Graham's Number - Numberphile

See our other Graham's Number videos: bit.ly/G_Number
A number so epic it will collapse your brain into a black hole! Yet Tony Padilla and Matt Parker take the risk of discussing its magnitude. Watch with caution.
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See also our video about the Googol and Googolplex at: • Googol and Googolplex ...
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[@NoriMori1992]

I love Wikipedia's description of how big Graham's number is: "It is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume … But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number-and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe."

[@philip8498]

this reads like something from the hitchhikers guide to the galaxy

[@RH-ro3sg]

And, while perfectly true, even that is an extreme understatement, in the sense that that description already is true for g1= 3↑↑↑↑3, the mere _initial number_ (with just 4 measly arrows), used to get up to Graham's number. Even for 3↑↑↑3 (three arrows), you'd have to repeat the 'number of digits' procedure several _trillion_ times to arrive at something humanly digestible (or at a number expressible within our observable universe as described in the quote). For 3↑↑↑↑3 (4 arrows) that number not only far exceeds the number of Planck volumes in the observable universe, but is utterly beyond human comprehension itself.

[@andrewbloom7694]

@@RH-ro3sg They are all well beyond human comprehension. You can try to define them with things like arrow notation sure, but you can't fundamentally UNDERSTAND something like that. Not even the smartest human can.

[@RH-ro3sg]

@@andrewbloom7694 I think it depends on how exactly you'd define 'comprehension' or 'understanding'. In a rather strict sense - intuitively _grasping_ and _feeling_ the magnitude of a number and immediately recognizing it without conscious thought, we as humans probably don't truly 'get' any number beyond approximately 7. Beyond that, we have to start counting (or approximating), both of which are already more indirect ways of appreciating a number. In the sense of being to able to _visualize_ a number in some manner, I'd say our comprehension ends at around a googol, if we're being very charitable (possibly the limit is much lower). You're talking about imagery such as 'a hundred million of our observable universes, filled to the brim with grains of sand' then. I suppose that visualization of such a type is what most people think of when they say they 'comprehend' a number. But it's not the only way to get to understanding. Numbers such as Graham's number can still be 'understood', but in a more indirect way, namely by the procedures used to obtain them. Finally, there are numbers so large that even the procedures to obtain them cannot be described anymore, they can only be _characterized_ . Rayo's number would be an example. Also, I'm not really sure I truly _comprehend_ even a number as low as three. (As in: what is the ultimate essence of 'three-ness'?)

[@vedantsridhar8378]

Not even the number of powers, not even the number of arrows actually!

[@petertimowreef9085]

Mathemathicians are so funny. "Imagine a number that's unimaginably high. And then the answer is between that number, and 11. Childsplay really, let's go to the pub."

[@MrCubFan415]

Actually, the lower bound is 13 now (and the upper bound has been reduced to 2^^^6).

[@stefanr8232]

where is link to proof?

[@arnavanand8037]

2 + 2 = Something between -∞ and ∞

[@arnavanand8037]

Or possibly between 5 and 5454545575454545457575757575757242454545454542424545454

[@robinlindgren6429]

to be fair, having reduced it to any range at all means they have narrowed it down to a ratio that approaches 0% of all numbers, that's practically being spot on!

[@theviniso]

g64/g64=1. That's the only operation that I can do involving this number.

[@panosm2007]

+Nastygamerx70 ­ (Yasser Moustaine) how about g64 * 0 = 0?

[@panosm2007]

+Грамматический нацист nice

[@connfdm]

g64÷0=error

[@connfdm]

3^^^^^^^^^^...(g64 arrows)3 = g65

[@connfdm]

g64-(g64-1)=1

[@ve4410]

"Can you give me a ballpark" "It's between 11 and Graham's number" "That's convenient".....

[@austinlincoln3414]

Lol

[@FatherManus]

Yeah that really narrows it down.

[@finmat95]

Ehy, previously it was between 6 and Graham's number, that's an improvement, you could at least thank me.

[@召命神弓]

REALLY convenient

[@nthgth]

"There's still an infinite number of numbers that're bigger than Graham's number, right? So frankly, we pretty much nailed it as far as I'm concerned." Lmao

[@Ida-xe8pg]

I actually know graham's number G64/G64 = 1 , G64-G64 = 0 , G64*G64 = G64^2 ,G64+G64 = G64*2!!

[@Ida-xe8pg]

Graham's Number! universe collapse

[@shyshka_]

so does it mean that the calculation is infinitely precise?

[@danielxu3594]

@Fester Blats And also every number is less than Grahams number at the same time.

[@zasharan2]

The thing is, can you actually express those bigger numbers without saying G64 + some other number, or without using that same strategy more times, and one guy named Rayo did that. He gave a statement that gave a number bigger than Graham’s number, without using the way graham got his number.

[@leisulin]

But even as they almost literally said: Graham's number is unimaginably large, but it's still closer to zero than it is to infinity! Which boggles the mind even more.

[@yam1146]

My brain is too small

[@AA-el7ot]

Infinity is not a number though

[@franchstar1]

doesn't really boggles the mind since infinity is not a number but a concept and all numbers would be closer to zero.

[@Crazytesseract]

What do you mean by "closer to infinity"? If you say 5 is closer to infinity than 3, or Graham's number is closer to infinity than one trillion, that's fine; but it makes no difference to "infinity". Graham's number can be imagined extremely few.

[@leisulin]

@@Crazytesseract I mean just what I said. Actually my comment comes from some cartoon that was forwarded to me (the name of which I don't remember) depicting a kid in bed saying to his dad "I'm not sleepy yet, could you tell me a bedtime PARADOX" (not story), and the dad says "every number is closer to zero than infinity, but still we approximate large numbers as infinite". Which knocks the kid unconscious from the paradoxical shock.

[@MordredMS]

I actually came up with an even bigger number. Graham's Number+1. I call it "Mr. Whiskers".

[@glass7923]

XD

[@prometheusxo6013]

I wish comments like this show up more. Now it seems like channel promotion and pepole asking for likes are tue only thing I see, stuff like this is what the internet is for

[@vlh371]

The reason Grahams number is special is because it was used to solve a problem. Grahams number plus 1 isn't useful.

[@valhalla4558]

I came up with a far bigger number. Grahams number to the power of googolplexian. I call it "Mr Puff"

[@glass7923]

Keyslam Games I call it "Lo Wang"

[@onebigadvocado6376]

"There's a very easy analogy" (Promptly fails the analogy)

[@1SLMusic]

Parker Analogy

[@123games1]

Graham's number is so insanely large that the number representing the number of digits in Graham's number would have an incomprehensible number of digits itself!

[@jakethornton7]

+123games1 That even starts to apply around G1.

[@RockerSkate1423]

+123games1 Yeah man, even the number of digits would be a mind-blowing number, it's just insane.

[@drinkingthatkool-aid3193]

+Andrés Ramírez Yep even 3^^5 already has 0.61 x 10^(3.64 trillion)....DIGITS. And you still need to go down 7.6 trillion 3's to get 3^^^3.

[@RH-ro3sg]

In fact, if you repeated that process (the number representing the number of digits of the number representing the number of digits of Graham's number), and then again, and so on, even the _number of times you'd have to repeat that process_ to arrive at a number comprehensible for average humans would _still_ form an incomprehensibly large number of digits. And probably repeating the process on _that_ number still would. And so on. As a commentator once put it: "Graham's number is far larger than most people's intuitive conception of _infinity_ . ((Coincidentally, taking 'the number of digits' approximately is what you are doing when taking the logarithm of a number, so essentially we are talking here about log(log(log((log(g64) and the number of 'logs' you'd need to arrive at something digestible)) ".

[@user-bc3ri8ez9c]

Even the universe isn't enough to make a 1%

[@X-3K]

So basically, this number happened because someone gave a Mathematician a coloring book.

[@bakedpotato3734]

LOL

[@tomlupien4896]

A higher-dimensional coloring book

[@codysangster7413]

ye

[@mohammednajl5950]

and tree 3 is because of colouring pencils

[@spyrex3988]

Graph theory isn't just about colouring points

[@livinlicious]

The first digit of Grahams Number is 1. (in Binary)

[@Gonzaga78]

Hurr Durr

[@chrisroberts4599]

The first digit of Graham's number is 1 in Unary, Binary and Ternary. What are the odds?

[@PattyManatty]

Chris Roberts In ternary it could be 2.

[@chrisroberts4599]

PattyManatty Nope, it's a one. 10^N always start with 1 in decimal, and 3^N will always start with 1 in ternary.

[@PrimusProductions]

Graham's number is odd Graham's number is divisible by 3,9,27 and all powers of 3 up to Graham's number, log(3,G64) is an integer The last digit of Graham's number is 1 in Binary (because it is odd).

[@The_Story_Of_Us]

What makes Graham’s Number so great is that despite its (literally) unfathomable size, we can using less than a page’s worth of word’s describe how to get there. We can describe what 3↑3 means, we can describe what 3↑↑3 means, what 3↑↑↑3 means and what 3↑↑↑↑3 means, then we can describe what G1 is, all the way up to G64, all of it a process of iteration. And using just the power of these symbols and descriptive iteration, we can arrive at a number with 100% precision that arithmetic literally can’t even come close to describing. So when we say that we can’t picture Graham’s Number, I think that’s doing our brains a disservice.

[@The_Story_Of_Us]

@Oak Tree but we do legally own it. Whereas a number like TREE(3) is just so big we can’t describe it all, we don’t know how to arrive at that number via iterative process.

[@The_Story_Of_Us]

@Oak Tree I mean obviously they’re there. If you just divide 1 by Graham’s Number for example, but in terms of something practically applicable like Tree 3 or Graham’s Number, then yeah, that’d be cool.

[@MABfan11]

@@The_Story_Of_Us Bird's Array Notation can reach TREE(3) and beyond

[@The_Story_Of_Us]

@@MABfan11 How do we even begin to know these kind of things?…

[@BokanProductions]

You guys know how the new Webb Satellite from NASA allowed us to see more of the observable universe? I believe it's only a matter of time before we can see enough of the universe that Graham's Number could theoretically fit it in it.

[@squirrelknight9768]

"Frankly, we pretty much nailed it!" Lol that cracked me up

[@NoriMori1992]

Same! And his face when he says it is priceless.

[@MrFrak0207]

SquirrelKnight I love that guy Hahahha

[@doemaeries]

In the next math test I just write 6

[@knox140]

tfw the answer is 5

[@JohannaMueller57]

aha

[@jabruli]

-G64

[@JohannaMueller57]

Jakob Lippig why not -infinity < x < infinity? you guys just lack brain so much.

[@jabruli]

Cuz infinity contains x

[@ottoweininger8156]

The bit where he said we've narrowed it in from between 6 and Graham's Number, to between 11 and Graham's Number made me laugh.

[@TheSpotify95]

yeah, both 6 and 11 are tiny compared to even g1, let alone g64

[@MABfan11]

the new lower bound is 13

[@AzertyWasTaken]

I believe that the answer to the problem is a huge number but proving lower bounds is very hard.

[@Dogebloxian]

"Graham's number is still closer to zero than it is to infinity"

[@bunnyloverplayz1371]

Well obviously all numbers are

[@jd9119]

Zero and Graham's number are both numbers. Infinity isn't a number. It's a direction on a number line.

[@jamesworley9888]

Space is the only thing that we know for sure must be infinite, even if the universe isn't the space beyond and within it is. The only exception would be if somewhere we were surrounded by an infinite brick wall, and again there must be an infinite amount of space to contain it , so space is and must be infinite, there is no other possibility.

[@jd9119]

@@jamesworley9888 That's not true. You're making an assumption.

[@jamesworley9888]

@@jd9119 There is no assumption, I never said ''the universe'' IE ''the stuff IN space is infinite. I said space itself is infinite and no 'one who can think for 5 seconds is able to disagree. Tell me what wall could exist that says ''space ends here'', such a thought is utter nonsense. Especially sense the wall couldn't exist without an infinite volume. Your head would have to be thicker than that wall to even think such a thing or second guess the logic. Tell me where the space ends and anyone can debunk you simply by asking what is beyond that??? The answer is and can only be more volume IE SPACE!!!! You DMF

[@ckmishn3664]

According to the holographic principle the most data (bits) that can be stored in a volume is equal to the area of a bounding sphere in Planck lengths squared divided by 4. The visible universe is about 10^26 meters in length and Planck length is ~10^-35, so very roughly the visible universe can contain something like 10^122 bits of data before being "full" and collapsing into a black hole. Writing out, or otherwise listing the full expansion of a number without resorting to exponents, arrow-notation, recursion or other methods of compression requires a number of bits equal to the log of the number. Saying that your brain would collapse into a black hole if you had all the digits of Graham's Number in your head is one of the all-time biggest understatements. The entire visible Universe actually can't even contain the expansion of 3(three arrow)3. In fact even if you use exponents but just insist on printing out the exponents you still can't print out the expansion of 3(four arrow)3. Even resorting to arrow notation I think it's impossible to print out the expansion for the number of arrows any more than three levels lower.

[@YanTales]

but we can imagine it, and we are imagining it with our physical brain so it can exist and it does.

[@ckmishn3664]

Gaming Power Cool. Please imagine it and tell me what the first digit of Graham's number is (in base 10).

[@YanTales]

Patrick Wise its between 0 and 9

[@ckmishn3664]

Gaming Power So you know for a fact it's not a 9? Well that's something I guess.

[@YanTales]

Patrick Wise my bad, between 0 and 9 including 9.

[@opmike343]

Well, that escalated quickly...

[@samarvora7185]

Congratulations, dear sir! You've summed up the entire video!

[@cate01a]

yes! I've just been learning about n^^x and then when you've 3^^^^3 I'm going 'woah mate calm down' but then he comes in with g2=3(3^^^^3 ^'s)3 and I mean that's worthy of a stupidly large immense number but then it's g64! woah!

[@Combobattle]

exponentiated quickly

[@robertjarman3703]

@@cate01a g64! would be Graham´s Number, factorial. Go Graham´s Number times (Graham´s Number-1), so on all the way down to one, which is a catastrophically large number, so much bigger than Graham´s number that G64 might as well be 0 compared to it.

[@karlfeldlager7662]

@@robertjarman3703 Had you said 1 instead of 0, OK. But 0? 0 is stupidly tiny, I should say. Anyway, G64! is WAY below G65, for starts.

[@megatrix500]

now... Gn↑↑↑↑↑...↑↑↑↑↑Gn. |---Gn times---| Let the universe collapse.

[@Daniel-dc5mr]

Megatrix500 wow

[@Scias]

Just writing that endangers the existence of the universe, be careful lol

[@eclipseskaters]

Still an infinite amount of numbers larger than that number.

[@ashen_cs]

Haven't even reached Aleph^1 yet

[@abacussssss]

Less than g66.

[@sproins]

Other mathematicians explaining big numbers: You'd run out of space to write down all the digits. Matt Parker: You'd run out of pens in the universe.

[@cameronpotter2493]

The real problem makes wayyyyy more sense than the weird analogy about the committees and people thing.

[@thomashudson9524]

Thank you

[@xCorvus7x]

Care to describe it, while you're at it?

[@NoriMori1992]

@@xCorvus7x Ron Graham describes it in another Numberphile video.

