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## #1 2006-09-28 16:22:29

ganesh
Registered: 2005-06-28
Posts: 28,910

### Number Giants

What is the biggest number you have heard of?
A billion, a trillion, a centillion?
A googol? A googolplex?
These numbers dwarf in comparison to Moser's.
And Moser's is no match for Graham's Number,
the number in the Guiness book of Records for
being the highest number to be used in a
mathematical proof.
But to understand these, you would have to
first know what is iteration in mathematics,
tetration, hypertetration etc.
The polygon notation, knuth's up-arrow notation etc. are too difficult to comprehend for a beginner. Search for these in any search engine, and look at the results. Don't worry if you cannot follow them.
It takes hours, days, months to understand them.
Even knuth's up-arrow notation is insufficient to express Graham's Number.
The best way to express Graham's Number is using JOHN CONWAY'S chained arrow notation.
It can be said using Conway's notation that Graham's number liest between two numbers, not exactly defining Graham's Number.
However, Graham's Number isn't acceptable to some mathematicians, saying his proof may well be untrue. Graham's Number is the outcome of Ramsay theory, and a pity, Ramsay is no more!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2006-09-28 16:56:33

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Number Giants

Is it larger than 10!!!!!!!!!? (each being a factorial)

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2006-09-28 18:04:17

MathsIsFun
Registered: 2005-01-21
Posts: 7,694

### Re: Number Giants

Ye Gods, that factorial is huge!

Up-arrow notation is also diabolically large.

What an interesting comparison - which one packs the most punch?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #4 2006-09-29 00:13:08

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

Think of 3^3^3^3^3
This number is just about the start of Graham's number which extends to 63 layers.
Please use a search engine and search for Graham's number.
Factorials are nothing compared to iterations.
I shall try to explain in a simple way.
3+3 is the first stage.
The next is 3x3.
The next is 3^3.
What next?
The stage is called tetration. (I think)
3^^3.
and then would come
3^^^3.
The number you have given is less than the first level of Graham's Number.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #5 2006-09-29 00:24:26

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

Oh, I forgot to tell you.
Ricky, your number dwarfs in comparison to Moser's, leave alone Graham's Number.
It has been proved that Graham's Number is greater than Moser's.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #6 2006-09-29 00:47:41

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Number Giants

Ricky wrote:

Is it larger than 10!!!!!!!!!? (each being a factorial)

I thought that adding extra factorials made the result smaller because some of the multipliers are skipped.

For example, 5!! is 5*3*1 = 15.
So 10!!!!!!!!! would be 10*2, which is 20.

Why did the vector cross the road?
It wanted to be normal.

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## #7 2006-09-29 00:52:49

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

No, mathysperson.
10!! would be the factorial of 3628800, which is 10!.
Anyway, in all probability, the number given by Ricky is less than a Googolplex.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #8 2006-09-29 05:06:13

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Number Giants

Ganesh, 10!!, as you said, is 3628800!.  That's 3628800*3628799*3628798*368797....

That's only the first four terms, and it's already 173,400,926,419,517,324,430,067,200

The first 4 terms of a 3,628,800 factorial is already 27 digits long, and that's only the 2nd factorial of 9.  Imagine just taking the factorial of the number above.

I tried to get mathimatica to do (10!)! but it keeps on crashing.  Anyone else have any luck?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #9 2006-10-09 20:25:41

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

Ricky,
Your number is much much smaller than 10^10^10^10^10^10^10^10.
Use a search engine and search for Graham's Number
You'd realize how smaller your number is.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #10 2006-10-10 15:48:43

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

Ricky,
I am sorry I was wrong in post # 9.
I didn't use Arithmetica or any computer,
just James Stirling formula which gives very close approximation for higher values of factorials.
The formula is n!=sqrt2*pi*n (n/e)^n
Therefore if the number of factorials is 10, your number may be greater.
However, I am sure it is lesser than 10^10^10^....20 times ^10.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #11 2007-05-21 05:16:06

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

I actually heard (or read) that the number of digits in Graham's Number exceeds the number of electrons in the observable universe.

Any idea whether this is actually true?

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #12 2007-05-21 05:20:16

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

By the way, in response to your question,"What an interesting comparison - which one packs the most punch?" concerning factorials and up-arrow notation, I'm pretty sure that up-arrow notation is the whopper, though I can't say for sure.

