Basic Ideas in Greek Mathematics

Michael Fowler UVa Physics Department

Closing in on the Square Root of 2

In our earlier discussion of the irrationality of the square root of 2, we presented a list of squares of the first 17 integers, and remarked that there were several “near misses” to solutions of the equation m2 = 2n2. Specifically, 32 = 2×22 + 1, 72 = 2×52 - 1, 172 = 2×122 + 1. These results were also noted by the Greeks, and set down in tabular form as follows:

3       2

7       5

17      12


After staring at this pattern of numbers for a while, the pattern emerges: 3 + 2 = 5 and 7 + 5 = 12, so the number in the right-hand column, after the first row, is the sum of the two numbers in the row above. Furthermore, 2 + 5 = 7 and 5 + 12 = 17, so the number in the left-hand column is the sum of the number to its right and the number immediately above that one.

The question is: does this pattern continue? To find out, we use it to find the next pair. The right hand number should be 17 + 12 = 29, the left-hand 29 + 12 = 41. Now 412 = 1681, and 292 = 841, so 412 = 2× 292 - 1. Repeating the process gives 41 + 29 = 70 and 70 + 29 = 99. It is easy to check that 992 = 2×702 + 1. So 992/702 = 2 + 1/702. In other words, the difference between the square root of 2 and the rational number 99/70 is approximately of the magnitude 1/702. (You can check this with your calculator).

The complete pattern is now evident. The recipe for the numbers is given above, and the +1’s and -1’s alternate on the right hand side. In fact, the Greeks managed to prove (it can be done with elementary algebra) that pairs of numbers can be added indefinitely, and their ratio gives a better and better approximation to the square root of 2.

The essential discovery here is that, although it is established that the square root of 2 is not a rational number, we can by the recipe find a rational number as close as you like to the square root of two. This is sometimes expressed as “there are rational numbers infinitely close to the square root of 2” but that’s not really a helpful way of putting it. It’s better to think of a sort of game - you name a small number, say, one millionth, and I can find a rational number (using the table above and finding the next few sets of numbers) which is within one millionth of the square root of 2. However small a number you name, I can use the recipe above to find a rational that close to the square root of 2. Of course, it may take a lifetime, but the method is clear!