What is the largest real number one can write within 200 characters?

[micromass]

∞-1

In which number system are you working when you say $\infty$ ?

 

[nolxiii]

11

also, to clarify, this number is not written in base ten but in some much larger base size

 

[micromass]

also, to clarify, this number is not written in base ten but in some much larger base size

Then you need to specify the base in your description.

 

[rootone]

In which number system are you working when you say $\infty$ ?

Lets say binary for simplicity, although I do realize that that a computer memory containing the number would require an infinite number of bits.

 

[nolxiii]

no one else specified their base size

 

[micromass]

no one else specified their base size

Do we really need to specify that 100% of humans nowadays work standard in base 10?

 

[micromass]

Lets say binary for simplicity, although I do realize that that a computer memory containing the number would require an infinite number of bits.

But $\infty$ is not a real number. So what kind of number is it? How is it defined?

 

[collinsmark]

For those interested in Graham's number, here is a video about it.

Also, Knuth's[/PLAIN] up-arrow notation is introduced in the video.

 

[ChrisVer]

what would happen in case I write:

lim_[n ->0] 1/n^2

??

 

[micromass]

what would happen in case I write:

lim_[n ->0] 1/n^2

??

Not a real number.

 

[ChrisVer]

Not a real number.

really?

 

[micromass]

really?

What real number would be the answer?

 

[nolxiii]

let G = graham's #
let ☺ mean G ↑'s in knuth notation
let ☻ mean G ☺'s
base G
13☻☻☻☻☻☻☻☻☻☻☻☻☻☻☻☻☻☻☻13

edit: well i guess we already kinda went there on page 1, but i'll keep my notation

 

[mrspeedybob]

If the subscript notation used in describing Gram's number is considered standard much larger numbers then Grams should be easily constructed thus...
g_{gn recursive subscriptsn}
Now you are left with describing the largest possible n with the remaining of the 200 characters.
This is just 1 example though of a rapidly increasing function recursed a large number of times, it may not be the best one to use.

More broadly, I think this will essentially come down to the most clever method of unambiguously describing 2 things...
1. The most rapidly increasing function
2. Vast numbers of recursions.
I'm sure someone has better ideas on how to approach both of those problems then I do, though they seem like they might be the same problem.

 

[micromass]

Time to come clean. I made this thread because I read a very interesting article about big numbers. It seems in this thread, many found their way to Graham's number and Ackermann function. But there is a function which increase even faster than those: the busy beaver function. Check it out:

http://www.scottaaronson.com/writings/bignumbers.html

 

[Dembadon]

Time to come clean. I made this thread because I read a very interesting article about big numbers. It seems in this thread, many found their way to Graham's number and Ackermann function. But there is a function which increase even faster than those: the busy beaver function. Check it out:

http://www.scottaaronson.com/writings/bignumbers.html

I really liked this part:

Could early intervention mitigate our big number phobia? What if second-grade math teachers took an hour-long hiatus from stultifying busywork to ask their students, "How do you name really, really big numbers?" And then told them about exponentials and stacked exponentials, tetration and the Ackermann sequence, maybe even Busy Beavers: a cornucopia of numbers vaster than any they’d ever conceived, and ideas stretching the bounds of their imaginations.

So it seems a very large number can use the BB function with BB(G) recursions? I know there is a more elegant and rigorous way to write it, but I don't think I'm clever enough.

 

[TheQuietOne]

googol, period

 

[Cruz Martinez]

googol, period

Even a googolplex is very very very small compared to graham's number.

 

[Cruz Martinez]

really?

yeah, suppose $\lim_{x \rightarrow 0} 1/x^2=b$. A theorem says that $\lim_{x \rightarrow a} f(x) = c$ if and only if for every sequence $x_n$ which converges to $a$, the sequence $f(x_n)$ converges to $c$. So take the sequence $\{1/n\}_{n \in \mathbb{N}}$, this sequence converges to 0, but $f(1/n)=n^2$ for $f(x) = 1/x^2$. This sequence does not converge to any real number, so it won't converge to $b$.

