Famous Numbers Part 6 – Pythagoras’s Constant

Kapitolinischer_Pythagoras
Pythagoras of Samos (570 BCE – 495 BCE) was a Greek mathematician and philosopher and founder of the Pythagorean school of thought

The Pythagorean school of thought, founded by Pythagoras of Samos, believed that all numbers were rational numbers. That is all numbers can be expressed as fractions. However no matter how long the Pythagoreans tried they couldn’t find a way to express \sqrt{2} as a fraction.

To them all numbers had to be expressed as fractions, especially a number that came about as a result of Pythagoras’s famous theorem. Try as they might, they failed. According to legend, the Pythagorean scholar Hippasus while at sea found a way to prove that \sqrt{2} is not a rational number, and also according to legend the scholar drowned at sea.

root2
Pythagoras’s Constant is the result of applying the Pythagorean theorem on this triangle

Legend or not, the Pythagorean Scholars not only proved that their notion of numbers was wrong but that there is an entire class of numbers that cannot be represented as fractions. These are the irrational numbers and \sqrt{2} was one of the first numbers to be proven irrational.

Even though the number is attributed to Pythagoras, approximations of Pythagoras’s constant are as old as Babylon. There are Babylonian clay tablets of right triangles with the inscription :

pythagoras constant

which translated is 1 + \frac{24}{60} +\frac{51}{60^2} + \frac{10}{60^3} which is about 1.41421296... which is a pretty good approximation of \sqrt{2}.

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