Commensurability/Historic/Motivation for real numbers/Remark
The question, whether there exist, beside the rational numbers, further useful numbers, has its roots in ancient Greece. The question was raised in the form whether two lengths, given in a natural way, are always commensurable to each other (whether there exists a third length such that the two given lengths are integer multiples of it). The Pythagorean school believed in the harmony of the universe, and this implied that all proportions between natural lengths must be expressed by a proportion of integer numbers. They found such integer proportions in music (music intervals), and thought that such proportions hold for the motions of the planets, and for geometry in general. Some people speculate that they knew the reasoning of fact showing the irrationality of (the incommensurability of the side length and the length of the diagonal in a square), but that they concealed this knowledge. Anyhow, later in ancient times the knowledge was accepted that there are irrational numbers.
