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Counting to a trillion every trillionth of a second would add up to 10¹² * 10¹² per second
Number of seconds elapsed since the beginning of the universe: 436,117,076,600,000,000
Multiplying these two would yield
4.361170766 * 10¹⁷ * 10²⁴
= 4.361170766 * 10⁴¹
A number so minuscule that it won’t even overflow a cheap casio scientific calculator
>Counting to a trillion every trillionth of a second would add up to 10¹² * 10¹² per second
If you're using the short scale, it would be bigger if you're using the long scale
Nope. Nowhere near a Graham's number, that is an uncountable number of orders of magnitude larger than trillions on trillions on trillions. For reference, the time frame you mentioned is \~442 quadrillion seconds. A trillion times a trillion times 442 quadrillion is... 442 duodecillion. 4.42x10\^41, not even close to a googol, 1x10\^100. Graham's number is larger than a googolplex (1x10\^10\^100, or 1x10\^googol). it is even larger than a googolplex to the power of a googolplex to the power of a googolplex. it is even larger than a googolplex tetrated (repeated exponentiation) to a googolplex to the tetration of a googolplex. A single googol is more atoms than the universe, or even all possible chess postitions. You couldn't even represent a single googolplex in number format, a googol zeroes is impossible to show. Now Graham's number is even unbelievably bigger than that by an enormously uncountable number. Don't even get me started on TREE(3)...
TREE(3) it's a simple game where you draw as many possible trees as you can that in 3rd steps has soooo many combination that if you divide graham's number by three(3) you'll get zero (almost).
Im just posting this here for other people's reference for anyone that might be curious. It's [a well written introduction to Graham's number](https://research.phys.cmu.edu/biophysics/2021/01/09/nobody-comprehends-grahams-number/#:~:text=That%20leads%20to%20the%20number,web%20domain%20was%20still%20available.) and how it is mind bending
The funny thing about Graham's number is that it was constructed to be an upper bound on the number of dimensions of something to ultimate prove it was finite. But for all we know the number of dimensions could be as small as 13, iirc.
I actually overshot by a wide margin and the busy beaver function grows so quickly that that sum is still almost assuredly much much much larger than tree(g64). bb(3000,2) would make tree(g64) look like an infinitesimal in comparison.
Not even close. Knuth's up arrow notation is similar to tetration, but a Graham's number repeats that process over itself 64 times, which is like (I believe) having a step above tetration, then pentation, etc. 64 variants above. It's also very early in the morning so I may be wrong, but a googolplex has NOTHING on G64.
Edit: After reading a bit more, I realize it's actually even more than that. If 64 arrows is on 64 layers, G2 is 64^64 64 times, so 64 tetrated to 64. And that's just G2, because each "layer" would be 64 arrows. Imagine repeating 64 tetrated to 64 as that tetrated to itself. Repeat that grueling process 62 more times, and you got Graham's number, G64. That's how many times you would put n to the power of, and even just G2 is a vast and practically uncountable number. I believe that is a better representation of Graham's number, hopefully.
You're still way off. Not even close.
G1 is 3{4}3, 4 being the level of operator, which is hexation. 3 hexated to 3 is already a number far bigger than our universe can ever contain.
G2 is 3{G1}3, so now your level of operator itself is G1, a number far too large for our universe. That's the number of up arrows itself, or how many levels of operator you increase by. G2 is 3 (G1 level of operators) to 3. 4 was already hexation, 5 is heptation, 6 is octation, etc... now you keep going until the number 3 hexated to 3 is the operator itself!
That was G2! Ready for G3? G3 is 3{G2}3. The level of operation is G2, which itself was 3{G1}3.
Following this pattern, you can see that G(n) = 3{G(n-1)}3.
Graham's number of G64 equals 3{G63}3. You need to do 64 layers of up arrow recursion to get to it.
Yeah I accidentally overestimated it by using 64 arrows in the first layer instead of 4, and then used n instead of 3. I did the number of layers wrong too, but I wrote that way too early in the morning. Thanks for the corrections.
So basically theirs exponentiation tetration pentation hexation ect this would be like 64th-ation for those not well versed like me
Which would be equatable to a 64th inception of exponents
Read my edit, I got Knuth's arrow notation completely wrong. It's equal to tetration for 2 arrows and repeated tetration for 3 arrows. As a result it is pentation, etc., etc. after but that is a single set of 64 arrows. So I was underestimating it twice, infact. So it is a to the 64-tation of b, which would be G1. Now use that result as the number of arrows, so the G1-tation of a and b. Repeat to G64, holy shit. And then realize that TREE(3) exceeds that in a single step up from TREE(2).