[@Kunal29Chopra]

they actually didn't do a great job here, explaining the committee analogy, with the switches between Tony and Matt, also the fact that they were saying the analogy right from their head, but if read in a paper, the analogy is actually very easy to follow.

[@adamqazsedc]

@@xCorvus7x Graham himself actually explained the number, the proper and more understandable way

[@user-gi3ro9rm9k]

i will give the man who tells me the entire graham's number a nobel peace prize for stopping the chaos going inside my head right now

[@delilahfox3427]

Kyu Hong Kim That's physically impossible.

[@vgamerul4617]

@@delilahfox3427 tf

[@vgamerul4617]

@strontiumXnitrate killed 2852 kids' hope

[@NotAGoodUsername360]

Actually, quantum mechanics forbids this.

[@Dexuz]

The universe may as well collapse and recreate itself a g63 times before that man ends.

[@StardropGaming]

Plot twist: Graham's Number + 2 is prime.

[@martinshoosterman]

+StarDrop +Rip proving that.

[@tannerearth0396]

(2^G)+1 is prime. I checked

[@dennismuller1141]

@TannerEarth03 - GTA Boss actually, (2^n)+1 can only be prime if n is a power of 2. G is a power of 3, so (2^G)+1 can't be prime. primes in the form of (2^n) + 1 are called Fermat-primes btw

[@reuben2011]

Wikipedia has a proof. The idea is that you can always factor a sum of odd powers (e.g. x^3+y^3). Now, if n were not a power of 2, then it has an odd prime factor p. So you can write n = kp where k is some integer. Thus, 2^n + 1 = 2^(kp) + 1 = (2^k)^p + 1^p and thus we've written 2^n+1 as a sum of odd powers (which factors).

[@NeemeVaino]

@@dennismuller1141 Fermat numbers are of form 2^2^n+1 and there is no known primes for n>4. Mersenne numbers are of form 2^n-1 and contain large primes but very sparsely.

[@grantmayberry7358]

8:30 "We pretty much nailed it as far as I'm concerned." Never mind the fact that that number is longer than the observable universe.

[@BokanProductions]

You guys know how the new Webb Satellite from NASA allowed us to see more of the observable universe? I believe it's only a matter of time before we can see enough of the universe that Graham's Number could theoretically fit it in it.

[@TheSpotify95]

@@BokanProductions Let's first try and find a way of writing down the full expanded value of 3↑↑↑3 (the tower itself reaches to the Sun), then go to 3↑↑↑↑3, then go from there.

[@BokanProductions]

@@TheSpotify95 Alright, I get it you don't need to explain more.

[@unclvinny]

I like to think about Graham's Number before I go off to sleep. Thanks, Numberphile!

[@hymnodyhands]

unclvinny I thought I was the only one... Why count sheep when you can count endless towers of threes?

[@blue9139]

I think of utter obvilion lol

[@idioting]

im definitely going to not sleep for 70 days after this

[@cate01a]

​@@hymnodyhands three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three three to the three to the three...

[@turicaederynmab5343]

I've got such a headache after watching this, just thinking about a number with 1 digit larger makes my stomach hurt.

[@chadcarl7554]

how ironic, my head hurts as well.

[@ryan2-518]

Suraj's opinion can die in a hole that's not ironic

[@equilateraltriangle8619]

This is an antidote (to end your life(no offense)) G64^^^^(G64^^^^G64xRayo’s number)^G64.

[@davecrupel2817]

Stop thinking with your stomach 🤣

[@jaredunrot717]

Sadly my mind has collapsed

[@marcelinozerpa3947]

I got lost at "committee"

[@FrostyLava]

The "truest" comment

[@TianXiaoMao]

I got lost at 27. 🥵

[@sebastianweigand]

Love the channel, keep up the great work!

[@CertifiedAnything]

$10.00

[@Oskar5707]

dumbass

[@denzelhobbs9982]

​@@Oskar5707 little boy is triggered

[@tornadoreaper]

ten HUMDRED dollarrrs ????? scream 😱🎵😱🎵😱😱🎵🎵🎵

[@ender5312]

GLOWING

[@IVAN3DX]

2:38 Matt.exe had stopped working.

[@JimmyLundberg]

That's when the balding process began. :(

[@achyuthramachandran7391]

IVAN3DX I was reading this EXACTLY when he said "that that that that" 😂😂😂😂 killed me 😂😂😂😂😂

[@SpaceChimpProduction]

IVAN3DX

[@dranreb2250]

Right after seeing this, youtube crashed...

[@mrsuperguy2073]

I didn't even notice!

[@alexdabeast1892]

(Graham's number)!

[@horrorandgames]

I think you would need a computer with a nuclear reactor for computing power 😂

[@alexdabeast1892]

:D

[@matthewdaws9877]

+AlexDaBeast g64! ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑ g64!

[@GarryDumblowski]

+Wout Kops A nuclear reactor doesn't make any difference. It's just a power source. You could power any old computer with a nuclear reactor.

[@TankleKlaus]

+MrAlen61 How about (number of sub-atomic particles in the observable universe)! ^googolplex ?

[@romanr9883]

"we pretty much nailed it, as far as im concerned" hrhrhr

[@emmeeemm]

lol, I love that Graham's Number is so huge that it takes multiple mathematicians to explain it in one Numberphile video.

