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

ChrisVer · Mar 27, 2016

micromass said:

Not a real number.

really?

micromass · Mar 27, 2016

ChrisVer said:

really?

What real number would be the answer?

nolxiii · Mar 27, 2016

Code:

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 · Mar 27, 2016

If the subscript notation used in describing Gram's number is considered standard much larger numbers then Grams should be easily constructed thus...
g_{g_{n recursive subscripts_n}}
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 · Mar 27, 2016

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 · Mar 27, 2016

micromass said:

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 · Mar 27, 2016

googol, period

Cruz Martinez · Mar 27, 2016

TheQuietOne said:

googol, period

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

Cruz Martinez · Mar 27, 2016

ChrisVer said:

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 · Mar 27, 2016

TheQuietOne said:

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 · Mar 27, 2016

2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{^{2^{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]

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{3^²}}}}}}}}}}}}}}}}}}

axmls · Mar 27, 2016

OrangeDog said:

2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{^{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]

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{3^[SUP]2}}}}}}}}}}}}}}}}}}[/SUP]

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 · Mar 27, 2016

OrangeDog said:

2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{^{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]

divided by .0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{3^[SUP]2}}}}}}}}}}}}}}}}}}[/SUP]

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 · Mar 27, 2016

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

micromass · Mar 27, 2016

OrangeDog said:

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 · Mar 27, 2016

micromass · Mar 27, 2016

OrangeDog said:

no you cant

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

OrangeDog · Mar 27, 2016

more lies

micromass · Mar 27, 2016

Code:

[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 · Mar 27, 2016

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

OrangeDog · Mar 27, 2016

I guess someone doesn't like trolls.

micromass · Mar 27, 2016

I do actually, especially when they fail.

jfizzix · Mar 27, 2016

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

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

micromass · Mar 27, 2016

jfizzix said:

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 · Mar 27, 2016

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

micromass · Mar 27, 2016

OrangeDog said:

Micromass takes Graham very seriously.

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

micromass · Mar 27, 2016

jfizzix said:

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 · Mar 27, 2016

micromass said:

Can't beat Graham.

Graham^Graham

axmls · Mar 27, 2016

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 · Mar 27, 2016

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!