Hippasus of Metapontum (/ˈhɪpəsəs/; Greek: Ἵππασος, Híppasos; fl. 5th century BC), was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence ofirrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods, for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Some modern scholars though have suggested that he discovered the irrationality of √2, which it is believed was discovered around the time that he lived.
Life[edit]
Little is known about the life of Hippasus. He may have lived in the late 5th century BC, about a century after the time of Pythagoras. Metapontum in Italy (Magna Graecia) is usually referred to as his birthplace,[1][2][3][4][5] although according to Iamblichus some claim Metapontum to be his birthplace, while others the nearby city of Croton.[6] Hippasus is recorded under the city of Sybaris in Iamblichus list of each city's Pythagoreans.[7] He also states that Hippasus was the founder of a sect of the Pythagoreans called the Mathematici(Greek: μαθηματικοί) in opposition to the Acusmatici (Greek: ἀκουσματικοί);[8] but elsewhere he makes him the founder of the Acusmatici in opposition to the Mathematici.[9]
Doctrines[edit]
Aristotle speaks of Hippasus as holding the element of fire to be the cause of all things;[10] and Sextus Empiricus contrasts him with the Pythagoreans in this respect, that he believed the arche to be material, whereas they thought it was incorporeal, namely, number.[11]Diogenes Laërtius tells us that Hippasus believed that "there is a definite time which the changes in the universe take to complete, and that the universe is limited and ever in motion."[2] According to one statement, Hippasus left no writings,[2] according to another he was the author of the Mystic Discourse, written to bring Pythagoras into disrepute.[12]
A scholium on Plato's Phaedo notes him as an early experimenter in music theory, claiming that he made use of bronze disks to discover the fundamental musical ratios, 4:3, 3:2, and 2:1.[13]
Irrational numbers[edit]
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, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is confused.
Pappus 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.[14] Iamblichus 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.[15] In another account he tells how it was Hippasus who drowned at sea for betraying the construction of the dodecahedron and taking credit for this construction himself;[16] but in another story this same punishment is meted out to the Pythagorean who divulged knowledge of the irrational.[17] Iamblichus clearly states that the drowning at sea was a punishment from the gods for impious behaviour.[15]
These stories are usually taken together to ascribe the discovery of irrationals to Hippasus, but whether he did or not is uncertain.[18] In principle, the stories can be combined, since it is possible to discover irrational numbers when constructing dodecahedrons. Irrationality, by infinite reciprocal subtraction, can be easily seen in the Golden ratio of the regular pentagon.[19]
Some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2. Plato in his Theaetetus,[20] describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had already proved the irrationality of √2.[21] A simple proof of the irrationality of √2 is indicated by Aristotle, and it is set out in the proposition interpolated at the end of Euclid's Book X,[22] which suggests that the proof was certainly ancient.[23] The proof is one of reductio ad absurdum, and the method is to show that, if the diagonal of a square is commensurable with the side, then the same number must be both odd and even.[23]
In the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a much more emphatic and colourful tale. Some writers have Hippasus making his discovery while on board a ship, as a result of which his Pythagorean shipmates toss him overboard;[24] while one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that √2 is an irrational number."
Geometric proof[edit]
Another reductio ad absurdum showing that 
is irrational is less well-known.[13] It is also an example of proof byinfinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the previous proof viewed geometrically.
is irrational is less well-known.[13] It is also an example of proof byinfinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the previous proof viewed geometrically.
Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, 
. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and the ∠BAC and ∠DAEcoincide. Therefore the triangles ABC and ADE are congruent by SAS.
Because ∠EBF is a right angle and ∠BEF is half a right angle, BEF is also a right isosceles triangle. Hence BE =m − n implies BF = m − n. By symmetry, DF = m − n, and FDC is also a right isosceles triangle. It also follows that FC= n − (m − n) = 2n − m.
Hence we have an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore m and n cannot be both integers, hence is irrational.
is irrational.Pythagorean Theorem Proof[edit]
Suppose is rational.
is rational.
That means that we can make a right isosceles triangle where the side lengths are natural numbers and the legs and the hypotenuse do not share any common factors (except 1). {1}
Since the legs are equal, so are their squares. So in order for the Pythagorean Theorem to work for this special right triangle, the square of the hypotenuse has to be an even number (and if we cut it in half once then we have the area of the square of the leg).
Recall that the square of an even number is even and the square of an odd number is odd. So if the square of the hypotenuse is even the hypotenuse is even as well. {2}
Remember that a square is a quadrilateral with 2 pairs of parallel sides which are equal in length and has 4 right angles. So both sides of the square of the hypotenuse are even.
So the square of the hypotenuse of this right triangle can be cut in half twice and still have integer area. Since we only want to cut it in half once, then we'll get an even number.
So the square of the leg is even. Now according to {2} the leg must be even
This contradicts our assumption at {1} that the leg and hypotenuse have no common factors (except 1). Because if they're both even they share a common factor of 2. So the assumption that was rational is false. Or in other words is an irrational number.
was rational is false. Or in other words is an irrational number.
Continued fraction representation[edit]
The square root of two has the following continued fraction representation:
The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408. The convergent p/q differs from the square root of 2 by almost exactly 
[citation needed] and then the next convergent is (p + 2q)/(p + q).
[citation needed] and then the next convergent is (p + 2q)/(p + q).Paper size[edit]
The square root of two is the approximate aspect ratio of paper sizes under ISO 216 (A4, A0, etc.). This ratio guarantees that cutting a sheet in half along a line parallel to its short side results in the smaller sheets having the same ratio as the original sheet.