[@asusmctablet9180]

And yet we know that Graham's Number has a Persistence of 2. Let THAT sink in.

[@nuklearboysymbiote]

well nobody says it HAS to start with a 3. So... I started with a 1. And my brain didnt become a black hole because the end result (g64) is 1.

[@norielsylvire4097]

NukeML wow dont say

[@KaichouClips]

Has this been happening a lot? I thought it was just my browser messing up

[@SAKURA8023o]

View all Graham's number replies

[@augustackerman9404]

*all 35 replies*

[@thecakeredux]

*click* *zoop* gone

[@GarrettBorden]

It's crazy how incomprehensible Graham's number is. It's a shame that some people can't grasp it. "Is a googolplex bigger?" Lol. G1 dwarfs googolplex. Like it's not even comparable. And G2 is exponentially larger than G1. And so on. G63 might as well be "1" compared to G64! It's just mind boggling but I love this stuff. I started watching stuff on horizontal arrow notation and it's just ridiculous how quickly numbers start growing!

[@sebastianschon3141]

And then realize that this number - Grahams number - Is ridiculously small - compared to G65.

[@danielw.4876]

If you walked a googolplex miles, and then you walked Graham's number miles, they would both feel like the same amount since your brain would have no way of remembering how long you had walked for.

[@lindsaytang1017]

G63 might as well be 0

[@nsprphg]

Are there more angles in a circle than G64?

[@ZyphLegend]

Honestly, saying that G2 is exponentially larger than G1 sounds like an understatement. I feel like we need a new word to describe the absolutely mind bobbling distance between the two.

[@PhilBagels]

I know the digits of Graham's number in base 3. They are 10000000...0000000.

[@PhilBagels]

And while I'm at it. the digits in Graham's Number in base 27 are also 100000...00000. And the same is true in base 3^3^3 (~7.6 trillion), and in base 3^3^3^3, etc.

[@erichernandez6102]

I know Graham's number in base Graham's number: It's 10.

[@coopergates9680]

Eric Hernandez That's nice, unless you attempt to write G2, G7, G33, etc, etc. in that base.

[@zoranhacker]

Eric Hernandez umm isn't it 1?

[@zoranhacker]

zoranhacker oh right, it's not lol

[@guepardiez]

Graham once taught a king how to play chess, and the king promised to give him g1 grains of rice for the first square on the chess board, g2 grains for the second square, g3 grains for the third square...

[@apollog2574]

And so the universe was annihilated

[@donovanshea3308]

And henceforth the Venezuelan currency was inflated beyond belief

[@bachpham5025]

Jokes aside. Even if the king promised to give him only 1 grain of rice for the first square, 2 grains for the second, 4 grains for the third, 8 grains for the forth…etc ; the king cant keep his promise with all the rice on earth!

[@SirBojo4]

@@donovanshea3308 Consequently Uncle Sam embargo'd Venezuela to space-time's fabric decay!

[@pholdway5801]

I read this story in a book on maths that i got for a Christmas present when I was 8 years old It was big and a reddish pink colour on it's hardback cover , I still have it. It also had Pythagorean triangle story with large coloured illustrations.

[@bastian_5975]

Sum up this video in one sentence. Graham's number... IS OVER 9000!!!!

[@coopergates9680]

Bastian Jerome You mean (((9000!)!)!)!, or four consecutive factorials? Even that is less than g1 lollol

[@bastian_5975]

ok so I am correct In my asesment.

[@coopergates9680]

Bastian Jerome What game invented that phrase?

[@bastian_5975]

it wasn't a game, it was a man,and it was called Chuck Norris. He gave it to a show called Dragon Ball Z though. Goku had the line. someone asked what Goku's power level was when he went super saiyan and he responded "It's OVER 9000!!!"

[@bastian_5975]

ok it came from the show Dragon Ball-Z.

[@Infinite_Omniverse]

I used to be a mathematician like you, but then I took a Knuth's Up Arrow in the knee.

[@blue9139]

Oh no there are too many

[@skair5425]

A FELLOW SKRYIMMER

[@Lordidude]

Gra'ms Noombah

[@utetopia1620]

There's a lot of math jokes here, but I laughed more at your comment, mainly because I'm not a mathematician.

[@hemanthgowda5269]

Lol

[@idkwhattoputhere616]

its just their accent

[@verdi8325]

This is my favourite KZbin video of all time. Absolutely blows my mind.

[@ckmishn3664]

Prof. Graham did a much better job of explaining the underlying problem directly than either Tony or Matt did with the "committee" analogy.

[@greatwhitesufi]

Well, he made the number.

[@tcocaine]

he neither made the number nor explored it. Anyone can simply do this themselves..

[@zoewells3160]

@@tcocaine Well no nobody "makes numbers" but you know what they meant

[@adamqazsedc]

Agree

[@l34052]

I'm really bad at maths, I mean really hopeless but I've been fascinated by grahams number since I first heard about it a few years ago. There's just something really intriguing and fascinating about large numbers and the maths behind them. This and quantum mechanics are the 2 things I'd most dearly love to understand in life.

[@andreasdluffy1208]

Now dont hate me. But I think quantum physics is much more important then math. This type of math is kinda useless in my opinion

[@abdulazis400]

@@andreasdluffy1208 useless type of math WILL BE useful given enough time.

