poiewhfopiewhf (It's a joke. He got Graham's number (double entendre here) but couldn't call it (due to the number of digits there were) (I ruined the joke by explaining it, I'm sorry...))
***** In fact, 3↑↑↑3 is _already_ unimaginably larger than the observable universe if you could store 1 bit in every planck volume. And that's not even G1!
In a previous video I "under explained" the true magnitude of Graham's Number. This is because my brain is small. Apologies to Professor Graham who has a big brain and unsurprisingly explained his number flawlessly. This video is designed as a correction that can also stand alone. It will be our last Graham's Number video (for a little while, at least!)
DyllonStej Gaming Grahams number might as well be infinity, it is so big we have have not enumerated anything that requires such a large number. The number of atoms in the observable universe is around 10^80. That is TINY compared to grahams number. The number of ways any atom in the universe can interact with any other atom in the universe: (10^80!) / 2 STILL less than grahams number. The number is larger than any conceivable measurement, so what is the difference? xD
Yeah, you can argue that G2 is just as big compared to G1 as G1 is compared to 4. In both instances, the former number contains the latter number of arrows.
+depenthene Math should always be explained in as simple and intuitive ways as possible, no matter what level it is. There is absolutely no reason whatsoever to make it any more complicated than necessary - people who try to make it look complicated on purpose just try to show off.
Daniel Cannata So you assume that just because some people have problems with math, they are by definition "dumb"? I know several people who find math difficult and confusing, and they are extremely bright in a lot of other things - they can solve problems, they can understand other people's emotions very well, and they can make very sensible decisions in various situations. And yes, all those things are signs of intelligence.
Exponentiation is repeated multiplication. Tetration is repeated exponentiation. Penetration is repeated tetration. Intercourse is repeated penetration.
+Izandai It shouldn't be very hard to understand or explain, because the principle behind the arrows is the same as the more familiar operations. Multiplication -> iterated addition, exponents -> iterated multiplication, arrow notation -> iterated exponents.
@PewDieMilestones basically, and then repeated tetration is pentation (triple arrow), repeated pentation is hexation, etc, etc. They are just operations which repeat the one before n number of times.
@@gordontaylor2815 the arrow notation is pretty simple yet pretty hard for one simple reason: we can imagine multiplication even of huge numbers, we can imagine exponentiation of huge (yet not so huge anymore) numbers but tetration is already really big so we can't imagine it for big numbers (sometimes big is just bigger than 10), and pentation goes even further, we only can imagine it in the same notation they were represented the operation itself is easy, but to understand how these things work as numbers you need to reconstruct your mind a bit
This is the best video on Graham's Number by Numberphile. The earlier Graham's Number videos of Numberphile had been a bit confusing. Thanks... Thank you very much...!!!
Hey Brady! I know you don't consider yourself to be a great teacher, but this is so far the best explanation of Graham's number you have put up so far. :)
@@numberphile Was it the mistake where you said 3^^^3 has 3.6 trillion digits? Because it has way more than 3.6 trillion digits. In fact, the top 5 3s in the power tower itself have a value that's way larger than a googolplex, but the power tower contains 7.6 trillion 3s. So the number of digits in 3^^^3 is itself way beyond comprehension. In fact, the number of digits in the number of digits in the number of digits.............. in the number of digits in the number of digits in 3^^^3 is 3.6 trillion. (The number of times you say 'in the number of digits' is itself 7.6 trillion!") With that, Graham's number must be so large!
Numberphile ★☆★☆★☆OMG This has nothing to do with the comment but I wanted to get your attention. PLEASE make a video on how to solve infinite (divergent specifically) series!!!...!(G64 !'s)☆★☆★☆★
I don't know why everyone's complaining about this video. I think it was a great compacted summary of the Graham's Number series, and having it in a prepared format made it, at least to me, finally sink in.
+Tim Fischer incorrect. Graham's Number is always that monstrosity explained in the video. The solution to the vertices problem, however, could be as low as 13. Or as high as Graham's Number.
Each level you go down you roughly get 1/2 (log 3 to be exact) as many digits as the exponent of the number above. For example, 3^3 = 27 --- 2 digits (close to half of 3) 3^^3 = 7.6 trillion --- 13 digits (about half of 27) 3^^4 = 3^7.6 trillion --- 3.6 trillion digits (about half of 7.6 trillion) 3^^5 = 3^3.6 trillion digit number (on the order of 0.6 x 10^(3.64 trillion) DIGITS) --- which is about half of the actual value of the 3.6 trillion digit number. And so on until 3^^7,625,597,484,987.
Graham's Number is truly my favorite number, by the time you get to G2 the number of arrows in-between numbers is already greater that what you could fit in the entire Universe (even if 1 arrow was only 1 planck length in size) and from there it just escalates beyond any of our comprehension. It is really the largest number that I understand the reasoning behind (as opposed to just a really large number).
I personally LOVE the Graham's number videos. Although just TRYING to comprehend the size of that thing makes my head want to turn into a black hole. Great video once again!
Ever since I was a child I have been obsessed with gigantic, ridiculous numbers, so every time I see Graham's number in the title, I must watch the video immediately. I love your channel!
To people who are trying to compare this to things in the universe, It doesn't matte how small a length in how large a universe in however many dimensions or whatever. There is a reason they came up with arrow notation and the G1 G2 notations, they did this becuase there is literally no way of expressing these numbers with multiplication(ie taking values of small lengths and comparing to larger)/exponents(more dimentions)/tetration(google^google google times) because for every arrow you add you just go up another one of these mathematical operations. and there is G1 up arrows just in G2. So just stop trying to make comparisons to anything within the universe because it doesn't matter how much time you spend advancing your analogies, unless you use a system as ridiculous as the up arrow/ g1,g2,g3 systems you will NEVER reach these numbers.