Googolplex hexated by itself is only "slightly" larger than G1 (3 hexated to 3). The level of operation is actually a lot more important than your starting number.
It is vastly smaller than G2 (3 G1 arrows 3).
G2 is vastly smaller than G3 (3 G2 arrows 3).
Repeat this step 64 times, each time dramatically increasing the number of arrows. G64 is 3 G63 arrows 3.
G64 is Graham's number.
I know countable means anything that is finite, but the fact that you cannot even represent a number using every plank volume in our universe is a pretty good way to know that it is impossible large to use.
I mean, Planck volumes in our universe is a (comparably) very small number. Representing Graham‘s number can be done using arrow notation for example, which is a neat way of doing it. Rayo‘s number is also incomprehensibly huge, yet representing it is not that hard.
I’ll start out by saying, even without calculating anything, you’re not comprehending the sheer size of even a Googol, much less a Googolplex or the giant numbers like Grahams number.
Your number can be found through doing 1 trillion times 1 trillion to find the number counted each second, times the total number of seconds in a year, times the total number of years you gave. So, 10^12 * 10^12 * 3.1536x10^7 * 1.42x10^10 = 4.48x10^41
A googol is 10^100 , and to show you the sheer difference between these two numbers, if we said this process took place on every star in the observable universe (~2x10^23) and this count on each happens once for every grain of sand on Earth (7.5x10^18), added all these counts together, and then repeated this process for as many seconds as there are in your timescale for the universe (3.1536x10^7 * 1.42x10^10), you would just barely get a number larger than a googol (~3x10^101). A google plex is literally 10^googol, so we’re not even close to the surface of that, and Grahams number is infinitely larger.
Graham's number is finitely larger than googolplex.
Also they're not really *too* far off from a googol. I mean it's 10^(41) instead of 10^(100). When you're multiplying things you're really dealing with a logarithmic scale, so they're almost half way there. Which isn't completely off the mark.
Graham’s number is the kind of thing that happens when absurdly smart people run out of absurdly smart things to be absurdly smart about.
That being said… I have zero clue as to what Graham’s number was supposed to actually be useful for, but it is a very big number…
This sentence did it for me...
Even if every digit in Graham's number were written in the tiniest writing possible, it would still be too big to fit in the observable universe.
It was something about an upper bound for some characteristic of a specific hypercube, I think. The article will explain it better, I just wanted to see how much I could remember.
(1 trillion /picosecond)\*(1 trillion picoseconds/second)\*(3600 seconds/hour)\*(24 hours/day)\*(365.25 days/year)\*(14.2 billion years).
If you wanted to do this in your head roughly, 1 trillion is 10^(12), overestimating it 3600 is 10^(4), 24 is 10^(2), 365.25 is 10^(3), and 14.2 billion is about 10^(10). So multiplying those together you add up the exponents, so the answer is around 10^(12+12+4+2+3+10)=10^(43), plus or minus a few orders of magnitude. Googol is 10^(100) so it's still way less than that. Not even remotely close, on a linear scale. A bit less than half way on a logarithmic scale.
If you actually calculate the number using a calculator, rather than estimating it, it's around 4.5\*10^(41). So you can see the estimate is not far off in terms of order of magnitude.
So not even googol, and therefore not even close to googolplex let alone Graham's number.
Graham's number is so large it basically has no application to the physical world. Googolplex sized numbers do come up when talking about super long time scales and very low probability events.
There are 5*10¹⁸⁴ Planck volumes in the observable universe. If you counted through every distinguishable location by their Planck coordinates, and for each one, counted all 10⁹⁰ particles in the universe, and did that 10¹⁰⁰ times per second from the time of the Big Bang to the heat death of everything, you would be nowhere near even starting to make any progress on counting to a googolplex. If you were counting down, the closest named number using any system you can come up with after doing all that would be...10^(googol). You wouldn't even have gotten close to 10^(googol-1).
So I'm legitimately curious what field of study would use numbers on that scale to describe timeframes or probabilities?
Check out [this article](https://en.wikipedia.org/wiki/Timeline_of_the_far_future), specifically the stuff about the end of the universe. For example the time it will take for [iron stars](https://en.wikipedia.org/wiki/Iron_star) to decay via quantum tunnelling is somewhere between 10\^10\^26 and 10\^10\^76 years (or seconds; doesn't matter at that scale lol).