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #13 2007-05-21 05:31:44

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

It also seems that this phenomenal number is just the upper bound answer to a problem, the determined lower bound being about 6.

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #14 2007-05-21 05:36:00

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Number Giants

Oh, definitely. Even just a googol is more than the amount of electrons in the universe, and a googol is absolutely titchy compared with some of the other numbers mentioned in this thread.

I would also agree that Up Arrow notation makes bigger numbers than factorials, but I'm also not entirely sure.

Welcome to the forum, by the way. Very interesting name you have there.

Why did the vector cross the road?
It wanted to be normal.

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## #15 2007-05-21 06:26:49

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

Thanks. I spend most of my time trying to solve thinking puzzles, so the name's appropriate.

The entire idea of Graham's number is absolutely mind-boggling! just the first two steps (out of 64) leave me pretty much gone.

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #16 2007-05-21 07:04:07

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

### Re: Number Giants

Yeah , it's pretty big ,

Numbers are the essence of the Universe

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## #17 2007-05-21 11:37:49

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

### Re: Number Giants

I don't know if I said this here, but if all the matter in the universe were converted into ink, it wouldn't be enough to fully write out Graham's number. It's pretty big.

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## #18 2007-05-21 19:56:30

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

Let me see if I can explain the first steps of the number here:

1 Take the first step : 3^^^^3
2 Calculate it (it's bigger than the number of atoms in the observable universe)
3 Now go back to step 1, this time using the number you just found of up-arrows between the 3s.

Now repeat this entire process 64 times.

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #19 2007-05-22 17:39:57

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

By the way, don't actually try to calculate the number. It would overload every single computer in the world if they tried to just calculate the first step together...

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #20 2007-05-22 19:45:15

mikau
Member
Registered: 2005-08-22
Posts: 1,504

### Re: Number Giants

can someone tell me what 3^^3 is?

Is it 3^(3^3) ???

A logarithm is just a misspelled algorithm.

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## #21 2007-05-22 23:18:12

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

If I got the system of Knuth's up-arrow notation right, then I'm pretty sure that, yes,
3^^3 = 3^(3^3).
Therefore, 3^^^3 = 3^^(3^^3) = 3^^(3^(3^3))
And 3^^^^3 = 3^^^(3^^^3) = 3^^^(3^^(3^^3)) = 3^^^(3^^(3^(3^3)))

These last two are HUGE numbers.

The last one is the first step of G (Graham's number).

Last edited by Laterally Speaking (2007-06-05 23:55:04)

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #22 2007-06-06 00:01:35

Laterally Speaking
Real Member
Registered: 2007-05-21
Posts: 356

### Re: Number Giants

Even Moser's number is pretty huge (I'm not sure exactly what it is). It obviously involves Moser's polygon notation, and I think it's something like this: a million inside a million-sided polygon, which is inside that number of that number-sided polygons.

If you don't quite get this, please tell me...

I feel like Euler Avenue is a deserted wasteland... nobody's posted for almost a month here.

Last edited by Laterally Speaking (2007-06-06 00:01:55)

"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

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## #23 2007-06-06 02:06:03

ganesh
Registered: 2005-06-28
Posts: 28,910

### Re: Number Giants

Laterally Speaking,
It has been shown that (proved that) Graham's number is far greater than Moser's.
The polygon notation generates number giants quickly, but Moser's is no match for Graham's Number.
PS:- This is no wasteland. Only, people don't post here too often

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #24 2007-06-06 06:48:24

ben
Member
Registered: 2006-07-12
Posts: 106

### Re: Number Giants

Laterally Speaking wrote:

I feel like Euler Avenue is a deserted wasteland... nobody's posted for almost a month here.

Then the solution is simple - start an interesting and challenging thread of your own. I can offer a host of these, if you want.....

Or are you just a consumer, rather than a producer?

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## #25 2007-06-06 07:09:15

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Number Giants

In one of the other threads, it was said that the number of possible states of the Universe was found by the amount of particles in the Universe multiplied by the amount of time that the Universe has existed for. I'm not sure that's entirely right, but even if a better formula was found then it would still only be an estimate because quantumness wrecks everything.

Anyway, that sounds like a diabolically huge number, so it got me wondering where it fits in relating to Moser's and Graham's numbers.

Why did the vector cross the road?
It wanted to be normal.

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