Yeah, to add to this: The plank volume, the smallest possible volume in quantum mechanics, would only fit inside the entire observable universe 10^185 times
Thank you so much, Brady. I hadn't understand well when Ron Graham explained, but you made an excellent work. Very clear and fluid explanation. Comparing with number 10 made it much more clear to me. Keep doing this great job!
I'm actually really glad you posted another video about this, Brady. For me, trying to get my head to really imagine this is a fun part of Math that I have not experienced in a while. I think there is a real benefit to having people understand the real limits to perception and imagination, and the constructs clever people have come up with to allow us to play with these unimaginable values. Thanks again, Brady.
Incredible, I often wonder how huge just 3↑↑↑3 would be if it was possible to calculate a power tower 7,625,597,484,987 threes high! Picture this: 3↑↑2=27 3↑↑3=7,625,597,484,987 3↑↑4=1.258014298121x10^3,638,334,640,024 (notice how the number skyrockets just by adding ONE 3!) Now try to imagine (if you can) how big 3↑↑↑3 (3↑↑7,625,597,484,987) must be if just the first 3s at the very top are already giving us massive numbers! The numbers from 3↑↑↑3 onwards are truly unimaginable!
@@darknessbr3209 I didn't, it's said that 3↑↑4 or 3^7,625,597,484,987 equals that ridiculously massive number (1.258014298121x10^3,638,334,640,024), now how they were able to calculate that is beyond me lol
Still remember the time when I first learn about a number called Trillion and that blown my mind and here are we now with Graham's number, TREE(3)...etc
Isn't it actually impossible to imagine Graham's number? I remember hearing that the human mind can only contain 10 to the power of 10 to the power of 75 bits of information...
Superblooner1 Yes, it does. But it would mean blowing up an already insanely large number to even larger proportions, and TREE(TREE(TREE(...TREE(G64))...)) would be about as close you could get to infinity. G64 TREEs
Every time I watch a video like this, I understand a bit more of how big Graham's Number is. Now I realize that even if you measured the diameter of the observable universe (estimated to be 93 billion light-years (approx. 8.7982914e+23 kilometers)) in planck distance (1.616199×10^−35 meters), the number you come up with (I came up with 1.421979e+62 not sure if that is right or even close) would not be as big as Graham's number. That is crazy. I feel like it doesn't even compare with G1. I really like thinking about it.
I've really enjoyed your mini-series on Graham's number. Trying to imagine it really puts the concept of infinity into perspective for me; even G64 is infinitely small compared to infinity!
This is the third or fourth video on Graham's number. And each time, I've either learned something new, or gained just a bit more appreciation for just how big it truly is. Absolutely loving the series. But yes, I agree with some other comments, I wouldn't mind a new topic soon either.
After watching many videos on this topic I think I finally understand the entire process now. Thanks! Your ability to explain complex topics to us plebs is a welcome talent.
I'm liking the way you are narrating the videos now. Your explanations most times are really good to make us understand the subject from another point of view, like in the Monty Hall problem videos.
It seems like 64 is such an arbitrary number, if it is the largest imaginable number, than why not up it one more, make it G65, than it's an even bigger number. Why stop there? Why not continue till you have something like G64G64 (or something) I mean we could do this into infinity, so why just 64?
It's not at all the largest imaginable number. The important part is that this number was used in a proof and Graham had to account for _every step_. He had to argue that G63 might not be enough, but that G64 is _definitely_ enough.
I do love how you almost used Matt Parkers words, word for word. The visuals used were very helpful. My favorite part of all of this though, Mr. Graham's humble attitude towards the number itself and his 'discovery' of the computing method to describe such an inhumane number of recognition. The sheer description of it's vastness is described by one phrase... ... Meh, my mind is mush.
Ergo Proxy As far as I know there aren't even enough planck volumes in the (observable) universe to write a googolplex down if you write a digit per planck volume. And Grahams number is just SO much larger than a googolplex.
Try G65. And yes, be careful when using '!'. Much worse, like '!!' or '!!!' Because '!!' itself can mean: - Double factorial, n!! = n(n-2)(n-4)...3.1 if n is odd n!! = n(n-2)(n-4)...4.2 if n is even - Nested double factorial, (n!)! Double factorial grows slower than factorial itself. 5!! = 5.3.1 = 15 While nested double factorial grows much much much faster. (5!)! = 120! = **196 digits long**
Its so crazy that such an enormous number was used to prove something seemingly so small. A scenario where you must have color all of the lines in one plane the same color... and as Graham said, it’s likely to already be true at just 13, yet the number he proves is sooooooooooo much higher than that. It’s so much higher that’s it’s surprising that his prove was even considered useful, isn’t it? It’s like saying “we know the Empire State buildings height is at least less then the width of the solar system.”
The difference between G64 and G63 is mindblowing. If you were to multiple G63 with all subsequent numbers working downwards to G1, you would still be no where close to how big G64 is.
Braingasm!! As for "what is this used for?" I use it all the time. This thing is a *quantifiable* amount that is larger than the Planck Units of the universe. Which is *huge*--and yet smaller than infinity. Now with that in mind, measure anything else and things look totally different. When asked "what's your favorite number?" I tell people it's this. I
10^170 is many trillions of trillions of times larger than the number of Planck volumes in the universe, and computers can use it, and I can write it on a A5 sheet of paper, it's not really that large of a number
One other way to look at this is that, even given how massive Graham's Number is, adding just 1 more arrow, not even G64 arrows, just 1 arrow, makes Graham's Number look like it's absolutely nothing.
