Saturday, May 25, 2013

History of Math: Irrational Numbers

I normally end each month with a post on a mathematical story from history. Since I've spent the last month discussing the Pythagorean Theorem, I thought I'd finish it off with a story from the life of Pythagoras.

Pythagoras was born in 570 BC in Samos (which is now in Vathy, Greece). Throughout his life, he dabbled in philosophy, mathematics, music, and religion (he actually was the founder of the religion known as "Pythagoreanism"). He founded an organization (sort of like a school) where lots of ideas were developed. It is unclear if many mathematical and philosophical ideas of the time were directly from him or from one of his students.

The Pythagorean Theorem was used by Babylonian and Indian mathematicians who preceded him. However, he or one of his students was the first to determine a proof of it. Prior to that, the theorem would just be a conjecture (for an idea to be considered a theorem, it must be proven; otherwise, it is a conjecture).

Pythagoras had a belief that all numbers could be expressed as a fraction or terminating/repeating decimal. This held true until Hippasus, one of his students, tried solving the Pythagorean Theorem with legs 1 and 1. This yielded a hypotenuse of length √2.

12 + 12 = c2
1 + 1 = c2
2 = c2
√2 = c

Hippasus then tried to write this number as a fraction. During his attempts to do this, he ended up proving that the number can not be written in this way (this might not be Hippasus's exact proof, but click here to see a proof of this).

After Pythagoras found out, he was outraged. He did not want to accept the fact that irrational numbers exist. He tried proving Hippasus wrong, but could not do it. Because of this, Hippasus got to take a "boat ride" in the lake, where he was thrown overboard and drowned.

I found it interesting that in that day in age, people reacted this way to the discovery of irrational numbers. Though nobody got killed when complex numbers were found in 1539, there was lots of discomfort. Newton and Euler are both recorded as calling them "impossible" numbers, and the letter i stands for imaginary, since this number could not possibly exist. Gauss was the first to recognize complex numbers as a real thing.

1 comment:

  1. Marvelous! More more!