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

[Ben Niehoff]

In 5 characters, using only standard operations, and obeying the standard order of operations (thus sometimes requiring parentheses!), I think the best you can do is

(9!)!

 

[TheQuietOne]

I don't care about Graham cracker's number! I just like googol!

 

[TheQuietOne]

I agree with jfizzix

 

[andrewkirk]

The following may seem similar to Berry:

The largest number for which any human will ever write a formula or algorithm in the past and future history of the Earth

But, because it doesn't contain the negation of the Berry/Russell version, it is perfectly well-defined and non-paradoxical. It's just that we'd have to wait for the destruction of the Earth for somebody to compute it.

If I put some Knuth arrows at the end of it (there were some characters to spare), it would become self-referential and paradoxical.

 

[DevilsAvocado]

The largest number for which any human will ever write a formula or algorithm in the past and future history of the Earth

Someone could express Rayo's[/PLAIN] number \text{Rayo}(10^{100}) on Mars, and then you lose ...

The smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less.

 

[DevilsAvocado]

Graham's number < googology.wikia.com/wiki/TREE(3) < googology.wikia.com/wiki/SCG(13)

You didn't say anything about links!

BIG FOOT: \text{FOOT}^{10}(10^{100}) is of course much bigger, but I guess it's disqualified ...

 

[ChrisVer]

def FindGraham(s):
    if 'G' in s or 'g' in s:
        return True
    return False

print FindGraham("***")

where *** I put your string, guess now ; the result was false..
by a small modification I even got your character count:
870

Maybe you can fool the eye but not the PC...

 

[nolxiii]

bit.ly/1RFruCp
bit.ly/1OtaqP9
b(x)=busy beaver function
f=BIGFOOT
v(x)=b(f) recursions of b
r(x)=v(f) recursions of v
z(x)=r(f) recursions of r

11 in base z(f)

 

[Isaac0427]

G=Graham's Number
G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G[G]G

 

[micromass]

What does [G] mean?

 

[Isaac0427]

What does [G] mean?

I figured hyperoperation was allowed without explanation as it goes directly with Knuth's notation.

https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation#Generalizations

 

[ShayanJ]

I think there should be a time limit to the contest too. Otherwise people just continue nesting things together.
Also I really can't wait to see how micromass is going to judge which of the numbers given in this thread is bigger!

 

[jack action]

How about 7, only with a font size of 9.999 x 10^92?

You did ask for the largest number, and not the number of greatest possible magnitude.

In the same spirit:

[itex]9.\overline{9}[/itex] is the largest (widest) number, as it has an infinite amount of characters to display

 

[micromass]

In the same spirit:

[itex]9.\overline{9}[/itex] is the largest (widest) number, as it has an infinite amount of characters to display

So is $10.\overline{0}$
The $0$'s are left out for convenience but really should be there.

 

[Patrick_Hutecker]

int y=100000000;
for(x=0;x<-1;x++)
{
y=y*y
}
(never said we could use code)

 

[nolxiii]

y is 32 bit int?

 

[Dembadon]

int y=100000000;
for(x=0;x<-1;x++)
{
y=y*y
}
(never said we could use code)

Since you initialize x to zero, the condition in the for-loop will be false and it will not run your statement.

 

[nolxiii]

ah that too

i'm convinced the true winner in this is some perl one liner though, busy beaver be damned

 

[m k]

While true n = n + 1

 

[micromass]

While true n = n + 1

Doesn't terminate so not a valid number.

 

[Pepper Mint]

Doesn't terminate so not a valid number.

We have defined a size of an infinite set a valid cardinal number, haven't we ? -/+infinity is an extended real number until you hit Control+C.

 

[Dembadon]

We have defined a size of an infinite set a valid cardinal number, haven't we ? -/+infinity is an extended real number until you hit Control+C.

Micromass would have to confirm, but I'm assuming that rule 1 for the contest implies extended real numbers are not permitted since it explicitly prohibits the use of infinity.

 

[micromass]

Micromass would have to confirm, but I'm assuming that rule 1 for the contest implies extended real numbers are not permitted since it explicitly prohibits the use of infinity.

Indeed, only real numbers are allowed.