Remember; multiplication is repeated addition and exponentiation is repeated multiplication. The thing that comes next as repeated exponentiation is called tetration, and so this notation is not as unintuïtive or unnatural as it seems. There is also a set of functions called the hyperoperation, wich takes two numbers and repeats an iteration of one, the other number of times, generalizing addition, multiplication, exponentiation, tetration, and something called the successor function, wich is just a number plus one, wich is fundamental to algebra, because iterating that a times is equal to adding a and b. 3^^^^3 is than just H_6(3,3)
The amazing thing is, in the original problem involving coloured lines and higher dimensions, Graham's number describes a theoretical number of spacial dimensions. We can't even properly imagine a fourth spacial dimension, and a fifth and sixth sound even more crazy. How in the world are we supposed to make sense of Graham's number of spacial dimensions?
It will forever be uncomputable. There is no sensible example in our universe that will give even a faint meaning to this beast of a number. Astronomical units are not even dwarfs by comparison. If all the particles in the observable universe would be digits, you wouldn't even be able to write G1, let alone calculate it. If the universe resets itself a -Poincare recurrence time- times, and all the particles in existence would arrange and combine in all their possible states every time, you still wouldn't be able to put Grahams number into perspective. That is how big it is. However, what I do find fascinating enough about big numbers is that they are finite and they do end, and even though all I talked about above is a journey so unfathomably big that nothing and nobody can take, we still know how it ends. I don't think there is a better metaphor for life. We know the final destination. Does it matter what we put in between? Not really. If we think at our lives as a string of numbers, those strings will undoubtedly exist somewhere in that bigger number. Whatever you put in between, you will be correct somewhere. Live your life the best way you can as long as you don't hurt others besides you. Believing a higher being is writing the string for you or not, will not make any difference because you will still be correct. I might be wrong though, that's the bitch with unprovable claims. But it doesn't matter. I am still correct even if I'm wrong. Sometimes when I think about silly stuff, I like to think that Grahams number in humanity's quest to find infinity is just simply number 1. Thank you Mr. Graham. Your number in particular taught me the true perspective on infinity.
How about G repeated G64 times.... then finally 64 so.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.....etc etc G64.
Finally a good description of Graham's Number! Now is it a functional number that can be used in a equation and not just a simple place holder like "x" or "a"?? If so can you do a video on that? If not what is the largest number to be used it an equation, that doesn't result in itself being spit out with extra bit attach to itself??
Brady, fantastic presentation on Graham's Number. It's quite astonishing.! I felt the audio mix was heavy handed. The particular audio frequencies were annoying for me. I persisted but this is just to let you know it wasn't your best. Keep on. and well done..
Hah, it's funny. It actually doesn't work much with 2. You can use any number of arrows, trillions if you wish, the result will be still just 4 :D Pretty interesting.
for the more technical answer we turn to Wikipedia "In 1971, Graham and Rothschild proved that this problem has a solution N*, giving as a bound 6 ≤ N* ≤ N, with N being a large but explicitly defined number , where in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation.[2] This was reduced in 2013 via upper bounds on the Hales-Jewett number to .[3] The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003,[4] and to 13 by Jerome Barkley in 2008.[5] Thus, the best known bounds for N* are 13 ≤ N* ≤ N'. Graham's number, G, is much larger than N: , where . This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977."
qorilla Based on what this video describes, I would say that's accurate. However, despite all the videos that have been posted, that hasn't been made explicitly clear using the power of 2. I would've thought 2^2 would be 4, 2^^2 would be 16, 2^^^2 would be 2^2 with 16 2's in the stack (well past a standard calculator), and so on. But if the end digit refers only to the stack height, it would only ever be 2 and therefore 4 is as big as it would get.
I can think of a bigger number, Graham's Number + 1 :-D But seriously I've watched all the Graham's Number videos and have never been able to comprehend past G2
James Herd As Ron Graham said in his video - G64 is so massive that even doubling it doesn't really make it any bigger. Adding 1 to G64 isn't really any bigger than G64.
Graham's number is important because it was for a time the biggest number that was used constructively in a mathematical proof. Adding one is well and all, but you have to show how it got there...Grahams number actually accounts for all those arrows and G1-G64 ridiculousness. ...the +1 would probably not be on the wedding invite, sadly.
In your other video you said that 3 tripple arrow 3 could be written as 10^(some big number), but already 3 double arrow 4 is ~10^(10^12) which means 3 double arrow 5 should be bigger than a googolplex.
I've watched each of these videos on Graham's Number, and (as far as I recall), not once was it mentioned that the notation being used is called Knuth's Up-Arrow Notation. Thankfully, Knuth's Up-Arrow Notation is something I'd already familiarized myself with a couple of years ago - otherwise I'm not sure I'd be able to fully grasp the magnitude of these large numbers (even despite the multiple attempts at explaining them on this channel).
is this bigger than Graham's 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samramdebest I don't think you're understanding the sheer size of these numbers... G10 probably makes your googolplex(googolplex arrows)googolplex look minuscule.
SkyRipzScar but googolplex is so much bigger than 3 so the arrows escalate even quicker, can you prove that this number is smaller than Graham's Number?
samramdebest Not our job to prove your number is smaller, it's your job to prove your number is larger. However, I can tell you with near absolute certainty that SkyRipz is correct. Even G2 or G3 is probably larger than your number, which also has no known usability in mathematics.
3^^^^3 is just so crazy large, I actually find it very comedic. Cannot imagine what a number with that many arrows would be like. If you could live forever, counting to Graham's Number would sure as hell be the most frustrating challenge ever.
I used to think that exponentiation made things grow unimaginably fast. But arrow notation is completely ridiculous. But to be honest, I've always theorized it myself but thought it "didn't exist." I saw that the basic operations were like dimensions. Addition is as 1-dimension, multiplication is as 2 dimensions, and exponentiation as the third dimension. A+B means you start at A, count up from it B times. AxB means you start at A, and add it to itself B times. A (to the power of) B means you take A and multiply it by itself B times. I've never vocalized it, but I always wondered if there was a defined mathematical operation that existed that meant that you started at A and gave it B powers of itself, and I finally found this. And the best part is, arrow notation is completely open-ended, so to go one "dimension" further, you just need to add another arrow. This video is a dream come true for a curious mind! By the way, has anyone else ever thought of this before hearing of arrow notation?
