# A couple of maths-related questions..

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Joe90
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07 Oct 2017, 1:48 pm

1. I've often read that a pack of cards can be shuffled into a different order every time. But what really confused me once was when I read something that said that there are more different ways to shuffle a pack of cards than there are grains of sand on every beach on the planet. I thought a pack of cards can be shuffled in perhaps a few million different ways, if that, but more than the amount of tiny grains of sand there is on every beach??? This Earth is ENORMOUS, much bigger than I can imagine, and there are many beaches that go on for miles, and the number of grains of sand feels as immense as something like a million trillion zillion.

2. I have so many posts on WP, a 5-digit number that I can't read (sorry, I'm no good with numbers). What if I post so many every year, that by the time I'm 80 it reaches such a high number and overloads the internet and starts up a world erruption, like how they predicted Y2K? Or, being more logical, it might reach a number that's so high that it can't go up any more and my account shuts down.

I'm in a curious mood at the moment and I have a zillion fascinating questions (no pun intended), and this is just two of them.

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Chichikov
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07 Oct 2017, 2:03 pm

The formula for how many ways cards can be shuffled is "52!", which is 52*51*50*49..... and so on which ends in a number that is I think about 67 digits long. So that's a lot of millions.

For the second question, assuming the database holds your post count as a signed int (the default value and many leave it as it is) you can have 2,147,483,647 posts. If you posted once a minute it would take 4,000 years to use those up, and for once a second it would take 68 years. If you did manage to do that when you personally posted you'd get an error message but it wouldn't affect anyone else.

naturalplastic
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09 Oct 2017, 10:53 am

duplicate post

Last edited by naturalplastic on 09 Oct 2017, 3:02 pm, edited 1 time in total.

naturalplastic
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09 Oct 2017, 10:59 am

8 times ten to the sixty seventh power. That's how many possible ways there is to shuffle the deck of 52.

The number atoms thought to exist in the know universe is on the order of ten to the 85th.

The number of grains of sand (bigger than atoms) on the beaches of just this planet has gotta be a lot less than the number of atoms in the whole universe. So its quite believable that it could be equal to all of the possible decks of cards.
++++++++++++++

Your post count is a little under 14 thousand. Which (according your profile) is "point two percent" of all posts ever made on WP. Or one part in 500. So you would have to increase your post count five hundred fold before it would even strain WP. Much less bring down the whole internet.

Joe90
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09 Oct 2017, 3:19 pm

The numbers confuse me because surely there's got to be a 'last' number somewhere. So I think of all the trillions of existing things in the whole universe and beyond, including air, and I have read that one human is made up of a billion atoms alone, so the whole number of all the atoms must be about a thousand digits long. Surely you would run out of words if you counted all those.

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naturalplastic
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09 Oct 2017, 6:58 pm

Joe90 wrote:
The numbers confuse me because surely there's got to be a 'last' number somewhere. So I think of all the trillions of existing things in the whole universe and beyond, including air, and I have read that one human is made up of a billion atoms alone, so the whole number of all the atoms must be about a thousand digits long. Surely you would run out of words if you counted all those.

You have heard of the concept of "infinity" haven't you?

There is no "last number".

You can count all that you want. Calculate the U.S. National Debt in Italian lira, measure the distances between the stars in inches (or in millimeters), count all of the atoms in the Universe, or whatever. You will never "use up" all of the numbers!

Tollorin
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11 Oct 2017, 9:00 pm

Joe90 wrote:
The numbers confuse me because surely there's got to be a 'last' number somewhere. So I think of all the trillions of existing things in the whole universe and beyond, including air, and I have read that one human is made up of a billion atoms alone, so the whole number of all the atoms must be about a thousand digits long. Surely you would run out of words if you counted all those.

From a big number you can always add digits to make it bigger, so there is no limit.

Here is the biggest number I heard about:
https://en.wikipedia.org/wiki/Graham's_number

naturalplastic
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12 Oct 2017, 3:38 am

Yes.

Ten raised to the power of 100 is a "googol".

The number of atoms in the universe is 10 to the 85th. So a googol is a thousand trillion times the number of all atoms in the Universe.

Ten raised to the power of the googol is called a "googolplex".

Maybe I should restate that. A one followed followed by 100 zeroes is a googol. A one followed by a googol number of zeroes is a "googolplex".

But even a googolplex is diddly squat compared to graham's number.

eric76
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01 Nov 2017, 5:04 am

One interesting thing to consider is how many digits of pi are actually useful in the real universe before any discrepancies between actual and calculated lengths would be less than the Planck length?

naturalplastic
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05 Nov 2017, 7:58 am

^
That would depend upon the size of the circle.

Youre combining peaches and pears: a measurement of relative distance (pi), with a unit of measuring absolute distance (the Planck Length). Pi is how many times the diameter of a circle divides into the circumference of the same circle (and because the two things don't divide into each other evenly you get the leftover amount with the endless digits).Its a ratio. And not an absolute size of anything.

In contrast the Planck Length is like a centimeter, or an inch. Its an absolute amount. The same length.

