joint with Richard Elwes
Sometimes you can learn a lot from an old piece of clay. This is a Babylonian clay tablet from around 1700 BC. It’s known as “YBC7289”, since it’s one of many in the Yale Babylonian Collection.
It’s a diagram of a square with one side marked as having length 1/2. They took this length, multiplied it by the square root of 2, and got the length of the diagonal. And our question is: what did they really know about the square root of 2?
Questions like this are tricky. It’s even hard to be sure the square’s side has length 1/2. Since the Babylonians used base 60, they thought of 1/2 as 30/60. But since they hadn’t invented anything like a “decimal point”, they wrote it as 30. More precisely, they wrote it as this:
Take a look.
So maybe the square’s side has length 1/2… but maybe it has length 30. How can we tell? We can’t. But this tablet was probably written by a beginner, since the writing is large. And for a beginner, or indeed any mathematician, it makes a lot of sense to take 1/2 and multiply it by to get .
Once you start worrying about these things, there’s no end to it. How do we know the Babylonians wrote 1/2 as 30? One reason is that they really liked reciprocals. According to Jöran Friberg’s book A Remarkable Collection of Babylonian Mathematical Texts, there are tablets where a teacher has set some unfortunate student the task of inverting some truly gigantic numbers such as 325 · 5. They even checked their answers the obvious way: by taking the reciprocal of the reciprocal! They put together tables of reciprocals and used these to tackle more general division problems. To calculate they would break up into factors, look up the reciprocal of each, and take the product of these together with . This is cool, because modern algebra also sees reciprocals as logically preceding division, even if most non-mathematicians disagree!
So, we know from tables of reciprocals that Babylonians wrote 1/2 as 30. But let’s get back to our original question: what did they know about ?
On this tablet, they used the value
This is an impressively good approximation to
But how did they get this approximation? Did they know it was just an approximation? And did they know is irrational?
There seems to be no evidence that they knew about irrational numbers. One of the great experts on Babylonian mathematics, Otto Neugebauer, wrote:
… even if it were only due to our incomplete knowledge of the sources that we assume that the Babylonians did not know that had no solution in integer numbers and , even then the fact remains that the consequences of this result were never realized.
But there is evidence that the Babylonians knew their figure was just an approximation. In his book The Crest of the Peacock, George Gheverghese Joseph points out that a number very much like this shows up at the fourth stage of a fairly obvious recursive algorithm for approximating square roots! The first three approximations are
The fourth is
but if you work it out to 3 places in base 60, as the Babylonians seem to have done, you’ll get the number on this tablet!
The number 577/408 also shows up as an approximation to in the Shulba Sutras, a collection of Indian texts compiled between 800 and 200 BC. So, Indian mathematicians may have known the same algorithm.
But what is this algorithm, exactly? Joseph describes it, but Sridhar Ramesh told us about an easier way to think about it. Suppose you’re trying to compute the square root of 2 and you have a guess, say . If your guess is exactly right then
But if your guess isn’t right, won’t be quite equal to . So it makes sense to take the average of and , and use that as a new guess. If your original guess wasn’t too bad, and you keep using this procedure, you’ll get a sequence of guesses that converges to . In fact it converges very rapidly: at each step, the number of correct digits in your guess will approximately double!
Let’s see how it goes. We start with an obvious dumb guess, namely 1. Now 1 sure isn’t equal to 2/1, but we can average them and get a better guess:
Next, let’s average 3/2 and 2/(3/2):
We’re doing the calculation in painstaking detail for two reasons. First, we want to prove that we’re just as good at arithmetic as the ancient Babylonians: we don’t need a calculator for this stuff! Second, a cute pattern will show up if you pay attention.
Let’s do the next step. Now we’ll average 17/12 and 2/(17/12):
Do you remember what 17 times 17 is? No? That’s bad. It’s 289. Do you remember what 12 times 24 is? Well, maybe you remember that 12 times 12 is 144. So, double that and get 288. Hmm. So, moving right along, we get
which is what the Babylonians seem to have used!
Do you see the cute pattern? No? Yes? Even if you do, it’s good to try another round of this game, to see if this pattern persists. Besides, it’ll be fun to beat the Babylonians at their own game and get a better approximation to .
So, let’s average 577/408 and 2/(577/408):
Do you remember what 577 times 577 is? Heh, neither do we. In fact, right now a calculator is starting to look really good. Okay: it says the answer is 332,929. And what about 816 times 408? That’s 332,928. Just one less! And that’s the pattern we were hinting at: it’s been working like that every time. Continuing, we get
So that’s our new approximation of , which is even better than the best known in 1700 BC! Let’s see how good it is:
So, it’s good to 11 decimals!
What about that pattern we saw? As you can see, we keep getting a square number that’s one more than twice some other square:
and so on… at least if the pattern continues. So, while we can’t find integers and with
because is irrational, it seems we can find infinitely many solutions to
and these give fractions that are really good approximations to . But can you prove this is really what’s going on?
We’ll leave this as a puzzle in case you’re ever stuck on a desert island, or stuck in the deserts of Iraq. And if you want even more fun, try simplifying these fractions:
and so on. Some will give you the fractions we’ve seen already, but others won’t. How far out do you need to go to get 577/408? Can you figure the pattern and see when 665,857/470,832 will show up?
If you get stuck, it may help to read about Pell numbers. We could say more, but we’re beginning to babble on.
You can read about YBC7289 and see more photos of it here:
• Duncan J. Melville, YBC7289.
• Bill Casselman, YBC7289.
Both photos in this article are by Bill Casselman.
If you want to check that the tablet really says what the experts claim it does, ponder these pictures:
The number “1 24 51 10” is base 60 for
and the number “42 25 35” is presumably base 60 for what you get when you multiply this by 1/2 (we were too lazy to check). But can you read the clay tablet well enough to actually see these numbers? It’s not easy.
For a quick intro to what Babylonian mathematicians might have known about the Pythagorean theorem, and how this is related to YBC7289, try:
• J. J. O’Connor and E. F. Robertson, Pythagoras’s
theorem in Babylonian mathematics.
We got our table of Babylonian numerals from here:
• J. J. O’Connor and E. F. Robertson, Babylonian numerals.
For more details, try:
• D. H. Fowler and E. R. Robson, Square root approximations in Old Babylonian mathematics: YBC 7289 in context, Historia Mathematica 25 (1998), 366–378.
We also recommend this book, an easily readable introduction to the history of non-European mathematics that discusses YBC7289:
• George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Princeton U. Press, Princeton, 2000.
To dig deeper, try these:
• Otto Neugebauer, The Exact Sciences in Antiquity, Dover Books, New York, 1969.
• Jöran Fridberg, A Remarkable Collection of Babylonian Mathematical Texts, Springer, Berlin, 2007.
Here a sad story must indeed be told. While the field work has been perfected to a very high standard during the last half century, the second part, the publication, has been neglected to such a degree that many excavations of Mesopotamian sites resulted only in a scientifically executed destruction of what was left still undestroyed after a few thousand years. – Otto Neugebauer.