I wrote a letter to Mister Graham and he told me to look him up and call him. I found his number but it's gonna take a while to dial.
I can't press "like" on mobile, but that was funny!
You can press like.......
Bálint Kurucz I have liked my own comment once
no one would be that dismissive to not give their number while at the same time ask you to call
poiewhfopiewhf (It's a joke. He got Graham's number (double entendre here) but couldn't call it (due to the number of digits there were) (I ruined the joke by explaining it, I'm sorry...))
You are trying to make Graham's Number of videos about Graham's number, right?
IMortage that would be quite an achievement
Numberphile
Go for it! Try to collapse google's server park into a black hole.
The universe would be dead before they finished, even if they made like one video a second. I assume.
***** In fact, 3↑↑↑3 is _already_ unimaginably larger than the observable universe if you could store 1 bit in every planck volume. And that's not even G1!
Nixitur He was talking about time, but yeah.
In a previous video I "under explained" the true magnitude of Graham's Number. This is because my brain is small. Apologies to Professor Graham who has a big brain and unsurprisingly explained his number flawlessly. This video is designed as a correction that can also stand alone. It will be our last Graham's Number video (for a little while, at least!)
Please pin this comment.
it is ok
It's funny how, no matters how many videos i've seen about Graham number, it's always interesting and mind blowing. Thanks, Brady
Igor Kovacs Biscaia cool - thank you for watching
Can I get 100 subs before 2021?
@@numberphile Hey! There is a way to do it with the ^ sign! Just replace ⬆️s with ^s!
false.
Any online calculator that I try just lists the triple arrow as infinity...
*And that's just the third arrow.*
DyllonStej Gaming it's a big number!
Not infinitely big, though!
It is kind of nonsense (in maths) defining big numbers, cause any number is really small compared to infinity...
But it's cool anyway...
DyllonStej Gaming Grahams number might as well be infinity, it is so big we have have not enumerated anything that requires such a large number. The number of atoms in the observable universe is around 10^80. That is TINY compared to grahams number.
The number of ways any atom in the universe can interact with any other atom in the universe: (10^80!) / 2 STILL less than grahams number.
The number is larger than any conceivable measurement, so what is the difference? xD
Richard Smith I must say, that is a fascinating comment.
It truly is stunning just how massive G2 is from G1.
And G2 from G3...
And G3 from G4...
And G4 from G5...
Yeah, you can argue that G2 is just as big compared to G1 as G1 is compared to 4.
In both instances, the former number contains the latter number of arrows.
@@anticorncob6 that make this all unbelievably crazy
This is the best explanation I have seen. Maybe because it is explained to normal people, not to math genius.
+depenthene
"If you can't explain it simply, you don't understand it well enough." -Einstein
+depenthene
Math should always be explained in as simple and intuitive ways as possible, no matter what level it is.
There is absolutely no reason whatsoever to make it any more complicated than necessary - people who try to make it look complicated on purpose just try to show off.
***** I will check it out. Thank you.
Daniel Cannata
So you assume that just because some people have problems with math, they are by definition "dumb"?
I know several people who find math difficult and confusing, and they are extremely bright in a lot of other things - they can solve problems, they can understand other people's emotions very well, and they can make very sensible decisions in various situations.
And yes, all those things are signs of intelligence.
@@Peter_1986 not just math,everything should be explained in that way
Exponentiation is repeated multiplication. Tetration is repeated exponentiation. Penetration is repeated tetration.
Intercourse is repeated penetration.
Pentation is not that close too Penetration but a good joke for these that are to slow to figure the incorrect prefix of pent (penet)
asdf30111 Do you go to parks and tell the children that Father Christmas doesn't exist?
Ovenman940 How do you know what I do on Tuesdays?
And sigulation is repeated intercourse
Now that's a sick joke if I ever saw one!
Slightly too loud music but great explanation still.
sorry and thanks
Numberphile No worries Brady, your voice was still perfectly audibile and comprehensible (and that's me saying as a German haha).
Draftgon same :D
+Draftgon and me as a french
false.
Finally, a video explaining how arrow notation actually works.
+Izandai It shouldn't be very hard to understand or explain, because the principle behind the arrows is the same as the more familiar operations. Multiplication -> iterated addition, exponents -> iterated multiplication, arrow notation -> iterated exponents.
@PewDieMilestones basically, and then repeated tetration is pentation (triple arrow), repeated pentation is hexation, etc, etc.
They are just operations which repeat the one before n number of times.
@@gordontaylor2815 the arrow notation is pretty simple yet pretty hard for one simple reason: we can imagine multiplication even of huge numbers, we can imagine exponentiation of huge (yet not so huge anymore) numbers but tetration is already really big so we can't imagine it for big numbers (sometimes big is just bigger than 10), and pentation goes even further, we only can imagine it in the same notation they were represented
the operation itself is easy, but to understand how these things work as numbers you need to reconstruct your mind a bit
This is the best video on Graham's Number by Numberphile. The earlier Graham's Number videos of Numberphile had been a bit confusing. Thanks... Thank you very much...!!!
Hey Brady! I know you don't consider yourself to be a great teacher, but this is so far the best explanation of Graham's number you have put up so far. :)
Darshan Pala that's very nice of you... although I only made this video because of my own stupid mistake last time! :)
Brady I love hearing you explain things and how you get so excited about the math :)
@@numberphile Was it the mistake where you said 3^^^3 has 3.6 trillion digits? Because it has way more than 3.6 trillion digits. In fact, the top 5 3s in the power tower itself have a value that's way larger than a googolplex, but the power tower contains 7.6 trillion 3s. So the number of digits in 3^^^3 is itself way beyond comprehension. In fact, the number of digits in the number of digits in the number of digits.............. in the number of digits in the number of digits in 3^^^3 is 3.6 trillion. (The number of times you say 'in the number of digits' is itself 7.6 trillion!") With that, Graham's number must be so large!
ow, my brain
James A Clouder oops, sorry
Numberphile ★☆★☆★☆OMG This has nothing to do with the comment but I wanted to get your attention. PLEASE make a video on how to solve infinite (divergent specifically) series!!!...!(G64 !'s)☆★☆★☆★
*****
You just broke the universe.
