The Pythagoreans and rational numbers

Pythagoras started a school in southern Italy around 500BC. The school was open to all people, to men and women, and they lived communally. They had some very strange rules like never eating black beans. They were vegetarians because they believed the human soul could pass from one body to another after death, to animals as well as humans. They were strict, drank no alcohol and took a vow of silence for five years. And they were very good at mathematics.

The Pythagoreans almost worshipped numbers. They believed “All is number.” The one thing that everything had in common, was that it could be counted. They believed that numbers were the underlying substance of reality. The Pythagoreans believed that all numbers were rational. They even based their society on this.

The Pythagoreans were interested in order and harmony and they viewed the cosmos, a word Pythagoras invented meaning universe, as a reflection of mathematical relations.

They noticed that two strings of the same thickness, made from the same material, on a musical instrument, at the same tension and length would produce the same note. If the string was half as long, it would produce a much higher note but a note that sounds in harmony with the first. If the second string was rds as long as the first then another note that produces a harmonic sound is produced.

There are ratios for the difference between the lengths of the two strings for three notes 1:1, 2:1 and 3:2. Other ratios that produced harmonic sounds were found.

String	1:1	2:1	3:2

Pythagoras believed that the harmonies that could be created from playing strings that have lengths that differ by certain ratios extended to the cosmos. He believed that the sun, moon and the planets created their own sounds depending of their distances from the Earth and that the sounds were harmonious because the distances between all of these bodies could all be defined by ratios. These sounds were called the “Music of the Spheres.”

However, a Pythagorean discovered that not all numbers can be expressed as ratios. Hippasus discovered that the number cannot be expressed as a ratio of two integers. The discovery of this irrational number caused great upset to the Pythagoreans. Hippasus drowned at sea is suspicious circumstances after spreading the news that not all numbers are rational.