10 September 2007

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 xy 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 ex 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.


Aaron said...

Essentially all "real" numbers are either "single-ten" numbers... or "double-ten" numbers...

Ooog... this sentence bothers me, because for the real numbers, the truth is exactly the reverse!

Anonymous said...

umm ... as i'm sure you already know,
the "caret" (^) is pretty standard
for mainline exponentiation.
wikipedia blames ALGOL.

Isabel said...


that's a good point.

In that case, why does one not see the caret used in that way more often?

Anonymous said...

In that case, why does one not see the caret used in that way more often?

Well established mathematicians are too used to using superscripts. Less established mathematicians are afraid of looking like uncouth computer programmers if they use the caret without explanation, and it's generally not worth the trouble of inserting an explanation.

rokirovka said...

I prefer to use powers of e to quantify large numbers. Here is how I put it: How many natural logarithms of the number do you have to take to get a negative number?

For a googol, the answer is 5 natural logs.

For a googolplex, it is 6 logs.

For Skewes' number, it is 7 logs.

Then there is Graham's number G. Recall that just the first term of the 64 terms of G is 3^^^^3 (four up-arrows in Knuth's notation).

Now consider the much smaller number 3^^^3 with only three up-arrows. It takes more than 7,625,597,484,987 natural logs of that number to get a negative number.

That's a vast gap between Skewes' number and a tiny part of the first term of Graham's number. I wonder what all those numbers in between are good for, that take 10 or 20 or 50 or 100 or 1,000 natural logs to get a negative number?