Well, i am a googologist, so i study numbers WAY bigger than grahams number. Including TREE3.
Superblooner1 3(GTREE3)3
“Philosophy is a game with objectives and no rules. Mathematics is a game with rules and no objectives.”
I don't know why everyone's complaining about this video. I think it was a great compacted summary of the Graham's Number series, and having it in a prepared format made it, at least to me, finally sink in.
I haven't seen even a single complaint comment.
The background music is a bit too loud. It's almost distracting.
I agree
+973terminator I agree. +Numberphile : please don't do this again.
I thought it fight the video quite well, a bit loud but fitting.
+Moaiz Shahzad fit**
+973terminator I'd say it's way too loud. No idea what went wrong there. Can't have been intentional.
You like making Graham's Number videos *almost* as much as I like watching them. :-)
k.
It could also be 13 tho
+Tim Fischer incorrect. Graham's Number is always that monstrosity explained in the video. The solution to the vertices problem, however, could be as low as 13. Or as high as Graham's Number.
+Acsabi44 That's pretty obviously what Tim meant.
+NoriMori
Incorrect, that's completely obviously what Tim meant.
Bigassboya
That's pretty unobviously what Nori meant
Uh what are yall talking about?
Each level you go down you roughly get 1/2 (log 3 to be exact) as many digits as the exponent of the number above.
For example,
3^3 = 27 --- 2 digits (close to half of 3)
3^^3 = 7.6 trillion --- 13 digits (about half of 27)
3^^4 = 3^7.6 trillion --- 3.6 trillion digits (about half of 7.6 trillion)
3^^5 = 3^3.6 trillion digit number (on the order of 0.6 x 10^(3.64 trillion) DIGITS) --- which is about half of the actual value of the 3.6 trillion digit number.
And so on until 3^^7,625,597,484,987.
OH...... MY...... TOSTITOS! luminardy canfeermed
That's a very interesting observation. Thanks for bringing it up.
makes since since 3 is about the square root of 10
Thanks, that’s cool. Does that mean that G64 has G32 digits? Or some other number?
Graham's Number is truly my favorite number, by the time you get to G2 the number of arrows in-between numbers is already greater that what you could fit in the entire Universe (even if 1 arrow was only 1 planck length in size) and from there it just escalates beyond any of our comprehension. It is really the largest number that I understand the reasoning behind (as opposed to just a really large number).
I personally LOVE the Graham's number videos. Although just TRYING to comprehend the size of that thing makes my head want to turn into a black hole. Great video once again!
If you could imagine Graham's Number in its entirety, your mind WILL turn into a black hole! XD
I swear this is like the 4th time I've seen an explanation for Graham's Number. It's getting a bit redundant.
AlphaBetaGamma it says 16 replies for me and i only see yours xD
G0-th time lol
what is with the loud music?..
Ever since I was a child I have been obsessed with gigantic, ridiculous numbers, so every time I see Graham's number in the title, I must watch the video immediately. I love your channel!
I'm happy you guys finally used an example for double-arrows where the second number is NOT 3.
To people who are trying to compare this to things in the universe, It doesn't matte how small a length in how large a universe in however many dimensions or whatever.
There is a reason they came up with arrow notation and the G1 G2 notations, they did this becuase there is literally no way of expressing these numbers with multiplication(ie taking values of small lengths and comparing to larger)/exponents(more dimentions)/tetration(google^google google times) because for every arrow you add you just go up another one of these mathematical operations. and there is G1 up arrows just in G2.
So just stop trying to make comparisons to anything within the universe because it doesn't matter how much time you spend advancing your analogies, unless you use a system as ridiculous as the up arrow/ g1,g2,g3 systems you will NEVER reach these numbers.
Yeah, to add to this: The plank volume, the smallest possible volume in quantum mechanics, would only fit inside the entire observable universe 10^185 times
Yeah, even 3^^^3 is way too large to be compared to anything in the universe
I feel like I'm obligated to say something.
I am glad you did. It needed to be said.
do it then..
Thank you so much, Brady. I hadn't understand well when Ron Graham explained, but you made an excellent work. Very clear and fluid explanation. Comparing with number 10 made it much more clear to me. Keep doing this great job!
Graham's number thinks it's pretty big but it's too small to comprehend compared to the amount of real numbers between 0 and (1/Graham's Number)
Privilege Wales not Whales :P but da vohl
You might as well make real -> rational on your comment.
This comment should be higher. Mind bending
I'm actually really glad you posted another video about this, Brady. For me, trying to get my head to really imagine this is a fun part of Math that I have not experienced in a while.
I think there is a real benefit to having people understand the real limits to perception and imagination, and the constructs clever people have come up with to allow us to play with these unimaginable values.
Thanks again, Brady.
Incredible, I often wonder how huge just 3↑↑↑3 would be if it was possible to calculate a power tower 7,625,597,484,987 threes high!
Picture this:
3↑↑2=27
3↑↑3=7,625,597,484,987
3↑↑4=1.258014298121x10^3,638,334,640,024 (notice how the number skyrockets just by adding ONE 3!)
Now try to imagine (if you can) how big 3↑↑↑3 (3↑↑7,625,597,484,987) must be if just the first 3s at the very top are already giving us massive numbers! The numbers from 3↑↑↑3 onwards are truly unimaginable!
