On 07/20/2012 03:28 AM, BartC wrote:
"Erik Max Francis" <m...@alcyone.com> wrote in message
news:gskdnwoqpkoovztnnz2dnuvz5s2dn...@giganews.com...
On 07/20/2012 01:11 AM, Steven D'Aprano wrote:
On Thu, 19 Jul 2012 13:50:36 -0500, Tim Chase wrote:
I'm reminded of Graham's Number, which is so large that there aren't
enough molecules in the universe to write it out as a power tower
a^b^c^d^..., or even in a tower of hyperpowers a^^b^^c^^d^^... It was
the
provable upper bound to a question to which experts in the field thought
the most likely answer was ... six.
(The bounds have since been reduced: the lower bound is now 13, and the
upper bound is *much* smaller than Graham's Number but still
inconceivably ginormous.)
You don't even need to go that high. Even a run-of-the-mill googol
(10^100) is far larger than the total number of elementary particles in
the observable Universe.
But you can write it down, even as a straightforward number, without any
problem. Perhaps a googolplex (10^10^100 iirc) would be difficult to
write it down in full, but I have just represented it as an exponent
with little difficulty.
These bigger numbers can't be written down, because there will never be
enough material, even using multiple systems of exponents.
But that's true for precisely the same reason as what I said. If you're
going to write a number down in standard format (whatever the base),
then the number of digits needed scales as the logarithm of the number
(again, whatever the base). log_10 10^100 is trivially 100, so a rough
order of magnitude in that form is easy to write down. But the log_10
10^10^100 is 10^100 = a googol, which is already more than the number of
elementary particles in the observable Universe.
(A few years ago the biggest number I'd heard of was Skewes' Number
(something like 10^10^10^34), but even that is trivial to write using
conventional exponents as I've just shown. Graham's Number is in a
different
class altogether.)
Anything's trivial to "write down." Just say "the number such that ..."
and you've written it down. Even "numbers" that aren't really numbers,
such as transfinite cardinals!
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
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