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“The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.” ― George Bernard Shaw in Maxims for Revolutionists (1903)
The Pythagoreans were a sect (circa 6th century BC) who believed that all things are number. They also believed that all magnitudes (or numbers) were commensurable, e.g., for any two lengths, you could find a common unit such that both given lengths would be a whole number of units. In modern parlance, for the Pythagoreans all numbers were rational.
Myth has it that one of them, Hippasus of Metapontum, discovered that √2 is irrational or incommensurable, i.e., you cannot find a unit length such that the side of a square and the diagonal of the same square were both a whole number of units.
In one version of the myth, distraught by the existence of an irrational number, the other Pythagoreans threw Hippasus into the sea, hoping to drown him and his unpalatable discovery. Another variant has the drowning as a punishment from the gods and that the irrational number was the golden ratio, relating the side length and diagonal of a regular pentagon. A third variant mentions a celebration sacrificing hundreds of oxen. No matter which variant is true, it shattered the foundations of the Pythagorean world, and someone died. Moreover, given that we use the word irrational to denote things that go contrary to our reasoning, irrational numbers could claim to be the first mathematical pun and serve as a reminder of the difficulty in conquering our prejudices.
The main reason though for recounting this story is that, as in Shaw’s quote, significant progress in mathematics may often be inseparable from such unreasonable or uncomfortable developments. In an earlier blog we outlined a five-fold path to mathematical wisdom. We considered √2 as an example, looking at it from four (of five) perspectives – formal, algorithmic, graphical and practical – leaving a unifying and historical perspective for now: that of √2 being irrational.
Unfortunately, irrational numbers are often treated as something esoteric and/or as a formal construct to complete the number system – and the traditional proof of the irrationality of √2 is also presented in an unduly abstract and algebraic manner. It is likely that the Pythagoreans had more accessible and illuminating proofs of the irrationality of √2. We outline two proofs that are candidates for the Pythagorean approach (further details can be found on Wikipedia).
Both proofs relate to geometric/practical aspects of √2. One proof considers √2 as the length of the diagonal of a unit square. √2 being rational would imply that you could find two whole numbers, a and b, with b being the side length of a square and a the length of the diagonal (√2=a/b). If it helps, consider a=577 and b=408.
In the other proof, if b gives the side length of a square, then a gives the side length of a square that has double the area.
Both proofs hinge on getting successive smaller variants of the constructs, giving pairs of smaller whole numbers that also define √2. The diagrams below show how the first descent is constructed.
Both diagrams tell us that if a/b=√2 then (2b-a)/(a-b)=√2 too. And one can iterate indefinitely! But, starting from a finite pair, one cannot have an infinite descent via whole numbers, hence the irrationality.
What is beautiful (and useful!) about these proofs is that one can reverse direction – iterating outwards to larger squares, one gets an iterative scheme for calculating √2. If a/b=√2 then (2b+a)/(a+b)=√2 too! Starting with as poor a pair as (1,1), i.e. √2 ≈ 1, we get
The above ‘Pythagorean’ approach has been used by artisans (and mathematicians) for millennia – using repeated subtraction of given quantities to find the greatest common measure, the same technique underlying the Euclidean algorithm for finding the greatest common divisor.
The technique is even more readily applied in the construction of a pentagon-pentagram pair, an icon used by the Pythagoreans, hence the possibility that the golden ratio could have been the first number to be proved irrational.
But there is even more mathematics to be mined here - by studying how the Pythagoreans recovered from the unexpected collapse of their foundations, embracing irrationality and accounting for it. Other cultures, Babylonian and Indian, had separately – and earlier – grappled with √2 and may have recognised its irrationality. Irrationals also provide a further perspective on the paradoxes of Zeno, which force one to engage with the inadequacies of discrete and continuous models for the world we inhabit. Intrigued? Then tune in next time for paradise lost, paradox regained.