How did you reach that 3^^4 result? (I don't know how to write these arrows)
@@darknessbr3209 I didn't, it's said that 3↑↑4 or 3^7,625,597,484,987 equals that ridiculously massive number (1.258014298121x10^3,638,334,640,024), now how they were able to calculate that is beyond me lol
i just have to ask, what kind of practical uses does that number has or will have.
It's the solution to an equation
Do a video on TREE(3)!
+Superblooner1 I agree with you here. I really don't understand anything on it but I'm fascinated by these large numbers.
Apparently it's such a big number that it's difficult to explain, it's difficult to even explain the notation of it
but it is such a fun number. TREE(2) is like 3, then suddenly, BAM, TREE(3) happens
Yesssss pleaseeee
challenge accepted.
I invented a bigger number.
G65.
Fight me.
Congratulations. In what proof do you use that number?
I can do you one better
G(G64))
+Sebastian Carrier no stop that's how you break the universe
Well guess what, I invented something bigger.
G66.
Fight me.
What if we do G(Graham's Number)?
Graham´s Monster just took my mind and climbed on the top of its power tower where I can not reach it. And just because you, Brady, unleashed it!
Still remember the time when I first learn about a number called Trillion and that blown my mind and here are we now with Graham's number, TREE(3)...etc
Isn't it actually impossible to imagine Graham's number? I remember hearing that the human mind can only contain 10 to the power of 10 to the power of 75 bits of information...
Just imagine G64! (Factorial)
Thats not even g(65)
Imagine TREE(G64)
Tree(64) doesnt exist
Superblooner1 Yes, it does. But it would mean blowing up an already insanely large number to even larger proportions, and TREE(TREE(TREE(...TREE(G64))...)) would be about as close you could get to infinity.
G64 TREEs
***** Well, you could always do TREE(G64) + 1...
Or better yet, TREE(G65)!
Maybe even TREE(G99999999999999999999999999999)...
I like the piano background music; it adds a nice touch.
Every time I watch a video like this, I understand a bit more of how big Graham's Number is. Now I realize that even if you measured the diameter of the observable universe (estimated to be 93 billion light-years (approx. 8.7982914e+23 kilometers)) in planck distance (1.616199×10^−35 meters), the number you come up with (I came up with 1.421979e+62 not sure if that is right or even close) would not be as big as Graham's number. That is crazy. I feel like it doesn't even compare with G1. I really like thinking about it.
I've really enjoyed your mini-series on Graham's number. Trying to imagine it really puts the concept of infinity into perspective for me; even G64 is infinitely small compared to infinity!
Now try to imagine GG64. Or GGGGGGG64!
Now it makes sense. I kept getting lost on how to break out the arrows into the lines but you explained it perfectly, Brady.
Please don't put music on talking videos. Trying to listen to your voice and ignore the music makes my brain hurt and I had to skip.
I know what you mean it made me feel nauseated.
This is the third or fourth video on Graham's number. And each time, I've either learned something new, or gained just a bit more appreciation for just how big it truly is. Absolutely loving the series. But yes, I agree with some other comments, I wouldn't mind a new topic soon either.
2:10 "Again, if the trailing number had been Satan"
O.o
+TwistyTie 666?
TwistyTie HAHAHAHAGAHAGGA
Now I can't un-hear that....thanks.
Said Ten*
After watching many videos on this topic I think I finally understand the entire process now. Thanks! Your ability to explain complex topics to us plebs is a welcome talent.
I think I reached Graham's Number in Cookie Clicker.
LOL
I'm liking the way you are narrating the videos now. Your explanations most times are really good to make us understand the subject from another point of view, like in the Monty Hall problem videos.
It seems like 64 is such an arbitrary number, if it is the largest imaginable number, than why not up it one more, make it G65, than it's an even bigger number. Why stop there? Why not continue till you have something like G64G64 (or something) I mean we could do this into infinity, so why just 64?
It's not at all the largest imaginable number. The important part is that this number was used in a proof and Graham had to account for _every step_.
He had to argue that G63 might not be enough, but that G64 is _definitely_ enough.
Because it's used to calculate something specific. He's not just making up big numbers for fun.
What is that specific thing
Lauren Hatfield
Dascription - Sandread
Nixitur
Thanks, that crealified it.
Im glad you made a follow-up explanation just for the magnitude.
Thanks, nicely done
Still a better love story than twilight.
The best video I have seen so far dealing with Graham's Number...
i'm gonna quit takin' LSD and start studyn' big numbers, it's a bigger trip!
do both! even more of a trip xd
I do love how you almost used Matt Parkers words, word for word. The visuals used were very helpful. My favorite part of all of this though, Mr. Graham's humble attitude towards the number itself and his 'discovery' of the computing method to describe such an inhumane number of recognition. The sheer description of it's vastness is described by one phrase... ... Meh, my mind is mush.
Someone, one day, needs to get a super computer to plot these numbers on a bar chart.
lets say 1 number is 1x1cm, i think the whole surface of the earth wouldnt be enough for this number
Ergo Proxy As far as I know there aren't even enough planck volumes in the (observable) universe to write a googolplex down if you write a digit per planck volume. And Grahams number is just SO much larger than a googolplex.
Cjreek wow that is some revelation, thanks :D
Ergo Proxy I don't think the entire galaxy would even fit.. probably bigger than the whole observable universe...
Probably, but if we did it across multiple universes, it might fit.
This channel is blowing my mind and reinvigorating my passion for maths.
Cheers!
G63 is such a small number compared to G64!
Watch out when u use ! symbol in a math comment xD
Try G65.
And yes, be careful when using '!'. Much worse, like '!!' or '!!!'
Because '!!' itself can mean:
- Double factorial, n!! = n(n-2)(n-4)...3.1 if n is odd
n!! = n(n-2)(n-4)...4.2 if n is even
- Nested double factorial, (n!)!
Double factorial grows slower than factorial itself.
5!! = 5.3.1 = 15
While nested double factorial grows much much much faster.