 

[Drakkith]

 The number of seconds required to calculate the number of digits in every number in this thread using an abacus, an unlimited supply of paper and pencils, and a wooden spoon.

 

[m k]

Doesn't terminate so not a valid number.

Still much more real than many.

 

[andrewkirk]

 The number of seconds required to calculate the number of digits in every number in this thread using an abacus, an unlimited supply of paper and pencils, and a wooden spoon.

Nice but, sadly, non-computable, owing to its self-referentiality generating an infinite regress. But if you added 'other than posts 104 and 124' it would work.

 

[DevilsAvocado]

Since you initialize x to zero, the condition in the for-loop will be false and it will not run your statement.

Doesn't matter, he and everybody else who try looping/recursive code tricks, will inevitable run out of RAM before even getting close to Graham's number [or TREE(3) or SCG(13) or BIG FOOT]. It won't work even if you turn the whole observable universe into a Samsung semiconductor plant ...

 

[Drakkith]

Nice but, sadly, non-computable, owing to its self-referentiality generating an infinite regress. But if you added 'other than posts 104 and 124' it would work.

That's okay. I still have my spoon.

 

[Patrick_Hutecker]

Since you initialize x to zero, the condition in the for-loop will be false and it will not run your statement.

I'm using java and that is an endless loop. The loop will run even if x is zero, if you think otherwise I don't think you know code.

 

[Patrick_Hutecker]

Doesn't matter, he and everybody else who try looping/recursive code tricks, will inevitable run out of RAM before even getting close to Graham's number [or TREE(3) or SCG(13) or BIG FOOT]. It won't work even if you turn the whole observable universe into a Samsung semiconductor plant ...

The number I'm producing is something and its called infinity. You can't show infinity on a computer, which doesn't make it false or not a concept. Just because their isn't enough ram to make my number doesn't mean it does exist.

 

[micromass]

The number I'm producing is something and its called infinity. You can't show infinity on a computer, which doesn't make it false or not a concept. Just because their isn't enough ram to make my number doesn't mean it does exist.

Infinity is not a real number.

 

[Patrick_Hutecker]

Infinity is not a real number.

ok its a concept you got me there

 

[Dembadon]

I'm using java and that is an endless loop. The loop will run even if x is zero, if you think otherwise I don't think you know code.

I think you should probably try and run your code before suggesting someone "doesn't know code". The fact that the condition in your for-loop will always be false is not the only problem with your "endless loop". Your program won't even run because you didn't define a data type for 'x' and you missed a semicolon at the end of your statement in the for-loop.

 

[Borek]

I think you should probably try and run your code before suggesting someone "doesn't know code". The fact that the condition in your for-loop will always be false is not the only problem with your "endless loop". Your program won't even run because you didn't define a data type for 'x' and you missed a semicolon at the end of your statement in the for-loop.

Not to mention the fact y can't hold number larger than MAXINT (whatever it is in the particular system - assuming Java it should be +9,223,372,036,854,775,807). If memory serves me well, Java won't even tell you there was an overflow.

Not to mention the fact OP never stated it is Java and it can be C or C++ as well (not that it changes the code behavior or problems it has).

 

[TeethWhitener]

Can we use busy beaver, or do the functions have to be computable?

 

[TeethWhitener]

Can we use busy beaver, or do the functions have to be computable?

Assuming that we can...here's a start:

Defs: [BB=busy beaver; f^n=f composed w/self n times; f→m→n=f^(m^(m…), m appears n times; g64=Graham’s #], number: (BB(g64)→BB(g64)→BB(g64))→(BB(g64)→BB(g64)→BB(g64))→(BB(g64)→BB(g64)→BB(g64))

8 characters to spare, just to leave room for someone to top it!

Edited to make BB^n definition clearer.

 

[DevilsAvocado]

The number I'm producing is something and its called infinity. You can't show infinity on a computer, which doesn't make it false or not a concept. Just because their isn't enough ram to make my number doesn't mean it does exist.

You are using "int y=100000000;" and the int64 unsigned range is 0 to 18,446,744,073,709,551,615 which is quite far from both infinity and Graham's number ...

Now, it's true that you can "show" many (virtual) things on a computer (including flying dogs), and one way to handle extremely large numbers is the binary64 double-precision floating-point format, with a max exponent of 1023, however the numbers you get is rounded approximations and does not qualify as a real number.

