## Biggest value

A forum for good logic/math puzzles.

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Bravemuta
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### Biggest value

How can you find out which is the biggest value you can define in a given number of characters, or less (say 15). Obviously, there's 999...99 15 times, but I bet there's a bigger number than that, like. 9^9^9^9^9^...^9 (15 characters, 8 "9"s and 7 "^"s). You can even use letters/words, as long as it's a character, like . This is linked to these two:

I think, first of all, you need to define a system and see what is permitted to write and what it's not. Otherwise you can say "&", where & is an arbitrarily high value, defined however you like.

Any thoughts on it?

dedalus
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### Re: Biggest value

1/0.
Beat that. =P
Check out the definitions for http://en.wikipedia.org/wiki/Graham's_number. With notation like that you can define numbers that are absolutely massive.

I think it's better if you define the set of characters, but then it generally becomes easy; take the operator with the largest result, apply that n number of times to the digit 9. If there's any digits left over, e.g. 16 characters, attach an extra 9 to the end (9^9^9^..^99 if we're talking about the indice operator).
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Happyjon
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### Re: Biggest value

http://blag.xkcd.com/2007/03/14/large-numbers/

dedalus
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### Re: Biggest value

http://en.wikipedia.org/wiki/Knuth's_up-arrow_notation << check that out. If you recursively applied that even twice you'd be well beyond the scale of any number that's actually been used by man.
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Bravemuta
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### Re: Biggest value

I was thinking about that. I think $9 \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 9$ is the biggest you can get, using that notation.

Notch
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### Re: Biggest value

Conway chained arrow notation let's us go even bigger.

$9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9$

Isn't there already a thread on this in forum games or somewhere like that?

GoC
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### Re: Biggest value

Here: http://echochamber.me/viewtopic.php?f=14&t=7469

Current leader is itaibn (though I keep claiming I have a bigger number and will get around to making it eventually )
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imatrendytotebag
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### Re: Biggest value

So... there are a lot of "biggest number" discussions, which usually resort to the ackermann function, the busy beaver function, and crazy recursion involving notation specially made for talking about big numbers, all leading up to "Let e be a number greater than any number posted in this thread," which sparks a flame war which comes to the conclusion that, invariably, all big numbers are nazi propoganda. (http://en.wikipedia.org/wiki/Godwin%27s_law).

In any case, we can basically describe a number arbitrarily big in 15 characters... as long as we can develop the requisite notation using established mathematics.

So what happens if we restrict our "vocabulary" to the following (and do not let ourselves use the characters to define new notation unless specified):

1) Just the digits 0-9 and concatonation? (Obviously 9999....)
2) The above and addition and multiplication? (Still 999....)
3) The above and function notation, separated with semicolons? (ie we can write f(n)=n*n;f(f(f(f(...f(9))...).)
4) The above and iterative notation? (ie if f(n) is a defined function, f^k(n) is understood to be f(f(...(f(n))...), f iterated k times.)

Anyway, these still leave room for creativity without people browsing wikipedia for big-number notation. Also, it gives more well-defined constraints to the problem (and perhaps keeps the numbers small enough that they can be compared).

Thoughts?
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GoC
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### Re: Biggest value

There are three non-trivial levels of restrictions you can apply. The first is that numbers be well defined. The second is that the numbers be finite. The third is that numbers be computable.
Any more restrictions either do nothing or mean there is a trivial way of making biggest numbers.
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### Re: Biggest value

there is probably a better way to arrange those but I don't care to calculate it

Notch
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### Re: Biggest value

I've got a feeling this problem might be very similar to BB in itself, if not actually being the exact same problem in disguise.

For few letters and only allowing simple math, the problem is trivial.
For more letters and more advanced math, the problem is massively difficult.

It might be interesting to try to solve this for some clearly defined space.
For example, with just basic math and one letter, the best I can do is "9". I've got a feeling this is optimal.

Oh, trying to calculate 9!! in calc.exe in windows xp makes it hang.

Nitrodon
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### Re: Biggest value

9!! is about 1.609714400410012621103443611 * 10^1859933. I'm not surprised that Calc couldn't figure it out.

Bravemuta
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### Re: Biggest value

Notch wrote:It might be interesting to try to solve this for some clearly defined space.
For example, with just basic math and one letter, the best I can do is "9". I've got a feeling this is optimal.

What qualifies under basic math? I can say "z" in base 50.

GoC
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### Re: Biggest value

How about defining your number in the framework of ZFC set theory and having to prove it is finite in the same?
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Notch
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### Re: Biggest value

Bravemuta wrote:What qualifies under basic math? I can say "z" in base 50.

Very good question. I was vague there as I have no idea how to define that.

Although, how would you make it clear that it's in base 50? Just a z might just as well be some random constant or variable. "z in base 50" might work, but that's more than just one character.

