We know that Pythagoreans in Ancient Greece discovered that the square root of two is an irrational number. Why was that discovery historically significant? What value was that knowledge to the Ancient Greeks?

I do not agree on some details of the interpretation regarding the discovery of the irrationality of $\sqrt{2}$ as a confutation of the

Pythagoreans [...] belief that all numbers could be constructed as the ratio of 2 numbers.

My undestanding is that all "archaic" Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two segments of lenght $a,b$, it is always possible to find a segment of "unit lenght" $u$ such that it measures both, i.e. such that [using modern algebraic formulae which are totally foreign to Greek math] :

$$ a=n\times u\ \text{and}\ b=m\times u\ \text{for suitable}\ n,m$$

From the above instance of the assumption, it follows that : $$ \frac{a}{b}=\frac{n\times u}{m\times u}=\frac{n}{m}$$

The assumption amounts to saying that the ratio between two magnitudes is always a ratio between integers (i.e. in modern terms: a rational number).

But note that for Greek math the only numbers are the natural ones and they must be distinguished from magnitudes : a segment, a square, ... which are "measured by" numbers.

For ancient Greeks there are no rational numbers; but only magnitudes measurable with multiples of a suitable unit one.

The discovery of the existence of irrational magnitudes, through the proof that the case where $a$ is the side of the square and $b$ its diagonal is not expressible as a ratio between (natural) numbers, leads Greek math to the withdrawal of the above (implicit) assumption, that we may call : "commensurability assumption" and to the axiomatization of geometry, i.e. the systematic effort to explicitly lists all the needed assumptions.

According to this link, Legend has it that Hippasus first discovered the irrationality of $\sqrt{2}$. The second link in fact mentions a legend that held that supporters of Pythagoras murdered Hippasus -- who allegedly discovered the irrationality of $\sqrt{2}$ on a boat in the middle of the sea -- by throwing him overboard immediately after he informed them of his discovery.

The Pythagoreans had the belief that all numbers could be constructed as the ratio of 2 numbers. (That they were rational) So basically it was a big deal because it flew in the face of knowledge. All of their work was based on the premise of rational numbers being all the numbers.

Any new evidence that completely overturns a fundamental truth has often been met with derision. Even in (relatively) modern times, Imaginary Numbers were considered "fictitious or useless, much as zero and the negative numbers once were."

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    Btw, a car can weigh $\pi$ tons, but can't $4+3*\sqrt{-1}$ tons. – peterh Feb 10 '15 at 17:40
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    I see your point. Technically it could if its flux capacitor was allowed to way 3i tons and the rest of it weighed 4 tons. – Carlos Bribiescas Feb 10 '15 at 22:47

These legends do exist, and have for along time. But few if any specialist historians of the subject believe Pythagoreans discovered irrationality of $\sqrt{2}$. See:

Pythagoras vs. the idea of Pythagoras

It is very hard to judge of Greek mathematics before Euclid, let alone before Plato, as there is so little evidence. The most widely read single study of it today is probably D. H. Fowler's Mathematics of Plato's Academy, and for what it is worth I think he may well be right. In short he argues that incommensurability was a well known topic easily handled by Greek mathematicians as far back as our evidence can take us.

The fact that $\sqrt{2}$ existed and is irrational was a blow to the ancient Greeks who only believed in numbers that they could calculate to a certain degree of precision whenever required. Or in other words, they were familiar with rational numbers. The fact that others numbers existed would have carried the same sort of feelings in them as and when we first encounter topics such as countability and uncountability and the continuum hypothesis in set theory. At first it may seem to be all some sort of circular and wrong argument but given some time we get used to it. And perhaps so did the Greeks.

As far as practicality is concerned, it would not have been much practical to them as they would not have been able to measure these new numbers to a degree of precision that they were used too. But the whole point of knowledge gathering is not where to put that knowledge to use, but why should that knowledge exist in the first place.

  • Note that irrational numbers can also be calculated to arbitrary precision using just rationals ($\mathbb Q$ is dense in $\mathbb R$). – Danu Dec 3 '14 at 10:14

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