If the circle in question is the size of the orbit of Saturn around the Sun it gonna be a lot more centimeters, and a lot more Planck lengths, than a car tire, or a circle than can fit into this post. So a small circle is going to hit the Planck lengths in fewer divisions than a big one. So the end point in the endless string of digits in Pi is going to vary by the size of the circle- if youre gonna drag the Planck length into the topic.

Timdil
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17 Nov 2017, 4:31 am

52! Is the number of possible combinations which is ~8*10^67 (8 with 67 zeros)

Michael829
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17 Nov 2017, 3:36 pm

Joe90 wrote:
1. I've often read that a pack of cards can be shuffled into a different order every time. But what really confused me once was when I read something that said that there are more different ways to shuffle a pack of cards than there are grains of sand on every beach on the planet.

It's true.

Quote:

I thought a pack of cards can be shuffled in perhaps a few million different ways, if that

It's a lot more than a few million.

The number would be written with 8 with 67 zeros after it.

That number is often written as 8E67. ...meaning 8 times 10 to the 67th power.

Let's just make a rough estimate of the number of grains of sand per foot of beach:

Say a grain of sand occupies a cube of space that's a half-millimeter across. Say that a typical beach extends 200 feet up from the ocean, and that the sand has an average depth of 4 feet.

So: Given those guesses, how many miles of beach would be needed, in order to have 8E67 grains of sand?

The answer is 1E53. 10 to the 53rd power. 1 followed by53 zeros. That's how many miles of beach would be needed.

That number of miles of beach needed could be worded as a trillion trillion trillion trillion, times a hundred-thousand. A lightyear is only about 6 trillion miles.

Now, you might say maybe the sand-grains are smaller, or that the beaches are broader, with deeper sand. But 1E53 is such a large number that you could change each of those numbers by a factor of a thousand, or a million, and the result would still point to the same conclusion:

The number of grains of sand on all the world's beaches can't be as large as the number of ways that a 52-card deck can be shuffled.

Quote:

, but more than the amount of tiny grains of sand there is on every beach??? This Earth is ENORMOUS, much bigger than I can imagine, and there are many beaches that go on for miles, and the number of grains of sand feels as immense as something like a million trillion zillion.

If the Earth's number of miles of beach were 100 times the Earth's circumference, then, based on my assumptions of half-millimeter grains, beaches 200 feet broad, with 4 feet deep sand, then the total number of sand-grains would be about 2E21.

That could be said as Two billion trillion.

That's a lot less than the number of ways a 52-card deck could be shuffled.

Again: Making the grains smaller, the beaches broader, and the sand deeper won't make enough difference to change the conclusion.

Michael829

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Michael829
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17 Nov 2017, 6:25 pm

I'd better clarify that, when I said "billion" and "trillion", I mean the American meanings for those words.

In American usage, a billion is a thousand millions, and a trillion is a million millions.
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So, the billion that I referred to would be called a thousand million, or a milliard, in official or traditional British usage.

And the trillion that I referred to would be called a billion, in official or traditional British usage.

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The U.S. uses the old French system for naming those numbers. But, some time after the U.S. adopted that, France adopted the British and German system, leaving us with a number-naming system different from the one shared by England, France and Germany.

But, to complicate it further, I've read that the American number-names are becoming increasingly used in England.

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I said that, by my estimates that I stated, there are about a Billion Trillion grains of sand on the world's beaches.

If the sand-grains each only occupy the volume of a cube 1/7 of a millimeter to a side, and if the beaches really average 1000 feet broad, and if the sand really averages 40 feet deep, then that would bring that number up to something more on the order of a trillion trillion.

But my estimate for the total length of the world's beaches--100 times the Earth's circumference, is almost surely a big exaggeration.

So my guess for the number of grains of sand on the world's beaches would be somewhere between a billion trillion, and a trillion trillion.

Michael Ossipoff

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17 Nov 2017, 6:51 pm

I've just checked on the Internet, and found two estimates of the number of grains of sand:

One estimate says that there are only 1/100 of a billion trillion grains of sand on the Earth's beaches and deserts combined.

The other estimate, from Scientific American magazine, says that the number of sand-grains on the Earth's beaches is about 5 times a billion trillion.

The Scientific American estimate uses the same grain-size estimate that I used, and almost the same beach width that I used (82% of it), but a higher estimate for the average depth, and a total beach length of 14 Earth circumferences instead of 100 (which I'd considered a high estimate anyway).

Michael Ossipoff

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kokopelli
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30 Nov 2017, 6:09 pm

Joe90 wrote:
The numbers confuse me because surely there's got to be a 'last' number somewhere. So I think of all the trillions of existing things in the whole universe and beyond, including air, and I have read that one human is made up of a billion atoms alone, so the whole number of all the atoms must be about a thousand digits long. Surely you would run out of words if you counted all those.

Let x be the "last number". Does that mean that x+1 is not a number?

Of course not. If x is a number, then x+1 is a number. If there is such a thing as a last number, then there is another number after that.

Last edited by kokopelli on 30 Nov 2017, 11:03 pm, edited 1 time in total.