[@dailybroccoli7538]

@@abdulazis400 and by those time, Quantum physics would have been printed in high school text books. Higher Maths is not useful period

[@newwaveinfantry8362]

You're really ignorant if you would generalize all of higher mathematics as useless.

[@MABfan11]

@@abdulazis400 wonder what Googology will be useful for...

[@miklemikemuster]

"pretty much nailed it". I love these guys.

[@rxhx]

Two questions though: 1) Why does Graham's number finish at that satisfying number G64? 2) Why/how do we know its last digits but not the first??

[@Machame08]

Given the hidden synchronicities prevalent in math I think it would have almost seemed stranger for it to finish at some arbitrary number

[@Travvypattyy]

Minecraf

[@karlfeldlager7662]

ad 2) Take powers of two: They end in 2,4,8,6,2,4,8,6 .. but start with 2,4,8,1,3,6,1,2,5,1,2,4,8,1 .. . At the end we can compute "modulo", at the front not.

[@T0rche]

Graham's Number ↑↑↑↑↑↑Graham's Number worth of arrows↑↑↑↑↑↑ Graham's Number

[@BradenBest]

G [G + 2] G From an abstraction of en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation where [N] = ↑(N-2)

[@norielsylvire4097]

T0rche (g65)

[@thehiddenninja3428]

Smaller than G66

[@bluey1328]

g64? dang even math trying to get in on that nintendo power...

[@STANKYCHEEZEMAYNE]

G TO THE POWER OF SIXTY FOOOOOOOOOOOOOOOOOOOOOOUR

[@therealjoediaz]

They have a stack of g

[@Attaxalotl]

The true biggest number: N64

[@VaraNiN]

Is there a way how Graham got to this stupidly big number, or has he just made it up and said the anwer just can't be higher than this?

[@DonSunsetAtDawn]

He probably proved it.

[@Maxuro]

Man really... is this supposed to be a serious comment? Or you are just trying to be fun? Because you're looking more stupid than funny. You really think that exists a mathematical theorem proven by just saying "Hey MAN! i made up this PRECISE and EXACT number, i'm sure that the solution of this question is under this number MAN because WHATEVER MAAAAAN, IT'S COOL!" Seriously?

[@VaraNiN]

Raumo Yes I am serious. Why cant Grahams Number be the same just with 4s or 2s or 5s or whaterver at the start? And why is it 64 times and not 63 or 65? I just don't see any way how you can come to such a gigantic number. Of course he had some theorys that said how large the number approx. has to be, but would it matter if I add or subtract 1? Or 2? Or a million? A trillion? A google? Or even a googleplex? Would this really change Grahams number in a way that it affects the whole theorem? That's what I meant to say with my original comment. But if you can explain to me why it starts with a 3 and has 64 iterations and that it WOULD matter if I would subtract 1 that's fine. I will be happy to accept it. (But please without starting to rage again, ok?) P.S: Our argument seems kinda' pointless, because I think someone has proven that the solution is between 13 and 2^^^6 (2 triple-arrow 6). Still a gigantic number but much, much, MUCH smaller than Graham's Number, I think we both can agree on that^^

[@gocity9]

obviously he proved it otherwise it wouldn't be so widely known.

[@Timmoppy]

That was explained in the video as to how he got there..

[@SnlDrako]

Math. Where you can put it "it's somewhere between 6 and Grahams Number" and be considered precise AF, while messing up two decimal points in an equation and still fail in class. I love math.

[@methanbreather]

things like this happen when you don't keep your mathemathicans busy.

[@dash0173]

Oh and what do you get when you multiply Grahams number by Grahams numer?

[@Infraclear]

graham's number^2...

[@dash0173]

wrong

[@bangpaf2328]

(Graham's number)^2

[@TheTruthandEmotia]

Grahams number to the power of grahams number

[@ligueardent]

2Graham's number

[@jagjitdusanjh8356]

What would be the final digit of Graham's Number in Base 12?

[@MrCubFan415]

Either 3, 6, 9, or 0. Not sure which, though.

[@daleftuprightatsoldierfield]

Mr. Cub Fan 415 I'm pretty sure it's 3

[@arnold84120]

it must be within this set s = { 0,1,2,3,4,5,6,7,8,9,A,B} where A and B are the eleventh and twelfth digit in base 12

[@FaceySmile]

you don't say

[@anwarinianwarini2660]

E

[@TheJaredtheJaredlong]

I still can't imagine what logical sequence of steps gives you such a massive number as an answer.

[@tristo2005]

Numbers can get really big really fast given the right equation

[@grainfrizz]

Infinity is larger than Grahams number but infinity is for sissies.

[@blazintitan277]

Yup! We totally nailed it guys! Time for a coffee break!