(5!)! = 120! = **196 digits long**
@@r.a.6459 Try GG64. Or G....G64 with G64 Gs!
@@vedantsridhar8378
try (((...((GG...GG64)!)!)!)!)...)!)!)! with G64 'G's and G64 '!'
Its so crazy that such an enormous number was used to prove something seemingly so small. A scenario where you must have color all of the lines in one plane the same color... and as Graham said, it’s likely to already be true at just 13, yet the number he proves is sooooooooooo much higher than that. It’s so much higher that’s it’s surprising that his prove was even considered useful, isn’t it? It’s like saying “we know the Empire State buildings height is at least less then the width of the solar system.”
Yeah, initially people thought that there's no upper finite bound. It's a breakthrough to realize that there does exist a finite bound.
Now imagine if you went up a "Graham's Number" of times instead of 64 times, as in G(G64)!!!!
Rayo(TREE(G64)) is about the largest number I want to think about.
@@as7riverf(Rayo(BB(D(SSCG(TREE(G(32!))))))) (f is Large Number Garden Number function)
The difference between G64 and G63 is mindblowing. If you were to multiple G63 with all subsequent numbers working downwards to G1, you would still be no where close to how big G64 is.
Yeah, it's like comparing 4 and G1, and that's also an understatement.
*Has anyone ever heard of SALTS number ?*
Braingasm!!
As for "what is this used for?" I use it all the time. This thing is a *quantifiable* amount that is larger than the Planck Units of the universe. Which is *huge*--and yet smaller than infinity.
Now with that in mind, measure anything else and things look totally different.
When asked "what's your favorite number?" I tell people it's this.
I
10^170 is many trillions of trillions of times larger than the number of Planck volumes in the universe, and computers can use it, and I can write it on a A5 sheet of paper, it's not really that large of a number
Soundtrack too loud, can't maths.
One other way to look at this is that, even given how massive Graham's Number is, adding just 1 more arrow, not even G64 arrows, just 1 arrow, makes Graham's Number look like it's absolutely nothing.
G64? I prefer C64.
+post #617166 Or N64
why C64? G means graham, whats the C mean?
+Francesco Rende commodore 64, it's an old computer
ohhh, yeah, ive heard of that
That comparison at the end gives me chills. G64 and 2^G64 are practically the same thing
It's over 9000!
It's WAY over 9000.
its _slightly_ over 9000
+aceman0000099 Compared to infinity? Yeah, you're right.
Well 9000! is 9000 factorial (9000 * 8999 * 8998...* 2 *1) which is pretty pretty big at around 1 followed by 15,846 digits.
+Kam Zero its still over 9000!
Remember; multiplication is repeated addition and exponentiation is repeated multiplication. The thing that comes next as repeated exponentiation is called tetration, and so this notation is not as unintuïtive or unnatural as it seems.
There is also a set of functions called the hyperoperation, wich takes two numbers and repeats an iteration of one, the other number of times, generalizing addition, multiplication, exponentiation, tetration, and something called the successor function, wich is just a number plus one, wich is fundamental to algebra, because iterating that a times is equal to adding a and b.
3^^^^3 is than just H_6(3,3)
It's still way smaller than "Marc's Number"
;))))))
Melodic Guitar Rock/Metal GuiltyGearRockYou and way smaller then my number
And there are stupidly larger numbers than Rayo's number
The amazing thing is, in the original problem involving coloured lines and higher dimensions, Graham's number describes a theoretical number of spacial dimensions. We can't even properly imagine a fourth spacial dimension, and a fifth and sixth sound even more crazy. How in the world are we supposed to make sense of Graham's number of spacial dimensions?
😅
Background music is awful and too loud
It will forever be uncomputable. There is no sensible example in our universe that will give even a faint meaning to this beast of a number. Astronomical units are not even dwarfs by comparison. If all the particles in the observable universe would be digits, you wouldn't even be able to write G1, let alone calculate it. If the universe resets itself a -Poincare recurrence time- times, and all the particles in existence would arrange and combine in all their possible states every time, you still wouldn't be able to put Grahams number into perspective. That is how big it is.
However, what I do find fascinating enough about big numbers is that they are finite and they do end, and even though all I talked about above is a journey so unfathomably big that nothing and nobody can take, we still know how it ends. I don't think there is a better metaphor for life. We know the final destination. Does it matter what we put in between? Not really. If we think at our lives as a string of numbers, those strings will undoubtedly exist somewhere in that bigger number. Whatever you put in between, you will be correct somewhere. Live your life the best way you can as long as you don't hurt others besides you. Believing a higher being is writing the string for you or not, will not make any difference because you will still be correct. I might be wrong though, that's the bitch with unprovable claims. But it doesn't matter. I am still correct even if I'm wrong.
Sometimes when I think about silly stuff, I like to think that Grahams number in humanity's quest to find infinity is just simply number 1. Thank you Mr. Graham. Your number in particular taught me the true perspective on infinity.
How about
G repeated G64 times.... then finally 64
so.. GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG.....etc etc G64.
this is approximately equal to g64 but still less than g65
Or G(G(G ... (G64)...) repeated G64 times :o
Graham's number is so wonderful. perhaps a whole CHANNEL devoted to it!
The music makes me stabby.
Finally a good description of Graham's Number! Now is it a functional number that can be used in a equation and not just a simple place holder like "x" or "a"?? If so can you do a video on that?
If not what is the largest number to be used it an equation, that doesn't result in itself being spit out with extra bit attach to itself??
The music is too loud, else nice video
Brady, fantastic presentation on Graham's Number. It's quite astonishing.! I felt the audio mix was heavy handed. The particular audio frequencies were annoying for me. I persisted but this is just to let you know it wasn't your best. Keep on. and well done..
Why this obsession with 3? Why not 2? I like 2.
Because 3 takes it to Graham's Number.
