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

[ShayanJ]

The product of the factorials of (the absolute values of) all the numbers that will be posted, then. :D

So I'll post ## \infty ##!

 

[Ben Niehoff]

A(G,G),

where A is the Ackermann function and G is Graham's number.

 

[mrspeedybob]

A(G,G),

where A is the Ackermann function and G is Graham's number.

Why stop there?

A(A(G,G),A(G,G))

is only 16 characters. If you include definitions...

A=Ackermann function
G=Gram's number
A(A(G,G),A(G,G))

Now it's still only up to 51, so, by expanding on the same idea and being slightly more concise with the explanation you can get...

Ackermann function
Gram's number
G↑↑A(A(A(A(A(A(G,G),A(G,G)),A(A(G,G),A(G,G))),A(A(A(G,G),A(G,G)),A(A(G,G),A(G,G)))),A(A(A(A(G,G),A(G,G)),A(A(G,G),A(G,G))),A(A(A(G,G),A(G,G)),A(A(G,G),A(G,G))))),A(G,G))

Which by my count comes to 200 characters, including spaces.

 

[mrspeedybob]

Or...

Ackermann function
Gram's number
B(n,x)=A performed recursively n times with arguments x. I.E. B(2,3)=A(A(3,3),A(3,3))
C(n,x)=B performed recursively n times with arguments x.
C(C(G,G),C(G,G))

 

[ChrisVer]

f(x)=10^x!
f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(9)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

 

[micromass]

f(x)=10^x!
f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(9)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

This is still dwarfed by Grahams number though...

 

[ChrisVer]

This is still dwarfed by Grahams number though...

maybe... I haven't seen that number...
But if it's big I could try to put it in the f(f(...f(f(G))...)), and in place of 10 in 10^x! : G^x!

 

[mrspeedybob]

maybe... I haven't seen that number...

Fair warning. It'll break your brain.
http://waitbutwhy.com/2014/11/1000000-grahams-number.html

 

[Dembadon]

[itex]G^G[/itex]↑^{[itex]G^G[/itex]}[itex]G^G[/itex], where G is Graham's number.

 

[micromass]

[itex]G^G[/itex]↑^{[itex]G^G[/itex]}[itex]G^G[/itex], where G is Graham's number.

I'm afraid that doesn't beat the hugeness of the Ackerman function. https://en.wikipedia.org/wiki/Ackermann_function

 

[Dembadon]

I'm afraid that doesn't beat the hugeness of the Ackerman function. https://en.wikipedia.org/wiki/Ackermann_function

Indeed. So I looked at mrspeedybob's post and cut a few characters out so I could fit in exponents for the arguments of C. Microsoft Word says it's 200 characters exactly including spaces:

Ackermann func.
Graham's #
B(n,x)=A performed recursively n times with args x. I.E. B(2,3)=A(A(3,3),A(3,3))
C(n,x)=B performed recursively n times with args x.
C(C(G^C(G,G),G^C(G,G)),C(G^C(G,G),G^C(G,G)))

 

[nolxiii]

11

 

[CynicusRex]

0

As I think most of the universe is empty space, 0 pretty much sums it all up.
This is probably incorrect on many levels, but I like the idea.

 

[nolxiii]

-1/12

 

[rootone]

∞-1

 

[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}##.