[@dragoncrystal24]

Thanks for explaining this! Graham's number is now my new favourite number, and I can't wait to see what my math teacher initially thought about it (he's guaranteed to have heard about it before, he's a math addict)

[@amogus5902]

I once heard an analogy to describe grahams number, and it kinda helps me to wrap my head around it- If you filled the entire universe with digits the size of a Planck length (0.00000000000000000000000000000161255 meters) and in those digits were universes filled with Planck length digits, you would not have enough digits to represent Grahams number. For reference, there are 10^186 Planck lengths in the universe

[@philip8498]

i dont think you would have enough digits in there to describe G1 in there. let alone G64

[@vedantsridhar8378]

@@philip8498 In fact there isn't even enough space to write down all the digits of 3^^^3! (^ stands for 'arrow'). There isn't even enough space to write down the number of digits in the number of digits. Even the number of digits in the number of digits in the number of digits. And you keep saying 'in the number of digits' 7.6 trillion times, before you get to a number which you can theoretically write down in our observable universe, because that number contains a few trillion digits.

[@TheSpotify95]

@@vedantsridhar8378 Indeed. Remember, 3↑↑4 contains 3.6 trillion digits (you'd need a whole library of books to be able to print this number in text), 3↑↑5 has a 3.6 trillion digit exponent (so already we can't describe the number of digits, as that number is more than the Planck volumes that could fit the Universe), and 3↑↑↑3 actually means 3↑↑(7.62 trillion). That's 7.62 trillion, not just 5.

[@mustafamkamel]

One of the things I don't understand: why did Graham stop at g64? I think it's already proven that you can't even imagine how big a number it is, so why don't go higher that 64? Also, Why is it based on 3?

[@Nebukanezzer]

Those questions you'd need to read his paper for.

[@alexanderbrucegill8091]

The reason is because g64 is not Somme randomly made number it’s the upper most awnser to a hyper dimensional cube problem and he made this notation too reperdant this

[@jarchibald14]

This is one of the best videos on youtube, I come back once every couple years and watch it to get again

[@michaelhartley6791]

My year 11 class enjoyed this!!!

[@Jiimys187]

Michael Hartley but you’re not even a teacher

[@d3generate804]

Have you graduated yet?

[@EliasMheart]

Funny way to threaten someone as a weird supervillain: "Hands up, or I'll think of Graham's Number, and this whole area will go down!!" xD

[@wheresmyoldaccount]

Even plain old 2^64 -1 from the chessboard rice problem is a very large number (18 quintillion and something) to imagine. Once we get to 3↑↑↑3 , which is 3 with a power tree of 3's 7.6 trillion digits high... my brain gives in. 3↑↑↑3 is a number bigger than 10^3000000000000, whereas 10^80 accounts for the number of atoms in the known universe. And that number 3↑↑↑3 is way way way way beyond minuscule compared with 3↑↑↑↑3 (G1) which is way way way way way beyond minuscule compared with Graham's number.

[@ecksdee9768]

and to think other numbers like TREE(3) and SSCG(3) make Graham's Number look like 0 in comparison really blows your mind on how big numbers can get

[@hyrumleishman3624]

In conclusion: Numbers are ridiculous.

[@TheSpotify95]

Actually, 3↑↑5 is bigger than your 10^(large number) that you describe, since 3↑↑5 is bigger than googolplex. At least you can actually wrote down the full tower length of 3↑↑5 on a piece of paper. You can't do that with 3↑↑↑3 (3↑↑7.62 trillion).

[@gupta-pw5xb]

*Infinity* : Here's my son

[@j.hawkins8779]

With TREE(3) being either the older or younger brother LOL

[@jfb-]

And what happens when you take g(graham's number) and apply the Ackerman function to it?

[@electroflame6188]

+IdontHaveAnyGoodNameIdeasButIHaveATaco You have no idea what the Ackerman function is, do you?

[@arkues1161]

jfb-1337 your just a kid thay thinks he learned something cool but doesn't actually gets it

[@halo4224]

it's still smaller than g_65

[@delrasshial7200]

You fuckers

[@GirGir183]

You get sued by Ackerman.

[@pcarlisi]

July 8 2020, RIP Ron Graham, the big number man...

[@9RedJohn9]

7:20 "This is just AH" best part!

[@yv1547]

Me trying to hype myself up to the tinder date 7:56

[@vladpintilei6204]

😂😂😂

[@martinshoosterman]

Is it possible to have negative arrows, or approximate the values for a non integer number of arrows. Because then you could define a function as F(x)=3(x arrows)3

[@drinkingthatkool-aid3193]

Hmm I wonder. I guess if you had infinitely many negative arrows that would be equivalent to take the nth root of a number n times

[@dailybroccoli7538]

There will be 'infinitesimal' problems that need to be solved.

[@karlfeldlager7662]

Fractional exponentials (in the sense of exp[1/2](exp[1/2](x)) = exp(x) and so on) indeed exist: Hellmuth Kneser's half-exponential function

[@claudioestevez1028]

I just realized how precise all my mathematical answers have been. I've been nailing it all my life.

[@hamedhosseini4938]

Mother: why don't you hang out with neighbors kid? Neihbors kid:

[@Graverman]

his IQ 1/g64

[@EpicB]

0:33 Just out of curiosity, I decided to calculate that entropy equation. Assuming r=4, here's what I got: Smax=A/4L^2 A=4πr^2 L=1.616*10-35 m A=201.06192982974676726160917652989 4L^2=1.0445824*10^(-69) 201.06192982974676726160917652989/1.0445824*10^(-69)= Smax=1.9248067919749247858436938678068*10^71 There you go.