Hah, it's funny. It actually doesn't work much with 2. You can use any number of arrows, trillions if you wish, the result will be still just 4 :D
Pretty interesting.
for the more technical answer we turn to Wikipedia
"In 1971, Graham and Rothschild proved that this problem has a solution N*, giving as a bound 6 ≤ N* ≤ N, with N being a large but explicitly defined number , where in Knuth's up-arrow notation; the number is between 4 → 2 → 8 → 2 and 2 → 3 → 9 → 2 in Conway chained arrow notation.[2] This was reduced in 2013 via upper bounds on the Hales-Jewett number to .[3] The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003,[4] and to 13 by Jerome Barkley in 2008.[5] Thus, the best known bounds for N* are 13 ≤ N* ≤ N'.
Graham's number, G, is much larger than N: , where . This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977."
qorilla Based on what this video describes, I would say that's accurate. However, despite all the videos that have been posted, that hasn't been made explicitly clear using the power of 2. I would've thought 2^2 would be 4, 2^^2 would be 16, 2^^^2 would be 2^2 with 16 2's in the stack (well past a standard calculator), and so on. But if the end digit refers only to the stack height, it would only ever be 2 and therefore 4 is as big as it would get.
Kevin Anderson I guess Im more ov a visual learner, because I still dont see how 2 doesnt work for this.
Can you please make a video about TREE(3) or the the Fish numbers? I would like to see your take on those super huge numbers!
I can think of a bigger number, Graham's Number + 1 :-D
But seriously I've watched all the Graham's Number videos and have never been able to comprehend past G2
well i bet with u that u cant think of it haha
Said I can think of a bigger number not that I could comprehend it...
James Herd As Ron Graham said in his video - G64 is so massive that even doubling it doesn't really make it any bigger. Adding 1 to G64 isn't really any bigger than G64.
Graham's number is important because it was for a time the biggest number that was used constructively in a mathematical proof. Adding one is well and all, but you have to show how it got there...Grahams number actually accounts for all those arrows and G1-G64 ridiculousness.
...the +1 would probably not be on the wedding invite, sadly.
In your other video you said that 3 tripple arrow 3 could be written as 10^(some big number), but already 3 double arrow 4 is ~10^(10^12) which means 3 double arrow 5 should be bigger than a googolplex.
i give u 10 $ if u calcutale sin(G64). I can even tell u its between -1 and 1, so its very easy.
I only calculate it for at least $G64
+Michael Dunne How do you figure that sort of thing out?
+345 345 I think he meant how he figured out the sine of G64, not how sine itself works.
+ProgHead777 Not really. The server would get an overflow error and stop at 9.999e100
But what if you´re working in radians?
I've watched each of these videos on Graham's Number, and (as far as I recall), not once was it mentioned that the notation being used is called Knuth's Up-Arrow Notation.
Thankfully, Knuth's Up-Arrow Notation is something I'd already familiarized myself with a couple of years ago - otherwise I'm not sure I'd be able to fully grasp the magnitude of these large numbers (even despite the multiple attempts at explaining them on this channel).
is there an SIMPLE way to explain why G64 is the upper bound to the SIMPLE line color problem?
thanks for the great video 😃
1) You should check out Conway notation. It makes Knuth notation look TINY.
2) If G2 was your screen resolution, would G1 even fill a pixel?
Great but I would lose the "10"s (they add nothing to the understanding) and shoo the cat off the piano.
My head hurts more now, I found this version easier to not comprehend Grahams number, cheers Brady;)
is this bigger than Graham's 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***** how about googolplex↑↑↑...↑↑↑googolplex with a googolplex↑
samramdebest I don't think you're understanding the sheer size of these numbers... G10 probably makes your googolplex(googolplex arrows)googolplex look minuscule.
SkyRipzScar but googolplex is so much bigger than 3 so the arrows escalate even quicker, can you prove that this number is smaller than Graham's Number?
samramdebest Not our job to prove your number is smaller, it's your job to prove your number is larger. However, I can tell you with near absolute certainty that SkyRipz is correct. Even G2 or G3 is probably larger than your number, which also has no known usability in mathematics.
yes
"This is such a big number that two to that power and that power are kind of hard to distinguish."
One of the most boss things I've ever heard here.
3^^^^3 is just so crazy large, I actually find it very comedic. Cannot imagine what a number with that many arrows would be like. If you could live forever, counting to Graham's Number would sure as hell be the most frustrating challenge ever.
Great explanation of arrow notation and Grahams number
Thank you for making this video! It is the perfect preface to your other Graham's number videos.
I used to think that exponentiation made things grow unimaginably fast. But arrow notation is completely ridiculous. But to be honest, I've always theorized it myself but thought it "didn't exist." I saw that the basic operations were like dimensions. Addition is as 1-dimension, multiplication is as 2 dimensions, and exponentiation as the third dimension. A+B means you start at A, count up from it B times. AxB means you start at A, and add it to itself B times. A (to the power of) B means you take A and multiply it by itself B times. I've never vocalized it, but I always wondered if there was a defined mathematical operation that existed that meant that you started at A and gave it B powers of itself, and I finally found this. And the best part is, arrow notation is completely open-ended, so to go one "dimension" further, you just need to add another arrow. This video is a dream come true for a curious mind!
By the way, has anyone else ever thought of this before hearing of arrow notation?
Probably the most mind-blowing thing for me is that we can calculate the trailing digits of Graham's number. It ends in a 7.
This stuff is great! Any chance numberphile would like to tackle the busy beaver/turing machine halting problem numbers or tree(3)?
Serious question though. Is the arrow notation ever used anywhere outside Graham's number? Because I've never seen it anywhere else.
I know I'm a nerd because my heart was pounding by the end of the video. OMG, what's wrong with me. I'm in shock.
Thanks for breaking that down, and now I'm mind is officially blown!
I believe my brain is beyond Graham's porridge...