>>106567244
>>106567968
>Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers.
>Pappus (c. 400 AD) merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning.[21] Iamblichus (c. 300 AD) gives a series of inconsistent reports. In one story he explains how a Pythagorean was merely expelled for divulging the nature of the irrational; but he then cites the legend of the Pythagorean who drowned at sea for making known the construction of the regular dodecahedron in the sphere.[22]
>These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain.[25] In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedra. Irrationality, by infinite reciprocal subtraction, can be easily seen in the golden ratio of the regular pentagon.[26]
>Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of the square root of 2. Plato in his Theaetetus,[27] describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of sqrt 3, 5, etc. up to 17, which implies that an earlier mathematician had already proved the irrationality of sqrt 2.[28] Aristotle referred to the method for a proof of the irrationality of sqrt 2,[29] and a full proof along these same lines is set out in the proposition interpolated at the end of Euclid's Book X,[30] which suggests that the proof was certainly ancient.[31] The method is a proof by contradiction, or reductio ad absurdum, which shows that if the diagonal of a square is assumed to be commensurable with the side, then the same number must be both odd and even.[31]