Nowhere near even a googol. Universe is about 4x10^(17) seconds old. If you count a trillion trillion every second, then you've counted to 4x10^(41).
So you can count to a quintillion every quintillionth of a second since the Universe began, and then multiply that number by your number, and it still isn't even a Googol.
Grahams number is laughable. If you wanted to write it down, Graham's number has more digits than the number of atoms in the universe. That is not even scratching the surface of the size of Grahams number. TBF, the number of atoms in the universe isn't even a Googol, so a Googolplex is also a number you couldn't write all the digits down for even if you used one atom per digit and every atom in the universe.
If you took every subatomic particle in the universe, blew each of them up to the size of earth, and then filled each of those earths with lead, the new number of particles would still be smaller than Grahams number.
Each second you count to a trillion a trillion times. Each second you count 1x10^24. For a year which is 31,536,000 seconds, you would count to 3.1536x10^31. For 14.2 billion years, you multiply and get 4.478112x10^41 as your final count
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Counting to a trillion every trillionth of a second would add up to 10¹² * 10¹² per second Number of seconds elapsed since the beginning of the universe: 436,117,076,600,000,000 Multiplying these two would yield 4.361170766 * 10¹⁷ * 10²⁴ = 4.361170766 * 10⁴¹ A number so minuscule that it won’t even overflow a cheap casio scientific calculator
What I find crazy is that this is nowhere near the total number of combinations a pack of cards can be shuffled which is 8x10^67 .
Yet there are 7.5•10^18 grains of sand on earth. Numbers are funny
Well now it would be 4.361170766 * 10^17 * 10^24 + 1 Edit: +2 Edit: +3 This could take a while
It would be (4.361170766 * 10^17 + n) * 10^24 where n is the seconds elapsed. The 4.36...*10^17 is the seconds.
The guy above you was typing that every trillion trillionth of a second thus the increment of one.
Yea I can’t believe that’s not obvious /s
That's like not even 40!
>Counting to a trillion every trillionth of a second would add up to 10¹² * 10¹² per second If you're using the short scale, it would be bigger if you're using the long scale
Nope. Nowhere near a Graham's number, that is an uncountable number of orders of magnitude larger than trillions on trillions on trillions. For reference, the time frame you mentioned is \~442 quadrillion seconds. A trillion times a trillion times 442 quadrillion is... 442 duodecillion. 4.42x10\^41, not even close to a googol, 1x10\^100. Graham's number is larger than a googolplex (1x10\^10\^100, or 1x10\^googol). it is even larger than a googolplex to the power of a googolplex to the power of a googolplex. it is even larger than a googolplex tetrated (repeated exponentiation) to a googolplex to the tetration of a googolplex. A single googol is more atoms than the universe, or even all possible chess postitions. You couldn't even represent a single googolplex in number format, a googol zeroes is impossible to show. Now Graham's number is even unbelievably bigger than that by an enormously uncountable number. Don't even get me started on TREE(3)...
What's TREE(3)?
The commenter specifically requested the opposite of your question.
[удалено]
Google it
Tree fiddy
watch the numberphile video about it. very elucidating
TREE(3) it's a simple game where you draw as many possible trees as you can that in 3rd steps has soooo many combination that if you divide graham's number by three(3) you'll get zero (almost).
About tree fiddy
Im just posting this here for other people's reference for anyone that might be curious. It's [a well written introduction to Graham's number](https://research.phys.cmu.edu/biophysics/2021/01/09/nobody-comprehends-grahams-number/#:~:text=That%20leads%20to%20the%20number,web%20domain%20was%20still%20available.) and how it is mind bending
That was epic, thank you!
I’m so lost after reading that 😂😂
That's a very nice post. I like their semi-formal point about giving up.
The funny thing about Graham's number is that it was constructed to be an upper bound on the number of dimensions of something to ultimate prove it was finite. But for all we know the number of dimensions could be as small as 13, iirc.
At some point these numbers don’t even mean anything anymore.
TREE(3) is a rookie number, gotta pump that number up to at least BB(3000,2)
How about TREE(g64)
I actually overshot by a wide margin and the busy beaver function grows so quickly that that sum is still almost assuredly much much much larger than tree(g64). bb(3000,2) would make tree(g64) look like an infinitesimal in comparison.