 

[axmls]

googol, period

Graham's number is so much bigger than googol, that it is impossible to write down Graham's number in exponential form if you could write a trillion numbers on every atom in the universe and you had one hundred trillion universes. Meanwhile, googol is just $10^{100}$.

 

[OrangeDog]

2^{22222222222222222222222222222222222222}

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2222222222222222232}

 

[axmls]

2^{22222222222222222222222222222222222222}

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2222222222222222232}

Not even close to Graham's number alone. If you can write it using exponents, then it's much smaller than Graham's number. Actually, unfathomably smaller than Graham's number.

 

[micromass]

2^{22222222222222222222222222222222222222}

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2222222222222222232}

Not only does it vastly go over the character limit, it is also vastly smaller than Graham's number. No matter how many exponents you put in, there's not enough space and time in the universe for the exponent tower to get anywhere near Graham.

 

[OrangeDog]

You can't actually read my text, so how do you know that each one of those tiny exponents isn't grahams number?

 

[micromass]

You can't actually read my text, so how do you know that each one of those tiny exponents isn't grahams number?

If you hit "QUOTE", you can see what you wrote.

 

[OrangeDog]

Lies

 

[micromass]

no you cant

...Yes... you can...

 

[OrangeDog]

more lies

 

[micromass]

[QUOTE="OrangeDog, post: 5423766, member: 584341"]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP][SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP]

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]2[SUP]3[SUP][SUP]2[/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/SUP][/QUOTE]

 

[micromass]

In either case, if you used Graham's number, you needed to specify its usage.

 

[OrangeDog]

I guess someone doesn't like trolls.

 

[micromass]

I do actually, especially when they fail.

 

[jfizzix]

How about, a googolplex "factorialed" a googolplex number of times?

e.g., 3 "factorialed" two times would be (3!)!, or 6! or 720

 

[micromass]

How about, a googolplex "factorialed" a googolplex number of times?

e.g., 3 "factorialed" two times would be (3!)!, or 6! or 720

Can't beat Graham.

 

[jfizzix]

wouldn't it take more than 200 characters to properly explain how Graham's number works?

 

[micromass]

Micromass takes Graham very seriously.

Aren't you in awe at the hugeness of this number??

 

[micromass]

wouldn't it take more than 200 characters to properly explain how Graham's number works?

I guess it would. But I allowed referencing to outside sources for explanations.

 

[rootone]

Can't beat Graham.

Graham^Graham

 

[axmls]

Graham's number is the most terrifying number I've seen, and that's why I love it. No amount of googolplexes anyone strings together will even come close to the might that is Graham's number. If $G$ is Graham's number, every number theory textbook ought to start out by saying "Let $\infty = G$..."

 

[jfizzix]

Maybe we can tighten up the competition to see what's the biggest number we can write with five characters without allowing outside references

e.g.,

9^99!

 

[micromass]

Maybe we can tighten up the competition to see what's the biggest number we can write with five characters without allowing outside references

e.g.,

9^99!

Then it should be stated clearly what is allowed and what not.

 

[jfizzix]

Then it should be stated clearly what is allowed and what not.

I'd use the same rules as in your original post, but with a 5 character limit instead

 

[micromass]

I'd use the same rules as in your original post, but with a 5 character limit instead

Sure, but what operations do you consider standard? Obviously numbers 0-9 are allowed (base 10), +, -, *, /, !, what else?

 

[jfizzix]

Sure, but what operations do you consider standard? Obviously numbers 0-9 are allowed (base 10), +, -, *, /, !, what else?

^

 

[micromass]

^

9^9! seems to be the largest I can think of then

 

[jfizzix]

9^9! seems to be the largest I can think of then

I had to look up what !, was, but from Wolfram Mathworld, 9! = 9*7*5*3*1, which would be less than 9!

 

[micromass]

I had to look up what !, was, but from Wolfram Mathworld, 9! = 9*7*5*3*1, which would be less than 9!

I just meant the factorial of a factorial really. I severely dislike the (admittedly) standard notation 9! = 9*7*5*3*1.

 

[mrspeedybob]

g_gggG

 

[axmls]

g_gggG

Okay now that is large.

 

[jfizzix]

Graham's number is too dam high