A 3700-year-old proof that the diagonal of a unit square has length √2 | Hacker News Reader

A 3700-year-old proof that the diagonal of a unit square has length √2

math.ubc.ca

54 pointssethg23 comments

In the 6th century BC Pythagoras' followers knew that the square root of two was irrational (couldn't be expressed as the ratio of two integers). The story goes that this messed with a lot of their theories so it was ordered to be kept a secret, and anyone who dared reveal the fact was killed.

It seems kind of funny to me that they actually had a cult/religion based on math. Instead of just accepting the existence of irrationals and attempting to update their theories - they tried to suppress the new evidence that contradicted their teachings.

The Stanford Encyclopedia of Philosophy has a long article on Pythagoras that doesn't mention the whole rutti-tutti-irrationality issue.

http://plato.stanford.edu/entries/pythagoras/

They do say that we don't have anything written by Pythagoras or his contemporaries--the earliest extant works mentioning him were written more than a century after his death--and then a lot of people attributed their own views to Pythagoras to give themselves the benefit of his fame. So it's really hard to know what the historical Pythagoreans did or believed.

I think it is safe to say that if the Pythagoreans kept the irrationality of √2 a closely guarded secret, it wasn't their only closely guarded secret.

But what would "accepting the existence of the irrationals" mean? As far as the Greeks knew, a "number" was a/b where a and b were integers.

"Accepting" that the length of a triangle was not a "number" could have all sorts of consequences, including rejection of their belief in validity of mathematical reasoning, so I'm not surprised they were scared of it.

The Greeks did know about irrational numbers. They could draw a right triangle whose legs were 1 and 1. They knew that the length of the hypotenuse (a number that we call the square root of two) couldn't be represented as the ratio between any two integers, yet it had a length.

Actually, length was primary since geometry was the foundation of their math.

In the book The Heart of Mathematics by Edward Burger and Michael Starbird, there is a quotation of a translated passage from Proclus, a 5th-century AD philosopher:

"It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."

Cantor was very troubled by his discovery of the transfinite numbers because they conflicted with the infinite nature of God (hence the name transfinite).

Shouldn't be any surprise to anyone that when a theory conflicts with _beliefs_ there's trouble.

I love how mathematicians use the term "discover" instead of "invent" the way anyone else would.

They use the term 'discover,' I would presume, because it's the most technically accurate term for what they do. As near as I can tell, as a relative layperson, mathematicians look at the existing state of mathematics, embodied in proven theorems, and consider the implications of those theories, looking for unanswered questions or outright contradictions. They then attempt to find the new theory that will provably answer the question or resolve the contradiction.

The only 'invention' that goes on seems, to me, to be on the order of brainstorming, i.e., inventing hypotheticals to test their implications or shed new light on an existing conundrum.

Fundamentally, I think the use of 'discovery' terminology, rather than 'invention' terminology arises from the nature of the new thing produced. Discoveries, in the scientific or mathematical sense, were 'always there' within the corresponding realm of inquiry (e.g., the set of axioms that comprise mathematics, the physical world for scientific discoveries, & etc.) and represent a mere formulation of an existing but previously hidden truth. Inventions, on the other hand, represent a novel application of existing principles for some external purpose. The implementation cannot be said to have flowed from existing knowledge in any meaningful way. Nor do inventions have any general theoretical utility. (Mousetraps are fun little inventions, but don't contribute to a 'Theory of Pest Control' and we certainly would not refer to the 'discoverer of the mouse trap.')

Additionally, other scientists do in fact use the terms 'discover' and 'invent' in precisely the same sense as mathematicians do, and for similar reasons. In fact, these are the commonly accepted notions of the terms even among nonspecialists.

Using 'mathematical invention' rather than 'mathematical discovery' erroneously (or maliciously) ascribes arbitrary subjectivity to mathematical thought.

they say "discover" because they believe it was present ever since. just like physics don't say they invented helium or oxygen, they just discovered it.

