How Many Noughts ?

[kuiperbelt]

A googol is an inconceivable number made up of a '1' followed by one hundred (100) noughts.

[ClareTG0ld]

Just to clarify (read: correct) my answer to gizmonster at 9.07, that's referring to writing out a googolplex digits, which is not the same as writing out a googolplex itself -- which can be done within the confines of the known universe, depending on how small your writing is and whether you're allowed to write multiple digits at once.

But it does raise a few questions about what it means to represent a number. We're talking here, for example, about writing the digits in base ten, which is obviously natural for a googolplex = 10^10^100, or similar numbers. But you could presumably vary this by either allowing for more "symbols" to represent unique numbers while still using a base system, in which case the time/space required would get smaller -- for example, 10^100, in hexadecimal has only 84 digits rather than 100, although they're no longer just zeroes and a leading one. On the other hand, if you assumed that a "Planck volume" is the smallest physically meaningful space -- a dangerous assumption, but let's run with it -- then there's no room within that space to represent anything other than a filled/empty state, meaning that you'd be stuck with binary, in which case you'd need more digits.

But at the other end of the scale, you also don't need that much information to represent a googolplex at all. It can be done in nine characters: 10^10^100 is precisely one googolplex; and two of those characters are anyway redundant in handwriting, because you can do the same job by writing in usual power notation. That being the case, a googolplex is quite a tiny "large number". It's even not that difficult to see how to get bigger ones: for example, let f(n) = 10^f(n-1), and let f(1) = 100, then f(2) is a googol, f(3) is a googolplex, etc, and the growth rate here is huge because, for example, f(4) has a googolplex digits and so on.

A famous example in this language would be Graham's number, which has the form 3^3^3^... ^3, but where the number of 3's is so unbelievably huge that it scarcely bears contemplating. And even *that* is small in the grand scheme of things.