The Square Root of 2 by David Flannery

By David Flannery

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The fraction 30547 21600 which, as you can see, is not 577 exactly. 408 Why, then, is it thought that they knew of “our” 577 ? 408 Because in base 60, 577 = 1; 24, 51, 10, 35 . . 408 it is suspected that they just truncated (shortened) the sexagesimal expansion of this fraction. To three places, as we’d say. Yes. How did the Babylonians find such approximations? It is not exactly known, but there is speculation that they knew of a method of approximation. Was it a different method from the one using the sequence of fractions we have discovered?

After all, these numbers are the only ones needed for commercial transactions, and they are equally adequate at describing various other physical phenomena. They also suffice for most measuring purposes that occur in practice. Even though they cannot be used to give the measure of the diagonal of a unit square. Yes, the issue about the new nature of 2 and its cousins, 3, 5, . . was a theoretical one rather than a concern with “practical” measurement. The Greeks were fully aware that even if fractions could not measure the diagonal of a unit square exactly, they could measure it to any desired degree of accuracy.

I did. Only those rectangles, the length of whose long side to the length of the short side are in the ratio 2 : 1, have this property. I’d be interested to see why. It will require some algebra. I thought it might! Let us begin with a labeled diagram of a rectangle: Here l stands for the length of the longer side and b stands for the length of the shorter side. I suppose one side has to be longer than the other for there to be any hope of success. You are right to question my assumption that one side is longer than the other.

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