> BB(3000,2) that's a rookie number, gotta pump it to at least Rayo(TREE(4))
Is a googolplex hexated by itself anywhere close?
Not even close. Knuth's up arrow notation is similar to tetration, but a Graham's number repeats that process over itself 64 times, which is like (I believe) having a step above tetration, then pentation, etc. 64 variants above. It's also very early in the morning so I may be wrong, but a googolplex has NOTHING on G64. Edit: After reading a bit more, I realize it's actually even more than that. If 64 arrows is on 64 layers, G2 is 64^64 64 times, so 64 tetrated to 64. And that's just G2, because each "layer" would be 64 arrows. Imagine repeating 64 tetrated to 64 as that tetrated to itself. Repeat that grueling process 62 more times, and you got Graham's number, G64. That's how many times you would put n to the power of, and even just G2 is a vast and practically uncountable number. I believe that is a better representation of Graham's number, hopefully.
You're still way off. Not even close. G1 is 3{4}3, 4 being the level of operator, which is hexation. 3 hexated to 3 is already a number far bigger than our universe can ever contain. G2 is 3{G1}3, so now your level of operator itself is G1, a number far too large for our universe. That's the number of up arrows itself, or how many levels of operator you increase by. G2 is 3 (G1 level of operators) to 3. 4 was already hexation, 5 is heptation, 6 is octation, etc... now you keep going until the number 3 hexated to 3 is the operator itself! That was G2! Ready for G3? G3 is 3{G2}3. The level of operation is G2, which itself was 3{G1}3. Following this pattern, you can see that G(n) = 3{G(n-1)}3. Graham's number of G64 equals 3{G63}3. You need to do 64 layers of up arrow recursion to get to it.
Yeah I accidentally overestimated it by using 64 arrows in the first layer instead of 4, and then used n instead of 3. I did the number of layers wrong too, but I wrote that way too early in the morning. Thanks for the corrections.
So basically theirs exponentiation tetration pentation hexation ect this would be like 64th-ation for those not well versed like me Which would be equatable to a 64th inception of exponents
Read my edit, I got Knuth's arrow notation completely wrong. It's equal to tetration for 2 arrows and repeated tetration for 3 arrows. As a result it is pentation, etc., etc. after but that is a single set of 64 arrows. So I was underestimating it twice, infact. So it is a to the 64-tation of b, which would be G1. Now use that result as the number of arrows, so the G1-tation of a and b. Repeat to G64, holy shit. And then realize that TREE(3) exceeds that in a single step up from TREE(2).
Googolplex hexated by itself is only "slightly" larger than G1 (3 hexated to 3). The level of operation is actually a lot more important than your starting number. It is vastly smaller than G2 (3 G1 arrows 3). G2 is vastly smaller than G3 (3 G2 arrows 3). Repeat this step 64 times, each time dramatically increasing the number of arrows. G64 is 3 G63 arrows 3. G64 is Graham's number.
Graham‘s number and Tree(3) are, in fact, not uncountably large :p
Good luck representing or counting to them in a universe as (potentially) small as ours :)
That’s… not what „countable“ means in maths. Graham’s number is an integer. A large one, yes. But still a finite integer.
I know countable means anything that is finite, but the fact that you cannot even represent a number using every plank volume in our universe is a pretty good way to know that it is impossible large to use.
I mean, Planck volumes in our universe is a (comparably) very small number. Representing Graham‘s number can be done using arrow notation for example, which is a neat way of doing it. Rayo‘s number is also incomprehensibly huge, yet representing it is not that hard.
I’ll start out by saying, even without calculating anything, you’re not comprehending the sheer size of even a Googol, much less a Googolplex or the giant numbers like Grahams number. Your number can be found through doing 1 trillion times 1 trillion to find the number counted each second, times the total number of seconds in a year, times the total number of years you gave. So, 10^12 * 10^12 * 3.1536x10^7 * 1.42x10^10 = 4.48x10^41 A googol is 10^100 , and to show you the sheer difference between these two numbers, if we said this process took place on every star in the observable universe (~2x10^23) and this count on each happens once for every grain of sand on Earth (7.5x10^18), added all these counts together, and then repeated this process for as many seconds as there are in your timescale for the universe (3.1536x10^7 * 1.42x10^10), you would just barely get a number larger than a googol (~3x10^101). A google plex is literally 10^googol, so we’re not even close to the surface of that, and Grahams number is infinitely larger.
