On another forum, someone said that she’d read or heard that the number of orders in which 52 cards can be arranged is greater than the number of grains of sands on all the worlds’s beaches. She said that can’t be true, and that it must be made up, and that it can’t be that a deck of cards can be arranged in billions of orders.
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I replied that it’s a lot more than billions. It’s a trillion trillion trillion trillion trillion. …times 80 million.
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A typical estimate of the number of grains of sand on the world’s beaches is roughly on the order of a billion trillion. Maybe there could be high estimates of a trillion trillion.
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Then she asked a good question that hadn’t occurred to me: Then in how many orders could a list of all the world’s people be arranged?
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The answer is 10 to the power of about 72 billion.
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10^(72 billion).
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How much space would you need to write that? If you used a (fairly typical) type-size in which each character occupies a square millimeter, then, to write the necessary string of “A trillon trillion trillion…trillion trillion trillion…trillion trillion trillion”, you’d need a piece of land about 718 feet square.
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(In this post, when I say that a number is about 10^N, I mean that the exponent rounds to N, when rounded to the nearest whole number. When I, instead say “roughly”, I mean only that it’s close to that.).
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That’s a large number. What could it enumerate? Let’s compare it to what’s arguably the largest local cosmic number, the number of Planck volumes in the observable universe. Let’s call that “Npvou” (pvou stands for Planck volumes, observable universe.)
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The observable universe is a little more than 90 billion lightyears in diameter.
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A Planck volume is about a trillionth of a trillionth of a trillionth of a trillionth of a trillionth of the volume of a proton.
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The number of orders in which a list of all the world’s people could be arranged is roughly the 400 millionth power of Npvou.
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You’d have to multiply the number of Planck volumes in the observable universe by itself roughly 400 million times, to get the number of orders in which could be arranged a list of the world’s people.
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But you can get a similarly large number, starting with a number a lot smaller than the world population.
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The number of orders in which120 objects can be arranged is roughly a trillion times the number of Planck volumes in the observable universe, give or take a factor of 10.
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You know those rectangular QR scan-patterns, consisting of an array of smaller squares, shaded and unshaded, which you can scan, to reach an advertiser’s website? A typical such array is 35X35 small squares.
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The number of possible combinations of shaded and unshaded squares in that array is roughly the square of the number of Planck volumes in the observable universe.
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But say that, not only do you chose some arbitrary combination, of any size, of small squares in that array, but you also number them in some order. The number of ways you can do that is the number of permutations (ordered combinations) of all sizes that can be taken from 1225 objects.
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Where N is very large, he number of permutations, of all sizes, that can be taken from N objects approaches e*(N!), where e is the base of the natural logarithms.
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With N = 1225, that number is about Npvou to the 17th power. You’d have to raise the number of Planck volumes in the observable universe to the 17th power, to get the number of permutations of all sizes that can be taken from the small squares in that 35X35 array.
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Maybe the number of Planck volumes in the observable universe could be called the largest local cosmic number. But there’s a larger cosmic number:
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Max Tegmark estimated that, if the universe is infinite, and if the physical constants are the same out to a sufficient distance, and if the spatial density of mass doesn’t systematically vary out to a sufficient distance, (What I say below depends on those assumptions,which might not really hold,out to those greater distances) then the most likely distance to the nearest Hubble volume whose things and events are identical to the things and events of our Hubble volume (whose center we’re at) is roughly 10^(10^118) meters.
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I’ll I’ll call that Dd (for “duplication distance”)
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Well, what if we numbered and listed all of the orders in which 120 objects can be arranged.
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In how many orders could that list be arranged?
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Roughly at least10^(10^200).
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What power is that of Dd? That would be 10^200 divided by 10^118. That would be 10^82, which is close to the estimate for the number of atoms in the observable universe. That’s how many times you’d have to multiply Dd by itself to get the number of orders in which could be arranged all of the orders in which 120 objects could be arranged.
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What about dividing 10^(10^200) by Dd? That would be 10^ ((10^200) – (10^118)). That’s very close to about 10^(10^200). In other words, dividing 10^(10^200) by Dd, isn’t much different from dividing it by 1.
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Compared to 10^(10^200), with respect to division, Dd doesn’t look different from 1.
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That doesn’t change if Dd is expressed in Planck lengths instead of meters.
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A Planck-length is about hundredth of a millionth of a trillionth of the diameter of a proton.
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How many Hubble volumes, scattered here and there, could be expected to be found out to a distance of 10^(10^200) meters? Dividing by Dd, and cubing the result, you get about 10^(10^200.477) Hubble volumes within that distance.
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In general doing relatively small things to a very large number doesn’t noticeably change that large number
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Of course all of that is also true of the number of orders in which could be arranged all of the orders in which a list of the world’s population could be arranged, except much moreso.
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Of course the reason why you can get such very large numbers from the number of orderings, combinations or permutations of relatively small numbers, is because all the many multiplications. …an enhancement that results in numbers larger than the enormous Npou, because Npvou is only a straight unenhanced count.
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I’m not interested in cosmic matters, because this universe is just what’s needed for us to be here. Naturally, when physicists investigate and examine the physical world, they’ll find something that’s consistent with our existence. The cosmic matters aren’t really part of our lives, and, for awe or wonder, what could match the amazing fact that this life started?
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..but there was no other way to say anything about how large those numbers are, except in comparison to cosmic numbers.
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Michael Ossipoff