By Isaac Asimov
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Extra info for Adding a Dimension
It might be necessary to draw a new line through a point and make it parallel or, perhaps, perpendicular to a second line. It might be necessary to divide a line into equal parts, or to double the size of an angle. To make all this drawing as neat and as accurate as possible, instruments must be used. It follows naturally, I think, once you get into the Greek way of thinking, that the fewer and simpler the instruments used for the purpose, the closer the approach to the ideal. Eventually, the tools were reduced to an elegant minimum of two.
Once TT was proved irrational, mathematicians were satisfied. The problem was over. And as for the application of IT to physical calculations, that problem was over and done with, too. You may think that sometimes in very delicate calculations it might be necessary to know TT to a few dozen or even to a few hundred places, but not so! The delicacy of scientific measurements is wonderful these days, but still there are few that approach, say, one part in a billion, and for anything that accurate which involves the use of TT, nine or ten decimal places would be ample.
If you do this, you will find that the more terms you use, the closer you get to 1, and you can express this in shorthand form by saying that the sum of that infinite number of terms is merely 1 after all. There is a formula, in fact, that can be used to determine the sum of any decreasing geometric progression, of which the above is an example. Thus, the series: 3 _[_ 3 _L 10 100 9 1000 -[_ 3 10000 _L 3 IOOOOO • *• adds up, in all its splendidly infinite numbers, to a mere $, and the series: i + J _ + _i_ + ^ 3 ^ + 2 20 200 2000 i 20000 " ' * adds up to f.
Adding a Dimension by Isaac Asimov