Biggest value

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

Bravemuta
Posts: 54
Joined: Thu Oct 11, 2007 3:48 pm UTC

Biggest value

Postby Bravemuta » Tue Apr 28, 2009 10:22 am UTC

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:
Berry's Paradox
Richard's Paradox

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?

User avatar
dedalus
Posts: 1169
Joined: Fri Apr 24, 2009 12:16 pm UTC
Location: Dark Side of the Moon.

Re: Biggest value

Postby dedalus » Tue Apr 28, 2009 10:45 am UTC

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).
doogly wrote:Oh yea, obviously they wouldn't know Griffiths from Sakurai if I were throwing them at them.

Happyjon
Posts: 7
Joined: Tue Apr 07, 2009 6:13 am UTC

Re: Biggest value

Postby Happyjon » Tue Apr 28, 2009 11:14 am UTC

There was a blag about this awhile ago.

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

User avatar
dedalus
Posts: 1169
Joined: Fri Apr 24, 2009 12:16 pm UTC
Location: Dark Side of the Moon.

Re: Biggest value

Postby dedalus » Tue Apr 28, 2009 11:41 am UTC

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.
doogly wrote:Oh yea, obviously they wouldn't know Griffiths from Sakurai if I were throwing them at them.

Bravemuta
Posts: 54
Joined: Thu Oct 11, 2007 3:48 pm UTC

Re: Biggest value

Postby Bravemuta » Tue Apr 28, 2009 12:00 pm UTC

I was thinking about that. I think [math]9 \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 9[/math] is the biggest you can get, using that notation.

Notch
Posts: 318
Joined: Tue Dec 12, 2006 5:52 pm UTC
Location: Stockholm, Sweden
Contact:

Re: Biggest value

Postby Notch » Tue Apr 28, 2009 1:17 pm UTC

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

[math]9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9\rightarrow9[/math]

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

GoC
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Tue Apr 28, 2009 3:15 pm UTC

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 :P )
Belial wrote:I'm just being a dick. It happens.

User avatar
imatrendytotebag
Posts: 152
Joined: Thu Nov 29, 2007 1:16 am UTC

Re: Biggest value

Postby imatrendytotebag » Wed Apr 29, 2009 10:44 pm UTC

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?
Hey baby, I'm proving love at nth sight by induction and you're my base case.

GoC
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Thu Apr 30, 2009 8:16 pm UTC

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.
Belial wrote:I'm just being a dick. It happens.

aduubian
Posts: 25
Joined: Thu Apr 16, 2009 10:32 pm UTC

Re: Biggest value

Postby aduubian » Fri May 01, 2009 12:08 am UTC

how about 9^9!^9!^9!^999!
there is probably a better way to arrange those but I don't care to calculate it

Notch
Posts: 318
Joined: Tue Dec 12, 2006 5:52 pm UTC
Location: Stockholm, Sweden
Contact:

Re: Biggest value

Postby Notch » Fri May 01, 2009 2:37 pm UTC

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. :D

Nitrodon
Posts: 497
Joined: Wed Dec 19, 2007 5:11 pm UTC

Re: Biggest value

Postby Nitrodon » Fri May 01, 2009 2:52 pm UTC

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

Bravemuta
Posts: 54
Joined: Thu Oct 11, 2007 3:48 pm UTC

Re: Biggest value

Postby Bravemuta » Fri May 01, 2009 3:11 pm UTC

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
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Fri May 01, 2009 3:59 pm UTC

How about defining your number in the framework of ZFC set theory and having to prove it is finite in the same?
Belial wrote:I'm just being a dick. It happens.

Notch
Posts: 318
Joined: Tue Dec 12, 2006 5:52 pm UTC
Location: Stockholm, Sweden
Contact:

Re: Biggest value

Postby Notch » Fri May 01, 2009 5:02 pm UTC

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
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Fri May 01, 2009 8:54 pm UTC

With respect to this thread I think:

BB(BB(BB(9!!)))
or
BBBBBBBBBBBB(9)
or
[math]BB_{BB_{BB_{BB_{BB}}}}(9!!)[/math]
Depending on notation allowed.
Belial wrote:I'm just being a dick. It happens.

User avatar
MHD
Posts: 630
Joined: Fri Mar 20, 2009 8:21 pm UTC
Location: Denmark

Re: Biggest value

Postby MHD » Fri May 15, 2009 3:39 pm UTC

*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.)
EvanED wrote:be aware that when most people say "regular expression" they really mean "something that is almost, but not quite, entirely unlike a regular expression"

User avatar
quadmaster
Posts: 192
Joined: Mon Apr 27, 2009 12:39 am UTC

Re: Biggest value

Postby quadmaster » Sat May 30, 2009 3:43 pm UTC

biggest # that can be expressed in 51 charachters+1
I... I didn't do it.
<- he did it, I swear

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

Re: Biggest value

Postby skeptical scientist » Sat May 30, 2009 8:04 pm UTC

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.
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

qetzal
Posts: 862
Joined: Thu May 01, 2008 12:54 pm UTC

Re: Biggest value

Postby qetzal » Sat May 30, 2009 8:30 pm UTC

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.

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

Re: Biggest value

Postby skeptical scientist » Sun May 31, 2009 2:10 am UTC

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.

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

qetzal
Posts: 862
Joined: Thu May 01, 2008 12:54 pm UTC

Re: Biggest value

Postby qetzal » Sun May 31, 2009 4:18 am UTC

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
Posts: 286
Joined: Tue Aug 22, 2006 10:35 pm UTC
Contact:

Re: Biggest value

Postby GreedyAlgorithm » Sun May 31, 2009 5:09 am UTC

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).
GENERATION 1-i: The first time you see this, copy it into your sig on any forum. Square it, and then add i to the generation.

GoC
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Sun May 31, 2009 11:49 pm UTC

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?
Belial wrote:I'm just being a dick. It happens.

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

Re: Biggest value

Postby skeptical scientist » Mon Jun 01, 2009 4:35 am UTC

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
Posts: 336
Joined: Mon Nov 24, 2008 10:35 pm UTC

Re: Biggest value

Postby GoC » Mon Jun 01, 2009 3:04 pm UTC

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.
Belial wrote:I'm just being a dick. It happens.

Agent_Irons
Posts: 213
Joined: Wed Sep 10, 2008 3:54 am UTC

Re: Biggest value

Postby Agent_Irons » Tue Jun 23, 2009 6:02 pm UTC

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?

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

Re: Biggest value

Postby skeptical scientist » Tue Jun 23, 2009 7:21 pm UTC

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


Return to “Logic Puzzles”

Who is online

Users browsing this forum: No registered users and 9 guests