Encapsulation, inheritance and polymorphism
Erik Max Francis
max at alcyone.com
Sat Jul 21 23:32:36 CEST 2012
On 07/20/2012 03:28 AM, BartC wrote:
> "Erik Max Francis" <max at alcyone.com> wrote in message
> news:GsKdnWOQPKOOvZTNnZ2dnUVZ5s2dnZ2d at 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 && max at alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Jabber erikmaxfrancis
She's your moon, she's your sun / She could even be the one
-- Nik Kershaw
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