old and Still Relevant
The history of mathematics includes many prominent figures who developed and described a consistent, accurate understanding of the nature of numbers and how they describe our universe. The Chinese, Egyptians and Babylonians were the first true mathematicians in recorded history and many of their contributions were spurred by the necessity to use mathematics to support trade and business. The Greeks played an immensely important role in the development of mathematics, many taking the different approach than their predecessors that mathematics and science were inextricably linked to philosophy. Pythagoras of Samos is one of several prominent Greek philosophers whose work is still used essentially without change. Pythagoras is often described as history’s first pure mathematician.
Born about.560BC, Pythagoras was a Greek philosopher and religious leader who was responsible for important developments in the mathematics, astronomy, and the theory of music. He migrated to Croton in southernItaly from Greece and founded a philosophical and religious school there that attracted many followers. Pythagoras is universally accepted by historians as an extremely important figure in the development of mathematics yet relatively little is known about his mathematical achievements. Unlike many later Greek mathematicians whose books and writings were preserved, none of Pythagoras's writings have been found. The Pythagorean Society, which he formed and led was half religious - half scientific and followed a code of strict secrecy. His inner circle of followers was known as the mathematikoi who lived permanently within the Society, held no personal possessions and were vegetarians. They were personally taught by Pythagoras, obeyed strict rules of conduct and believed(3)
(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all members of the order should observe strict loyalty and secrecy.
Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The members of the outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.
Every evening each member of the Society had to reflect on three questions:
- What good have I done today?
- What have I failed to do today?
- What have I not done today that I should have done?
Pythagoras and the mathematikoistudied mathematics in a pure philosophical sense, not as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems. (4) Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. We accept and recognize a number such as 2 as an abstract quantity. However, in the time of Pythagoras, it was a remarkable step from the concrete observation that 2 ships + 2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc.
Pythagoreans believed that all relations could be reduced to number relations ("all things are numbers"). They developed this theory from certain observations in music, mathematics, and astronomy. For example, the Pythagoreans noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. In astronomy, the Pythagoreans were well aware of the periodic numerical relations of heavenly bodies.
Much of the Pythagoreans life style was embodied in their beliefs about numbers. They considered the number 1 the essence of reason, the number 2 was identified with opinion, and four was associated with justice since it is the first number that is the product of equals. Of numbers greater than 1, odd numbers were masculine and even numbers were feminine so 5 represented marriage, the union of the first masculine and feminine numbers (2 + 3).
The Pythagoreans knew, as did the Egyptians before them, that any triangle whose sides were in the ratio 3:4:5 was a right-angled triangle. The Pythagorean theorem, that the square of the hypotenuse (longest side opposite the right angle) of a right triangle is equal to the sum of the squares of the other two sides, may have been known in Babylonia, where Pythagoras traveled in his youth. The Pythagoreans, however, are usually credited with the first proof of this theorem.
The Pythagorean Theorem, as described in your Dugopolski text on page 364-5, is a 2,500 year old still valuable tool for solving a number of real-life applications problems. Following are twoexamples; the first is solved as a radical problem as you have worked with in Chapter 5 and the second as a quadratic equation that you will be working with in week 3:
1. Carpenters stabilize wall frames with a diagonal brace as shown in the figure. If the bottom of the brace is attached 9 feet from the corner and the brace is 12 feet long, how far up the corner post should it be nailed?
12 ft
HH =
9 ft
2. Two cars left an intersection at the same time, one heading due north, and the other due west. Some time later, they were exactly 100 miles apart. The car headed north had gone 20 miles farther than the car headed west. How far had each car traveled?
Let x = distance traveled by the car headed west so x+20 = distance traveled by the care headed north. The cars are 100 miles apart so the hypotenuse of the right triangle equals 100.
c2 = a2 + b2(Pythagorean Theorem)
1002 = x2 + (x + 20)2 = x2 + x2 + 40x + 400
0 = 2x2 + 40x -9600
0 = x2 + 20x – 4800 = (x + 80) (x – 60)
x = 60, -80 (discard)
Car headed north traveled = 60 miles; car headed west traveled = 80 miles
References:
- Calter, Dr. Paul and Dr. Michael A. Calter, Technical Mathematics, New York: John Wiley and Sons, 2000.
- Lial, Margaret, John Hornsby, and Charles d. Miller, Intermediate Algebra with Early Functions and Graphing, 6th Edition, ReadingMA: Addison Wesley Inc, 1998.
- “Pythagoras of Samos” retrieved March 2, 2005 from
- “Pythagoras of Samos , retrieved March 2, 2005from
5. Smith, Karl J., The Nature of Mathematics, Eighth Edition, Brooks-Cole Publishing, 1998.