July 26, 2017
“Dipa, do you know the Pythagorean Theorem was known long before Pythagoras?” my colleague Tim asked me one day during a conversation on our department’s hallway.
This intrigued me to investigate the claim…
When I think about this theorem, I vividly remember my high school math teacher, Mr. Abdur Rahman. We addressed him as Rahman Sir. Rahman Sir was a middle-aged, good looking gentleman with black and white hair. He always wore an ironed white panjabi and pyjama. With his French cut beard and white clothing, he looked like a saint to us. We admired and respected him dearly. Even with a class of fifty plus teenage girls, who giggled all the time for no reason at all, he had complete control of the class.
I remember Rahman Sir with his deep voice explaining the Pythagorean Theorem to us…
In a right triangle, if we draw a square on the hypotenuse and draw squares on two other legs, then the area of the square on the hypotenuse will be equal to the sum of the areas of the squares on the other two legs. In other words, if c is the length of the hypotenuse and a, b are the lengths of the other two sides of a right triangle, then a^2 + b^2 = c^2.
He also told us that any three numbers (a, b, c) satisfying the Pythagorean Theorem is called a Pythagorean triple.
I think many of you—like me—may become nostalgic thinking about the Pythagorean Theorem. It takes us to care-free high school days and brings back memories of the teachers who guided us, cared about us and, most importantly, implanted the seeds of learning in us.
Pythagoras of Samos was born in Samos of Greece around 572 BCE.
In those days in Greece, people’s names reflected the city where they were born. This custom is still going on in some parts of India. Two philosophers, Thales and his student Anaximander of Miletos, introduced mathematical ideas to Pythagoras. Thales advised Pythagoras to travel to Egypt to learn more mathematics and astronomy. Pythagoras took his advice and traveled to Egypt. When he was in Egypt, Egypt was invaded by the Persian King. Pythagoras was taken as a prisoner to Babylon. Thus Pythagoras’ philosophy and mathematics were greatly influenced by the Egyptians and Babylonians.
After gaining his freedom, Pythagoras returned to Samos and set up a school, which he called the “Pythagorean Semicircle.” It was a philosophical and religious school. One of the mathematical doctrines of this group was that numbers are the substance of all things. In other words, numbers—positive integers—formed the basic organizing principle of the universe. What the Pythagoreans meant by this was not only that all known objects have a number, or can be ordered or counted, but also that numbers are the basis of all physical phenomena.
For example, a constellation in the heavens could be characterized both by the number of stars that compose it and by its geometrical form, which itself could be represented by a number. The motions of the planets could be expressed in terms of ratios of numbers. Musical harmonies depend on numerical ratios: Two strings with ratio of length of 2:1 give an octave when plucked; two with a ratio of 3:2 give a fifth; and two with a ratio of 4:3 give a fourth. Out of these intervals an entire musical scale can be created. Finally, the triangles whose sides are in the ratio of 3:4:5 are right-angled, establishing a connection between number and angle.
To the Pythagoreans, numbers were either positive integers or ratios of positive integers.
They did not believe in numbers which were not ratio of integers. Hippasus of Metapontum was a Pythagorean. He noticed that if two sides of a right triangle have unit length, then according to the Pythagorean Theorem, the length of the hypotenuse would be √2. No one was able to express √2 as the ratio of two positive integers.
Hippasus was obviously clever, but not that clever… While on a boat with Pythagoras and many other Pythagoreans, he announced that he’d found a way to demolish Pythagoras’ religion that all numbers are ratios of positive integers. Legend has it that Pythagoras tipped him over the side, drowned him, and swore the rest of the group to secrecy.
The philosopher Anaximander of Miletos, a mentor of Pythagoras, did not approve of his philosophical and mathematical theories and accused him of being a troublemaker and infidel. Pythagoras rejected these accusations, but was forced by his enemies to take refuge in a cave of Mount Kerkis, and then to leave Samos. He settled in Crotona, a Greek town in Southern Italy, and continued his religious, philosophical, and mathematical work. There is no historical document on his death. He is a legend to the mathematical community, especially to those who are number theorists.
Greeks were the first to learn not to accept what had been handed down from ancient times. Instead, they began to ask “Why?”
Greek thinkers figured out that the world around them was not mystic, that they could discover its characteristics by rational inquiry. Hence they were anxious to discover and expound theories in such fields as physics, biology, medicine, and politics. The Greeks believed, however, that mathematics was central to rational thought. Although Western civilization owes a great debt to the Greeks in literature, art, and architecture, it is the Greek idea of mathematical proof that is at the basis of modern mathematics—and by extension, at the foundation of our modern technological civilization.
Pythagorean triples (as they are now known) were known to Babylonians more than two thousand years before Pythagoras. Babylonians used the base sixty number system. A number 27,325 (= 27*10^3 + 3*10^2 + 2*10^1 + 5*10^0) in base ten number system, which we now use in our day-to-day life, is 7,35,25 (= 7*60^2 + 35*60^1 + 25*60^0) in base sixty number system.
No one definitively knows the reasons for using base sixty. The speculations are i) sixty is divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 and so trades were easy, ii) they considered 360 days in a year which is divisible by 60, iii) angle measurements is easier in base sixty, hence they were the pioneers of astronomy, and iv) each of human fingers has three parts except the thumbs; if we count three on each of the fingers and toes, we can count sixty at one time.
By using the base sixty number system, Babylonians were able to construct eighteen Pythagorean triples. Mathematical scholars figured this fact out by deciphering the writings on the Babylonian tablet Plimpton 322, in the Plimpton Collection at Columbia University.
In the base ten number system using a Babylonian algorithm, Pythagorean triples can be formed by using any number a…
Consider any number a and set v + u = a. Define v – u = 1/a. By solving v + u = a and v – u = 1/a, we get v = (a^2 + 1)/2a and u = (a^2 – 1)/2a.
Then (a^2 – 1, 2a, a^2 + 1) is a Pythagorean triple.
Pythagoras had exposure to Babylonian mathematics. He knew about the triples. As a Greek mathematician, he was not satisfied with the numerical answers he needed to prove that the answer was correct. He stated and proved the Theorem which is now known to us as the Pythagorean Theorem.
Dipa Sarkar-Dey, Ph.D., is an associate professor of mathematics and statistics at Loyola University Maryland, where she has taught for 30 years. Also by Dipa Sarkar-Dey.