Asimov on large numbers

Asimov on large numbers, including Skewes' number. (By the way, Skewes is pronounced in two syllables, which I didn't know, and Littlewood's middle name was apparently Edensor.)

Asimov also writes x^y as x\y, which is a nice trick when the exponents get complicated, as they do here. Why is there not a non-subscript notation for powers, other than exp(x) for e^x when the base happens to be e?

You can take any real number greater than 1 and write it as 10\10\...\10\k for some number of 10's and some constant 1 < k < 10; Skewes' number is 10\10\10\34, or 10\10\10\10\1.53, so Asimov refers to this as a "quadruple-ten number". If I remember correctly Douglas Hofstadter uses this same idea of quantifying numbers by the number of exponentiations needed in one of his Metamagical Themas columns, although more correctly Asimov had it first.

Essentially all "real" numbers are either "single-ten" numbers (between 10 and 10¹⁰) or "double-ten" numbers (between 10¹⁰ = 10\10\1 and 10¹⁰¹⁰ = 10\10\10)); note that the number of particles in the observable universe is much less than 10\10\2. (It's something like 10\10\1.90, which is actually much smaller than 10\10\2 even though it doesn't sound like it.)

There's a sense in which, say, 10\x and 10\y are close to each other if x is close to y. This is reflected in some asymptotic notation I've seen, where one writes f(x) ~ g(x) if (log f(x)/log g(x)) approaches 1 as x approaches infinity, so, for example, exp(x²) and exp(x² + x) are "close together" in this sense. I don't know of cases where one can only show that log log f(x) and log log g(x) are asymptotically equal; this seems like too coarse a definition of "equality" to be of much use.