Saturday, January 03, 2009

More googol fun

Earlier today I posted on twitter that you would need only 333 bits to store the value 10^100 (one googol), which is really just saying that there are about a googol states that 333 bits can be in (actually there are quite a few more ways, but to a rough order of magnitude it's accurate).

That, of course, got me wondering how many bits it would take to represent a googolplex, or 10^googol = 10^10^100. Actually, in my enthusiasm I asked my computer to display a googolplex, since it had so readily displayed a googol (which, written out, is really just 101 characters). But I realized my folly and quickly aborted that request.

But I was still wondering how many bits it would take. It turns out there's actually a straightforward way to solve this problem:

Let's start with the derivation for the number of bits needed to store the value googol.

x = 10^100 = 2^a , where a will represent the needed number of bits

log_10(x) = log_10(10^100) = 100 = log_10(2^a)

How to get the 'a' out? By remembering that log_w(s^t) = t * log_w(s); that log_w(w) = 1, and that we can convert bases as follows:

log_g(p) = log_h(p)/log_h(g)


100 = a * log_10(2)

100/log_10(2) = a * log_10(2)/log_10(2) = a * 1 = a

and so 'a' is ~332.1928 which we'll round UP (since we have to work in whole bits) to 333.


But what about our friend the googolplex?

We can use the same derivation as above, but substitute 100 for 10^100, so that we find

10^100 / log_10(2) = a

or, about 3.32 * 10^100 bits. That's right, we'd need 3 googol bits to store a googolplex. And there aren't that many bits. In the whole universe.

Now, of course, if we used something with more states, then we could represent a googolplex in fewer of whatever we're using; for example, with a three-state object would only need 2 googol of them; with a 60-state object, only 5.6 * 10^99, or about half a googol. Unfortunately, things don't improve very quickly due to that log_10. With a million-state object (10^6), we're only at 1/6 googol, which is still WAY WAY too many.

Anyway, there's not really a point to this rambling, I just think that really large numbers are cool.
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