Pythagoras and the Square Root of 2

Pythagoras

Pythagoras of Samos (569-500 BCE) was an actual person, but was also the founder of the Pythagoreans. He was a political figure and a mystic. Beyond this, he stood out in his time as he involved women as equals in his activities. The Pythagorean society focused on mathematics, but also had some religious properties which include a rigorous list of tenets. Among these were,

  • To abstain from beans
  • Not to pick up what has fallen
  • Not to touch the white cock
  • Not to stir the fire with iron

and many more. Not only this but the Pythagoreans had great devotion to Number:

“Bless us, divine Number, thou who generatest gods and men.”

and

Number rules the universe.

The Pythagoreans also believed that each number held great meaning all numbers held a gender and meaning. Odd numbers were male, and even numbers were female. For example,

 

  • The number one : the number of reason.
  • The number two: the first even or female number, the number of opinion.
  • The number three: the first true male number, the number of harmony.
  • The number four: the number of justice or retribution.
  • The number five: marriage.
  • The number six: creation
  • The number ten: the tetractys, the number of the universe

For more on the beliefs of the pythagoreans click this link. One may recall the name Pythagorean from a geometry course. In fact, it is one of the most well remembered rules learned in math. This is of course,

Pythagorean Theorem: In a right triangle, the sum of the square of the legs is equal to the square of the hypotenuse. That is, a^2 + b^2 = c^2.

Right_Triangle_With_Labeled_Sides

Evidence has been found that the Babylonians and Chinese both used this theorem 1000 years before Pythagoras’ time, however Pythagoras is thought to be the first to offer a proof of it. As previously stated, this is easily one of the most recognized theorems in the world. It has over 50 proofs. One of these by former U.S. President James Garfield.

James Garfield

Pythagoras noticed, however, that if he takes a triangle with two legs of lengths 1 he comes out with c^2 = 2. He then wondered if there was a rational number to satisfy this identity. As the title suggests, his conclusion was

Theorem: There is no rational number whose square is 2.

Proof: We shall offer a proof of this theorem using modern mathematics. By way of contradiction, suppose there is indeed a rational number c = \frac{p}{q} (where p and q have no common factors) so that c^2 = 2, but this means,

\frac{p^2}{q^2} = 2 which implies p^2 = 2q^2.

By definition the right side is even (2k is even when k is an integer) Hence the left side must also be even. Then we find that p = 2m for some integer m. Then,

(2m)^2 = 2q^2 which implies 2m^2 = q^2. Thus q must also be even. But this is a contradiction to our hypothesis that p and q have no common factors as they are both divisible by 2. Thus we cannot have that c is a rational number.

Q.E.D.

This revelation was very surprising to a society that worshiped Number. Everyone was wedded to the idea that numbers were rational, but here is irrefutable proof that there are numbers that are irrational (it was found much later by Cantor that there are in fact “way more” irrational numbers than rational numbers). For a while, the Pythagoreans treated this as a secret. As the legend goes, Hippasus of Metapontum was murdered for revealing this secret. A nearby community started to become wary of the growing popularity of the Pythagoreans. So much so that war was waged on them. It is thought that Pythagoras perished in the fire of his own school. Numbers in the past were important, and they are just as important today, but is it possible that a mathematical idea can still be so outrageous that it could spark a war?

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