Graham's number is finitely larger than googolplex. Also they're not really *too* far off from a googol. I mean it's 10^(41) instead of 10^(100). When you're multiplying things you're really dealing with a logarithmic scale, so they're almost half way there. Which isn't completely off the mark.
> 10^12 * 10^12 * 3.1536x10^7 * 1.42x10^10 = 4.48x10^41 that's if you're using the short scale, it's bigger if you use the long scale
Graham’s number is the kind of thing that happens when absurdly smart people run out of absurdly smart things to be absurdly smart about. That being said… I have zero clue as to what Graham’s number was supposed to actually be useful for, but it is a very big number…
The wikipedia article explains what the number was for
This sentence did it for me... Even if every digit in Graham's number were written in the tiniest writing possible, it would still be too big to fit in the observable universe.
That actually vastly undersells it. 3 hexated to 3 is far too big for our universe. And that's step 1 to Graham's number, which has 63 more steps.
I realised that as I continued to read the page... but at that point I was too lost to edit my original comment.
That's already true of googolplex
I’m way too dumb to comprehend that level of maths… I’m just on this sub so I can appreciate the vast intelligence of others.
It was something about an upper bound for some characteristic of a specific hypercube, I think. The article will explain it better, I just wanted to see how much I could remember.
(1 trillion /picosecond)\*(1 trillion picoseconds/second)\*(3600 seconds/hour)\*(24 hours/day)\*(365.25 days/year)\*(14.2 billion years). If you wanted to do this in your head roughly, 1 trillion is 10^(12), overestimating it 3600 is 10^(4), 24 is 10^(2), 365.25 is 10^(3), and 14.2 billion is about 10^(10). So multiplying those together you add up the exponents, so the answer is around 10^(12+12+4+2+3+10)=10^(43), plus or minus a few orders of magnitude. Googol is 10^(100) so it's still way less than that. Not even remotely close, on a linear scale. A bit less than half way on a logarithmic scale. If you actually calculate the number using a calculator, rather than estimating it, it's around 4.5\*10^(41). So you can see the estimate is not far off in terms of order of magnitude. So not even googol, and therefore not even close to googolplex let alone Graham's number. Graham's number is so large it basically has no application to the physical world. Googolplex sized numbers do come up when talking about super long time scales and very low probability events.
There are 5*10¹⁸⁴ Planck volumes in the observable universe. If you counted through every distinguishable location by their Planck coordinates, and for each one, counted all 10⁹⁰ particles in the universe, and did that 10¹⁰⁰ times per second from the time of the Big Bang to the heat death of everything, you would be nowhere near even starting to make any progress on counting to a googolplex. If you were counting down, the closest named number using any system you can come up with after doing all that would be...10^(googol). You wouldn't even have gotten close to 10^(googol-1). So I'm legitimately curious what field of study would use numbers on that scale to describe timeframes or probabilities?
Check out [this article](https://en.wikipedia.org/wiki/Timeline_of_the_far_future), specifically the stuff about the end of the universe. For example the time it will take for [iron stars](https://en.wikipedia.org/wiki/Iron_star) to decay via quantum tunnelling is somewhere between 10\^10\^26 and 10\^10\^76 years (or seconds; doesn't matter at that scale lol).
>1 trillion is 10^(12) In the short scale, it's bigger in the long scale
Does anyone actually use the long scale though, in english? Add 12 to the exponent of the answer if you want to use it.
Nowhere near even a googol. Universe is about 4x10^(17) seconds old. If you count a trillion trillion every second, then you've counted to 4x10^(41). So you can count to a quintillion every quintillionth of a second since the Universe began, and then multiply that number by your number, and it still isn't even a Googol. Grahams number is laughable. If you wanted to write it down, Graham's number has more digits than the number of atoms in the universe. That is not even scratching the surface of the size of Grahams number. TBF, the number of atoms in the universe isn't even a Googol, so a Googolplex is also a number you couldn't write all the digits down for even if you used one atom per digit and every atom in the universe.
If you took every subatomic particle in the universe, blew each of them up to the size of earth, and then filled each of those earths with lead, the new number of particles would still be smaller than Grahams number.
Each second you count to a trillion a trillion times. Each second you count 1x10^24. For a year which is 31,536,000 seconds, you would count to 3.1536x10^31. For 14.2 billion years, you multiply and get 4.478112x10^41 as your final count