Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: thedoc on 23/10/2012 16:41:43
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Hi Chris,
I hear there is a maths radio show this weekend so the Spriggs family has a maths question for you (it may also have some physics in it...!):
A concept that we enjoy in the Spriggs household is that of the googol, i.e. 10 to the power 100 or a 1 followed by a hundred 0s.
I believe that the googol may be a pretty useless concept, if there isn't a googol of anything in the universe... I said this to my boys and added that you don't need a number to describe something that doesn't exist...
Since that day, when our older son was about 7,our two boys have been trying to come up with a googol of something. They started with the grains of sand in the Sahara Desert. That is a very very small number. We eventually reached the number of photons in the universe. I don't know the answer to this, but I got to 10 to the power 87. Which leaves it 10 trillion times smaller than a googol. But I have never been able to prove this. It would be good if the Naked Scientists could oblige?! And could they get us past a googol of something, anything?
If nothing else, it has taught our boys the laws of indices!
Best regards,
Andy
Asked by Andy Spriggs
Visit the webpage for the podcast in which this question is answered. (http://www.thenakedscientists.com/HTML/podcasts/show/20121021/)
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We answered this question on the show...
Matt - A googol is a great number in that, it is a number that exists just because it has a nice name. There's nothing that we really have a googol of, but someone thought, “wouldn’t a 1 with 100 zeros be a great number?” so they called it a googol. You could hear him emphasising gooGOL because it’s spelled G O O G O L. The Google search engine is named after this but they spelled it differently. He’s absolutely correct. If you're going to try to count objects, you won't find a googol of anything. So, even going right up to number of protons in the universe, which I think is – physicists argue between 1079 to 1081, so it’s a 1 with somewhere between 79 and 81 zeros. But I have found one thing which we have more than a googol of in the universe.
Chris - Well, should we ask Ginny if she can guess what it is?
Ginny - I don’t know. I mean, I was thinking about neurons because they make an awful lot of connections, but I think when we are thinking about this, it probably wouldn’t be quite that many, even if you include all the animals on the Earth.
Chris - 1011 nerve cells in the brain, 1,000 connections per nerve cells per brain, so that would probably be about 1014,so we’re still a long way short of Matt’s googol.
Matt - Actually, you're not far off from what I decided on because when you're talking about connections, well suddenly, you're looking at different ways of combining things. And so, I looked at how many ways you can shuffle a pack of cards. So if you got a pack of 52 cards and you shuffle it, you can work out there's about 1067, It’s 8 times 1067 ways to shuffle a pack of cards which is a lot more than the number of grains of sand in the Sahara. In fact, if you’ve got a pack of cards, with just 27 cards, you would have about 1028 different ways to shuffle them. If you want a googol, if you get a pack of cards with 70 cards – so you're going to need your normal pack and then get 18 from a pack maybe with a different colour on the front or the back so you can tell them apart – if you shuffle 70 cards, There's a googol of possible ways to arrange them.
Chris - Beautiful. I was thinking of a similar example, Matt because I was looking at the power of passwords on the internet. Someone was asking a question on our forum and they were discussing the power of encryption and saying, “if I took a brute force attempt to crack a password just by taking random choices, how many possible combinations of a password encrypted with a certain number of bits of encryption?” And you end up with something which will take trillions of centuries to solve and the number of solutions I think is way more than a googol.
Matt - Yes, and again, you're looking at number of possible combinations. So in fact, a password is pretty close to a pack of cards because you’ve got the alphabet, upper and lower case and then you do actually get extra symbols, and bits and pieces. So, if you’ve got a password of even just 8 characters, you're already up in the very high tens of thousands and much bigger than that, you're into millions, billions and well off very quickly.
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A Planck length is 1.6162×10−35 meter, so a Planck volume is 4.22x10^-105 m3.; a cubic meter contains 2.37x10^104 Planck volumes. That's 23,700 googols of Planck volumes in every cubic meter.
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The number of possible games of Go is comfortably larger than a googol.
http://en.wikipedia.org/wiki/Go_and_mathematics
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The number of different photos you could take on you cellphone is much larger than a googol.
"1 with 100 zeroes in decimal" is approximately equal to "1 with 332 zeros in binary".
So any device which can produce a binary string independently varying in at least 332 bit positions can produce a googol of different outputs.
Given that even a very old cellphone produces image files containing a million or more bits*, that is well over a googol of (potentially) different image files. In reality, your cellphone will become obsolete and be discarded well before it reaches anywhere near this number of actual images.
*The JPEG image encoding standard is pretty effective at removing redundant information from an image, so most of these bits can vary independently, given different image inputs.
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I notice a pattern here...
It seems that if you're talking about physical objects then no, there isn't a googol of anything in the observable universe (except maybe strings or other small enough quantum things). But if you allow numbers of combinations, it's quite common and there are much bigger numbers as well.
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... So, if you’ve got a password of even just 8 characters, you're already up in the very high tens of thousands and much bigger than that, you're into millions, billions and well off very quickly.
If you include at least one each of uppercase, lowercase, number and special characters in an 8 character password there are 1015 permutations possible ...
[ Invalid Attachment ]
https://www.grc.com/haystack.htm
So an eight character password is OK* for an online login, but offline could be cracked by brute-force within few days on a bog-standard computer.
[ * don't use words which could be found in a dictionary ]
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Maybe the number of virtual particles present in quantum vacuum fluctuations Universe-wide?
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How about all of the VIBRATIONS of the elementary particles ever in the universe. I think that should cover it.
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Since pi is infinite... There you go, more numbers in pi than there is googol.
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What about the number of photons! They are omited from stardls and they filled all over universe?