Beyond that, there is Arbitrary-precision arithmetic, that was originally implemented in Lisp and nowadays is supported in most modern languages like Java, Ruby, C/C++, etc. Using this technique, the digits of precision are limited only by the available memory of the host system (including integer). The processing speed is naturally much slower, since fast CPU registers can not be utilized as normal.

This brings us back to the "Observable Universe Samsung Semiconductor Plant", which is the final inexorable bottleneck for digital precision computing.

Why!?

Well, let's say Samsung finds a very clever way to utilize every particle in the observable universe for storing one digital bit (1/0) — how man many bits can we store?

10^{90} bits

Normally the approximation is 10^{80} atoms in the observable universe, but Seth Lloyd (MIT) has published a paper Computational capacity of the universe where he calculate the number of bosons and fermions (particles/antiparticles) and thus get the bigger number (he also take gravitational energy into account but that's overkill = 10^{120} bits).

So, how big is the digital number Samsung/Seth Lloyd can ever store in the entire observable universe?

2^{10^{90}}

That's a very big number, and I tried the Big Online Calculator (with 2048 bits mantissa and 256 bits exponent) but it halted at 2^{10^{76}} and returned:

1.75\times 10^{301029995663981195213738894724493026768189 8814621085413104274611271081892744}

Not giving the full picture, that's a hefty number ...

But how hefty is it in comparison with the Real Monsters? Well, we can directly see that the 'tiny' (in this league) Googolplex 10^{10^{100}} is already much bigger.

Conclusion: If you use a digital computer to find the largest number — you're doomed to lose.

(And that goes for seconds and abacus as well, unless you plan to live longer than the universe! )

 

[nolxiii]

Defs: [BB=busy beaver; f^n=f composed w/self n times; f→m→n=f^(m^(m…), m appears n times; gr=Graham’s #], number: (BB(gr)→BB(gr)→BB(gr))→(BB(gr)→BB(gr)→BB(gr))→(BB(gr)→BB(gr)→BB(gr))

11 in base^

(changed g64 to gr to save a few characters)

can we use BIG FOOT or TREE(3) without links? (don't really understand either but pretty sure they're exact finite numbers)

 

[ProfuselyQuarky]

(And that goes for seconds and abacus as well, unless you plan to live longer than the universe! )

Darn, I was going to use my abacus

 

[DevilsAvocado]

 

[Fizica7]

0.3(3)

 

[Drakkith]

(And that goes for seconds and abacus as well, unless you plan to live longer than the universe! )

I don't know about you, but I plan to live forever.

 

[ShayanJ]

I don't know about you, but I plan to live forever.

Hoping is different from planning. And I'm pretty sure you only hope to live forever!

 

[ProfuselyQuarky]

Hoping is different from planning. And I'm pretty sure you only hope to live forever!

And planning is different from knowing

Drakkith knows that he's not going to live forever (I hope)

 

[Drakkith]

Drakkith knows that he's not going to live forever (I hope)

Way to dash my hopes and dreams...

 

[ProfuselyQuarky]

Way to dash my hopes and dreams...

There, there. My dream was to prove that narwhals are really unicorns undercover, but I soon got over it

 

[Isaac0427]

There, there. My dream was to prove that narwhals are really unicorns undercover, but I soon got over it

Who says they're not?

 

[ProfuselyQuarky]

Who says they're not?

Aha! A believer! Try telling that to a biology teacher

**I, for one, have done just that and do NOT recommend it**

 

[Vanadium 50]

but I plan to live forever

Or die trying?

 

[DevilsAvocado]

I don't know about you, but I plan to live forever.

No worries, mate. I'm almost a little bit jealous ... imagine 4-8 billion* years from now, when the Milky Way and Andromeda merge into one ... man, talk about fireworks.

On the other hand, the Dark Era in 10^{100} years, doesn't look that tasty ... photons, neutrinos, electrons, and positrons flying from place to place, hardly ever encountering each other, and on top of that; a terribly unstable abacus and wooden spoon + the knowledge that you have only come halfway in this hair-raising PF Contest ...

*Of course, that little hiccup of the Sun turning into a red giant, can easily be fixed with a wooden spoon and some perkiness.