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View Full Version : Enough volume to store Large numbers at a planck volume?

Sam99
2011-Oct-05, 01:49 AM
If each digit of a googolplex was stored in Planck sized cubes, rather then being written out in straight line at font of planck scale. What volume would you need to fit a googolplex?

I read that a graham's number wouldn't fit in our universe, Does anyone known how many universes one would need to fit a graham's number?

Im not very good with understanding these numbers, thanks

korjik
2011-Oct-05, 02:03 AM
10-35 cubed is 10-105 time 10100 equals 10-5m3 or 10 cc

WayneFrancis
2011-Oct-05, 02:54 AM
If each digit of a googolplex was stored in Planck sized cubes, rather then being written out in straight line at font of planck scale. What volume would you need to fit a googolplex?

I read that a graham's number wouldn't fit in our universe, Does anyone known how many universes one would need to fit a graham's number?

Im not very good with understanding these numbers, thanks

You can only "store" 2 numbers in a volume that size...0 or 1...actually you can pick any 2 arbitrary values. Either that volume contains something or not.

That said
1\times10^{100000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 000000000000}

1.6162\times10^{\neg35}m or
4.222\times10^{\neg105}m^{3}
2.3687cm^{3} would contain a googol

hmmm brain hurts trying to do a googolplex. Gut feel is the visible universe still isn't big enough.

korjik
2011-Oct-05, 04:35 AM
It would be about 101030 meters in radius. That is a big enough number that the conversion from meters to ly is about 1 in comparison, so yeah, just a little bigger.

Sam99
2011-Oct-05, 08:04 AM
It would be about 101030 meters in radius. That is a big enough number that the conversion from meters to ly is about 1 in comparison, so yeah, just a little bigger.

If i'm too understand, the volume needed for googolplex storing each digit in a planck volume would be 101030 meters in radius?

If so how many observable universes is that? thanks

grapes
2011-Oct-05, 11:14 AM
It's a lot smaller, if you use a compression algorithm on the googolplex digits first. :)

korjik
2011-Oct-05, 04:15 PM
If i'm too understand, the volume needed for googolplex storing each digit in a planck volume would be 101030 meters in radius?

If so how many observable universes is that? thanks

To first order, 1030. That is a number so big that dividing it by the radius of the universe isnt going to make it much smaller. Basically you would be subtracting 24 from 1000000000000000000000000000000.

antoniseb
2011-Oct-05, 05:25 PM
The radius (?) of the universe is not known. We know about how big the observable universe is. Let's call it 1011 light years radius (slightly higher than real numbers). Let's say there are 1016 meters in a light year, and 1035 Planck lengths in a meter. So, not counting a squeezing of Planck volumes that might or might not happen in black holes, there are about 10186 Planck volumes in the observable universe (the number is steadily growing, in ten billion years it might be 10187. One Googalplex is 10Googal, which is so much higher than the number of Planck volumes that even if we put a new one of whatever we're counting into each Planck volume, once every Planck time since the beginning of the universe, we still wouldn't have gotten noticeably beyond zero by now.

WayneFrancis
2011-Oct-06, 12:56 AM
It's a lot smaller, if you use a compression algorithm on the googolplex digits first. :)
compression 1010100 there you go

Sam99
2011-Oct-06, 04:50 AM
Could anyone roughly work out with the probabilities, how many times the monkey in the monkey theorem would type the whole of a shakespear's play word for word if the monkey typed away for a googolplex years?

Would you think it would happen a few times? or a really high number?

antoniseb
2011-Oct-06, 12:30 PM
... Would you think it would happen a few times? or a really high number?

Let's assume that there are 100 possible characters the monkeys could push on the keyboard, and a million characters in a Shakespeare play. So the chances would be 1001,000,000 (AKA 102,000,000) (divided by 38, the number of known Shakespeare plays) that any sequence of a million characters would be a Shakespeare play. This is so small a number compared to a Googalplex (though still much larger than Googal) that it should happen an almost uncountable number of times. n.b. my analysis did not include bias effects, such as monkeys tending to hit the same set of keys over and over, OR ... if you are merely using a random character generator, the advantages you might achieve knowing the relative frequency that certain letters (such as e, t, and s) are used.

Sam99
2011-Oct-06, 12:36 PM
Let's assume that there are 100 possible characters the monkeys could push on the keyboard, and a million characters in a Shakespeare play. So the chances would be 1001,000,000 (AKA 102,000,000) (divided by 38, the number of known Shakespeare plays) that any sequence of a million characters would be a Shakespeare play. This is so small a number compared to a Googalplex (though still much larger than Googal) that it should happen an almost uncountable number of times. n.b. my analysis did not include bias effects, such as monkeys tending to hit the same set of keys over and over, OR ... if you are merely using a random character generator, the advantages you might achieve knowing the relative frequency that certain letters (such as e, t, and s) are used.

Mind blown.:eek: thanks :razz:

Grey
2011-Oct-06, 02:04 PM
Im not very good with understanding these numbers, thanksFor the record, I don't think anyone is really good at understanding these numbers. That is, we can work with them mathematically just fine, but a googolplex is such a staggeringly large quantity that it's very hard (maybe impossible) to comprehend it in a meaningful way. Antoniseb does a good job of pointing out that counting the number of Planck volumes in the visible universe doesn't even make a scratch on a googolplex.

Sam99
2011-Oct-07, 06:56 AM
Im confused, on the wikipedia article it says to write down the number like 10000000.... etc would take about 3.5 x1096 metres in one-point. Surely a typo? far too small?

Kuroneko
2011-Oct-07, 10:18 AM
Im confused, on the wikipedia article it says to write down the number like 10000000.... etc would take about 3.5 x1096 metres in one-point. Surely a typo? far too small?
Wikipedia is correct. This thread is a bit confused, with people alternatively talking about writing it out in base-1, base-2, and base-10. These are very different (especially the first one).
In base-10, a googolplex has a googol digits (+1, but that's not important here). That's 10100 planck volumes ~ 177 cm³.
In base-2, a googolplex has log2(10) googols of digits. That's 588 cm³.
In base-1, a googolplex has a googolplex digits. At that scale, it doesn't really matter whether we're talking about planck volumes or m³ or cubic lightyears. The order of magnitude of the corresponding volume in m³ is log(101E100*1.77×10-104) ~ 10100 - 104 ~ 10100.

Note that taking a cube root of a googolplex gives a number closer to 10^{10^{99}} rather than 10^{10^{30}}.

On the other hands, a black hole with information of googolplex bits would have horizon area A = 4\times 10^{10^{100}} planck areas, and a mass off
M = \sqrt{\frac{Ac^4}{8\pi G^2}} = \sqrt{\frac{10^{10^{100}}}{2\pi}}\left(\frac{c^2l_ p}{G}\right)
which is still on the scale of a googolplex kg.

Ivan Viehoff
2011-Oct-07, 12:21 PM
For the record, I don't think anyone is really good at understanding these numbers.
The late mathematician noted for his very high productivity, Pal Erdos, said something along the lines that we know very little about very large integers. Maybe, he said, very large integers are really interesting, but we just don't know. By very large integers I believe he had in mind integers much, much larger than the circa 100-digit integers used in public key coding.