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When did the Pythagoreans discover irrational numbers?

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Hippassus of Meropontum discovered irrational numbers in the fifth century B.C., when he developed the Pythagorean theorum for right triangles and discovered that hypotenuse length could not always be expressed as a rational number. The Euclidean proof of irrational numbers did not appear until 300 years later.

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For the Pythagoreans, proving the potential for irrational numbers amounted to a mortal sin of sorts. Pythagoreans worshipped numbers as the foundation of both their philosophy and religion. They believed that all of cosmology, physics, ethics and spirituality was based in, and explainable through, rational numbers, particularly because of the infinite existence of rational numbers. Having something that was inexplicable through such a foundation was, at a minimum, disturbing for Pythagoreans.

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