Reminds me of one of my favorite films

"Donald Duck in Math-Magic Land" (Yes, I own the DVD)

Pythagoras and his secret crew have a cameo.

you can watch this on youtube. i recommend it.

The Pythagoreans had an elaborate philosophy based on the idea that the harmony of the universe was due to relations between the natural numbers. For example, Pythagoras discovered that musical notes and harmonies was based on relations between real numbers. He concieved the notion that the relations between the movements of the planets would create a similar harmonius symphony, the music of the spheres.

All that beauty and harmony fell apart when it was discovered that irrational numbers existed.

Not convinced that that's what it is. It's certainly a ~3700-year-old statement that the diagonal has length sqrt(2), but there's no evidence that it is, or was intended to be, or demonstrates that anyone had, a proof.

The author of those pages says "It amounts to a dissection of the square on the hypotenuse of an isosceles right triangle into pieces which can be reassembled to make up the two squares on the sides, and I can't see why the figure is exactly what it is if it weren't understood to demonstrate this." but it seems to me that even if all you want to do is write down that the diagonal is sqrt(2) then you'll need at least the square and one diagonal, and adding the other could as well be motivated by love of symmetry as by having noticed that with it there you can dissect-and-reassemble into a 2x1 rectangle or whatever.

Remarkable, none the less.

If you know that 1/2 * base * height = the area, then the second diagonal can be seen as the height of the triangle cut by the first diagonal. Since b = 2 h,

1/2 b h = 1/2 ==> b h = 1 ==> 1/2 b b = 1 ==> b b = 2 ==> b = sqrt(2).

I can't read cuneiform so I don't know if it makes this argument.

You'd first have to prove that the diagonals bisect each other (so that you can say b = 2h). I'm not sure whether that result was known at the time, but it's not too hard to prove.

You're right. That's pushing it for those handful of scratches on the tablet. Who knows, maybe they had some lemma tablets lying around.

From my own limited experience, ancient documents tended to be very elliptical; when everything had to be copied out by hand the reader was expected to do a little more work to understand what documents meant. So this tablet may have been the equivalent of notes accompanying a lecture: "As you can see from the triangles on this diagram..."

The tablet just gives the lengths of the sides and diagonals. There is no sign of any explicit reasoning on it. The author of the pages linked to from here thinks that the maker's decision to show both diagonals of the square indicates that it's meant to illustrate a proof. I am not convinced.

Could also be a failed attempt at drawing the "nikolaus hut" (not sure what it is called in english: http://www.kreudenstein-online.de/AppletNikol/AppletNikol.ht...).

That, or their alphabet is REALLY expressive.

Where is the proof?

There is no proof specifically this is really annoying about most of the Babylonian Texts, they never go into method very much.

However from the numbers they got (30, 1;24,51,10, and 30*1;24,51,10=42;25,35) it looks less like they used a proof as they did use an iterative approximation method as follows:

let a be some number

let a1 be an approximation of sqrt(a) such that a1 > sqrt(a)

then B1 = a/a1 is also an approximation of sqrt(a) but deviates in the other direction s B1 < sqrt(a).

we have now bracketed sqrt(a)

so a new better bracketing can now be made by

a2 = (a1 +B1)/2 and B2 = a/a2

and then

a3 = (a2 + B2)/2, B3 = a/a3

etc. the answer for sqrt(2) of 1;24,51,10 happens to equal a3 in this case.

note that the above is only speculation, but the method DOES produce the Babylonian numbers so and it uses only arithmetic that we know they had, so it seems pretty likely

(All this information and more can be found in Mathematical Cuneiform Texts by Otto Neugebauer)

I don't think there is one. At least from what I can gather, this isn't quite a proof that the diagonal of a unit square has length √2. The tablet tells us that the Babylonians knew the length was √2, but it doesn't tell us how they knew, or how they could prove they were right.

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