Adding a Dimension by Isaac Asimov

By Isaac Asimov

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M. class was calculus, which he as steadfastly refused to take—so we had to separate at eleven and meet at twelve. As it happened, his sociology professor was a scholar who did things in the grand manner, holding court after class was over. The more eager students gathered close and listened to him pontificate for an additional fifteen minutes, while they threw in an occasional log in the form of a question to feed the flame of oracle. Consequently, when my calculus lecture was over, I had to enter the sociology room and wait patiently for court to conclude.

To be sure, the series worked out for the evaluation of TT are none of them decreasing geometric progressions, and so the formula cannot be used to evaluate the sum. In fact, no formula has ever been found to evaluate the sum of the Leibniz series or any of the others. Nevertheless, there seemed no reason at first to suppose that there might not be some way of finding a decreasing geometric progression 45 that would evaluate TT. If so, TT would then be expressible as a fraction. A fraction is actually the ratio of two numbers and anything expressible as a fraction, or ratio, is a "rational number," as I explained in the previous chapter.

Expressions involving numbers raised to an irrational power, such as x#, are transcendental. In fact, if you refer back to Chapter 3, you will understand me when I say that it has been proved that the algebraic numbers can be put into one-to-one correspondence with the integers, but the transcendental numbers can not. This means that the algebraic numbers, although infinite, belong to the lowest of the transfinite numbers, J^ , while the transcendental numbers belong to the next higher transfinite, ££ .

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