Number Giants

Jai Ganesh · 2006-09-28 16:22:29

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!

Ricky · 2006-09-28 16:56:33

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

MathsIsFun · 2006-09-28 18:04:17

Ye Gods, that factorial is huge!

Up-arrow notation is also diabolically large.

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

Jai Ganesh · 2006-09-29 00:13:08

Ricky, your number is peanuts.
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.

Jai Ganesh · 2006-09-29 00:24:26

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.

mathsyperson · 2006-09-29 00:47:41

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.

Jai Ganesh · 2006-09-29 00:52:49

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.

Ricky · 2006-09-29 05:06:13

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?

Jai Ganesh · 2006-10-09 20:25:41

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.

Jai Ganesh · 2006-10-10 15:48:43

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.

Laterally Speaking · 2007-05-21 05:16:06

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?

Laterally Speaking · 2007-05-21 05:20:16

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.

Laterally Speaking · 2007-05-21 05:31:44

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

mathsyperson · 2007-05-21 05:36:00

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.

Laterally Speaking · 2007-05-21 06:26:49

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.

Stanley_Marsh · 2007-05-21 07:04:07

Yeah , it's pretty big ,

Zhylliolom · 2007-05-21 11:37:49

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.

Laterally Speaking · 2007-05-21 19:56:30

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.

Laterally Speaking · 2007-05-22 17:39:57

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...

mikau · 2007-05-22 19:45:15

can someone tell me what 3^^3 is?

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

Laterally Speaking · 2007-05-22 23:18:12

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)

Laterally Speaking · 2007-06-06 00:01:35

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)

Jai Ganesh · 2007-06-06 02:06:03

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.
For more onMoser's, click here.
PS:- This is no wasteland. Only, people don't post here too often

ben · 2007-06-06 06:48:24

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?

mathsyperson · 2007-06-06 07:09:15

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.

Math Is Fun Forum

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

Number Giants

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

Re: Number Giants

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

Re: Number Giants

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

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

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

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

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

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

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

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