GoC
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### Re: Biggest value

With respect to this thread I think:

BB(BB(BB(9!!)))
or
BBBBBBBBBBBB(9)
or
$BB_{BB_{BB_{BB_{BB}}}}(9!!)$
Depending on notation allowed.
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MHD
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### Re: Biggest value

*runs off to dig up the bignum thread...*

*Ahem*

The Steinhaus–Moser notation could prove useful for this, as you could theoretically enclose [imath]9 \rightarrow 9 \rightarrow 9 \rightarrow 9 \rightarrow 9 \rightarrow 9 \rightarrow 9[/imath] in two polygons with an arbitarily number of sides... (if a polygon is considered a character.)
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### Re: Biggest value

biggest # that can be expressed in 51 charachters+1
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skeptical scientist
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### Re: Biggest value

GoC wrote:The third is that numbers be computable.

All numbers* are computable. What you mean is that there is a clear algorithm for computing the number from the given description. (For example, BB(100) is computable, but you can't directly translate the description into an algorithm for computing it.)

*Natural numbers. This is also true of integers and rational numbers, but not of course true of arbitrary real numbers.
Last edited by skeptical scientist on Sun May 31, 2009 2:12 am UTC, edited 2 times in total.
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qetzal
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### Re: Biggest value

skeptical scientist wrote:
GoC wrote:The third is that numbers be computable.

All numbers are computable.

I didn't think that was true. E.g., see the discussion on BB numbers here.

skeptical scientist
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### Re: Biggest value

qetzal wrote:
skeptical scientist wrote:
GoC wrote:The third is that numbers be computable.

All numbers are computable.

I didn't think that was true. E.g., see the discussion on BB numbers here.

The busy beaver function is not computable. The number 1 is clearly computable, as is the number 2, as is the number 875476105, and so on. One of these numbers must be BB(100), so BB(100) is computable. This is not what we mean when we say the busy beaver function is incomputable.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

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qetzal
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### Re: Biggest value

In order to say that BB(100) is computable, don't you have to be able to compute the actual value of BB(100)?

GreedyAlgorithm
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### Re: Biggest value

qetzal wrote:In order to say that BB(100) is computable, don't you have to be able to compute the actual value of BB(100)?

No. Just like my claim that the 2^10000th prime is odd, the claim that BB(100) is computable is true regardless of whether I know which integer is the 2^10000th prime or which integer is BB(100).
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GoC
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### Re: Biggest value

skeptical scientist wrote:
GoC wrote:The third is that numbers be computable.

All numbers* are computable. What you mean is that there is a clear algorithm for computing the number from the given description. (For example, BB(100) is computable, but you can't directly translate the description into an algorithm for computing it.)

*Natural numbers. This is also true of integers and rational numbers, but not of course true of arbitrary real numbers.

Indeed. How about any number that can actually be compared to the others in this thread by a skilled mathematician?
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skeptical scientist
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### Re: Biggest value

GoC wrote:Indeed. How about any number that can actually be compared to the others in this thread by a skilled mathematician?

I think typically a skilled mathematician has better things to do than worry which of two numbers posted on a message board is bigger.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

GoC
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### Re: Biggest value

skeptical scientist wrote:
GoC wrote:Indeed. How about any number that can actually be compared to the others in this thread by a skilled mathematician?

I think typically a skilled mathematician has better things to do than worry which of two numbers posted on a message board is bigger.

I'll be a judge then. I'm pretty good with large numbers.
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Agent_Irons
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### Re: Biggest value

GreedyAlgorithm wrote:
qetzal wrote:In order to say that BB(100) is computable, don't you have to be able to compute the actual value of BB(100)?

No. Just like my claim that the 2^10000th prime is odd, the claim that BB(100) is computable is true regardless of whether I know which integer is the 2^10000th prime or which integer is BB(100).

Computability isn't transitive, and integers are things not processes. A formula giving an arbitrary integer is computable, for all integers. a->b; c->b; a!->c. Notation struggles, but hopefully you see what I mean. And we can redefine 'computable' to mean what you are implying, then the term becomes almost meaningless.

Shall we limit the thread to "those numbers which we could conceivably write down using conventional digits on a piece of paper of finite length" to avoid ambiguity?

skeptical scientist
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### Re: Biggest value

Agent_Irons wrote:Computability isn't transitive
I have no idea what you mean. Computability is not a binary relation.
Agent_Irons wrote:Shall we limit the thread to "those numbers which we could conceivably write down using conventional digits on a piece of paper of finite length" to avoid ambiguity?
All numbers can conceivably be written down using conventional digits on a piece of paper of finite length, too. Just start by writing 1, then 2, then 3, and so on, and you will eventually have written the desired number. I really think my first comment is the best way of capturing what we really want:
skeptical scientist wrote:All numbers* are computable. What you mean is that there is a clear algorithm for computing the number from the given description. (For example, BB(100) is computable, but you can't directly translate the description into an algorithm for computing it.)
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson