High School A Mind-Boggling Number Comparison

[peanutaxis]

The universe is ~10^17 seconds old.

There are ~10^23 atoms in just 12g of carbon.

 

[Drakkith]

And the observable universe is about 8.8x10^23 km in diameter.

 

[BWV]

there are around 10^10^6 books in the library

 

[sysprog1]

there are around 10^10^6 books in the library

Got me there $-$ at first I thought, typo? (presumably yes, for an actual number of books in an actual library $-$ either ${({10^{10}})}^6$ or $10^{({10^6})}$ is way too big) $-$ but $10 \times 10^6$ ($=10^7$, or 10,000,000 = ten million, is a goodly number of books for a very large library, to be sure, so maybe it's a decades-old estimate for the number of books in the Library of Congress (current estimate is over 39 million) $-$ then I checked the link $-$ oh, the Borges library $-$ (presumably) there's a copy of that set of volumes quietly growing explosively on any bookshelf in the (Hilbert's Grand) Hotel.

 

[BWV]

It’s the larger number - on the order of 10^1,000,000 ($25^{1,312,000}$ to be exact)

(The story is about a library that contains all possible 410-page books with a character set of 25 characters (22 letters, spaces, periods, and commas), with 80 lines per book and 40 characters per line)

 

[BWV]

a copy of that set of volumes quietly growing explosively on any bookshelf in the (Hilbert's Grand) Hotel.

One wing of the hotel a holds a Graham’s number of copies of the library, a single volume in each room of the wing placed beside Gideon’s Bible

 

[pinball1970]

TREE 3

 

[pinball1970]

More on the subject of the OP

Someone posted this about Grahams number on PF giving comparisons of physical things in the universe.

"10^113 – The number of hydrogen atoms it would take to pack the universe full of them.
10^122 – The number of protons you could fit in the universe.

10^185 – Back to the Planck volume (the smallest volume I’ve ever heard discussed in science). How many of these smallest things could you fit in the very biggest thing, the observable universe? 10^185. Without being able to go smaller or bigger on either end, we’ve reached the largest number where the physical world can be used to visualize it."

 

[DaveC426913]

I once wrote a short story about a guy named Carl who tried to visualize, extant, in one object, a Googolplex. His attempt caused the unverse to throw an error, stopping gravity, light propagation. and everything else - which then had to be fixed. It was called "Stack Overflow".

 

[phinds]

The universe is ~10^17 seconds old.

There are ~10^23 atoms in just 12g of carbon.

And I'm 5' 9' tall. What's your point? If you're looking for large numbers in the physical world, your choices are incredibly weak.

 

[BWV]

The number of microstates of the universe is the largest number with physical meaning?

$S_{univ} = \frac{k_B c^3 A}{4 G \hbar} \approx 10^{122} k_B$

So the number of microstates is
$Ω≈e^{10^{122}}$

 

[Cerenkov]

How many zeros is that, BWV?

 

[BWV]

How many zeros is that, BWV?

10^120, assuming that 10% of the digits in exp[10^122]~10^(10^121) are zeros ;)

 

[Cerenkov]

10^120, assuming that 10% of the digits in exp[10^122]~10^(10^121) are zeros ;)

Thanks BWV.

So is that a 10 followed by 120 zeros or is there more to it than that?

 

[Ibix]

So is that a 10 followed by 120 zeros or is there more to it than that?

1 with 121 zeroes after it is the number of zeroes in the number stated. Just as $10^5$ has five zeroes, $10^{10^{121}}$ has $10^{121}$ zeroes.

 

[BWV]

1 with 121 zeroes after it is the number of zeroes in the number stated. Just as $10^5$ has five zeroes, $10^{10^{121}}$ has $10^{121}$ zeroes.

So if you printed it in 10 point font on typical printer paper (3000 zeros per page) the stack would be ~10^97 light years high. But per above, the number of atoms in the ink would exceed the number in the entire universe

 

[DaveC426913]

So if you printed it in 10 point font on typical printer paper (3000 zeros per page) the stack would be ~10^97 light years high. But per above, the number of atoms in the ink would exceed the number in the entire universe

And it's still effectively zero compared to Graham's number.

 

[BWV]

Was asking Gemini trying to understand the relationship between physical and informational entropy - and they are closely related? So if $Ω≈e^{10^{122}}$ is the number of microstates of the universe it is also the number of bits required to encode a particular state? so the storing all the possible different configurations of the universe would take $Ω^Ω$ bits?

 

[pinball1970]

And it's still effectively zero compared to Graham's number.

which is effectively zero compared to TREE3

 

[pinball1970]

Numberphile on TREE3

 

[pinball1970]

and Grahams number.

 

[pinball1970]

Just to add, a big number is not particularly interesting, it is how these things are noted, can grow and are used in proofs.

 

[DaveC426913]

Just to add, a big number is not particularly interesting, it is how these things are noted, can grow and are used in proofs.

That was what intirgued me about Graham's Number. Tree(3) is big, but is it useful?

 

[sbrothy]

Numberphile on TREE3

OK I tried to wrap my head around this but I might as well have tried to wrap my head around a rock.

EDIT: Looking it up on Wiki didn't help.

 

[DaveC426913]

OK I tried to wrap my head around this but I might as well have tried to wrap my head around a rock.

EDIT: Looking it up on Wiki didn't help.

He goes through it pretty quickly. This is why i prefer reading over watching.

Read the wiki slowly, taking a moment to absorb one step before moving on to the next.

 

[sbrothy]

Yeah. Me too. That's why I tried Wiki. But it's simply beyond me, I must admit.

 

[pinball1970]

Yeah. Me too. That's why I tried Wiki. But it's simply beyond me, I must admit.

The concept or Grahams number the question is, how many dimensions do I need before I have to make a square of one colour in a plane?

For TREE (3) the question is, how many trees can I make from three different coloured seeds, before the forest has to die? based on the rules he sets out at the beginning.

The rules are just the rules of the game, the “common ancestor” one being the least obvious.

Not being trained in this sort of thing, I would be scratching my head as how to play this game once the trees started to get larger trying to work out if I was repeating myself.

What is crazy is that introducing just one more seed means the sequence goes from 3 to this gargantuan number.

TRRE (1) =1

TREE (2) = 3

TREE (3) = V, V large number dwarfing Grahams number, which itself is an extremely large but finite number.

There is another explanation of this where the presenter says that TREE(3) is the fastest growing “hierarchy” nothing can beat it.

 

[pinball1970]

Here is the comparison of TREE (3) and Grahams number with some more info on Knuth notation, the enthusiasm of this guy is infectious.

 

[sbrothy]

My mind immediately went to red/black data trees but that's a different thing I guess. I figure it an example of chaos as in simple rules results in incredibly complex behavior?

 

[sbrothy]

So what about TREE (n > 3)? Does that make sense at all?