[@EpicB]

Cooper Gates I don't think even Oliver Queen could handle one arrow.

[@coopergates9680]

***** Who's that? 3↑1 = 3 and 2↑2 = 4 haha

[@msolec2000]

+Naveek Darkroom That is definitely something Twilight would do.

[@aczepllin]

Love this got 2 likes and they both probably did it because they assume it's correct. Haha

[@TheLuckOfTheClaw]

Why would r be 4? Shouldn't it be something around 10^-1 m or less?

[@TheAed38]

The crazy thing is that as Carl Sagan puts it "A googolplex is precisely as far from infinity as is the number 1." As big as it is, the same thing goes for Graham's number.

[@jamesa6693]

Police officer: excuse me sir do you know how fast you were going? Mathematician: the speed limit is 15. Police officer: you were doing 157 Mathematician: nailed it😂

[@trentedwards6444]

I actually thought about something like this during class the other day, I was seeing the highest number I could get on the calculator with the least number of digits. This was how I did it ^-^

[@merloon]

Just have to ask the question that has to be on everyone's mind... how does Graham's number stack up against the googolplex?

[@numberphile]

no contest

[@93MHD]

the ratio between Graham's number and googolplex is approximately equal to Graham's number ;)

[@93MHD]

if you raise googolplex to the googolplex power googolplex times that wouldn't compare the googolplex root of Graham's number!

[@dementy9]

Timotheus24 well i could replace 3 with gogolplex an do the g64 with it right? :D

[@Quendarth]

DenolcTV Why stop there when you can replace 3 with googolplex and then do g(googolplex)?

[@scaper8]

I once heard in regards to Graham's Number, that there are more digits in it in standard notation than there are estimated protons in the universe. Fantastic, fascinating, and fabulous!

[@coopergates9680]

scaper8 You only need 3↑↑↑4 to do that, lolz

[@professorgrimm4602]

"The answer is between 11 and Graham's number" Wow thanks, that narrows it down so much. Any day now we'll have the precise answer.

[@NeemeVaino]

Explaining this to kids: Forget about g64, let's talk g1, the 3↑↑↑↑3: Smallest thing that can theoretically have any meaning is Planck length cube, largest meaningful volume is observable Universe. How much could one contain others? Well, something less than googol², not even googolplex that is 10^googol. So, googolplex is a nice number that we can tell how big it is - it has googol digits. About g1 we cannot do that. We cannot even tell how big is the number that tells how big it is. If we start to ask how big is the number that tells how big is the number that tells how big is the number ... so on, for how long? We cannot tell how long. How big is the number that tells how long it takes? No. How big is the number that tells how big is the number that tells... ... how long it takes. Still no. We cannot tell that. Meaning of words do not last that long. That's just g1, kids.

[@MKD1101]

*I am already struggling to find g spot and now you want me to figure out g64 as well!!!!!!!*

[@sdrtyrtyrtyuty]

"You'd run out of pens in the universe" Couldn't we just make more pens as we write?

[@swagmonee5699]

We would probably run out of resources in the universe before we could write it down. Not just in the pens, but we would also need to write this number down on the fabulous brown paper the people at Numberphile famously use.

[@evilcam]

+sdrtyrty rtyuty Not unless the universe itself turns out to be infinite or nearly infinite, and the materials which make up pens and paper and ink was also infinite or nearly infinite. Well, infinite is not a useful bound in this case (because Graham's Number is still finite) , but we certainly need more atoms, and space itself to be bigger than what we currently observe. As we currently understand, there are around 10^80 individual atoms in the observable universe. Now, don't scoff at that number, it is immense. That is a lot of atoms. However, 10^80 is much smaller than Graham's Number. So, at least as per the estimation of how many atoms exist in the observable universe, there are nowhere near enough atoms themselves to write out even a small fraction of Graham's Number. We would have to find more atoms to convert into ink, pens and paper to write it out. There simply is not enough atoms in the known universe to write it down, even if you made the integers only the size of one atom. Likewise, there is not enough space in the Observable Universe to write it out. Keeping in mind that there is a whole lot of space that we can measure, it's still nowhere near enough. The Planck Length, which is the smallest computable region of space (at least where quantum energy scales can form wavelengths we can comprehend) is pretty damn small. Smaller than any atom, smaller than anything which makes up the things that make up the things that make up atoms. Even if we counted them and assigned one per digit of Graham's Number so that every Planck Length corresponded to just one digit, there is not enough of them in the Observable Universe to write it out. The best I could find with a quick google search is that there are 7.04 x 10^64 Planck Lengths in the radius of the Observable Universe. I found a very rough and very approximate calculation on some physics forum which said there were 10^186 cubic Planck Lengths (thank you to Ilya for doing this for us). Which is still much smaller than Graham's Number. We would need G64 Planck Lengths, cubed, in the universe to get that ratio. Unfortunately, we don't actually know how big Graham's Number is in any sense which would tell us how big the Universe would be if we had G46's worth off Planck Lengths, so I can't really give you an idea of how big that would be, because I don't know it and have absolutely no way to go about thinking about it.

[@steve1978ger]

+sdrtyrty rtyuty - Not just pencils. The current estimate of atoms in the universe is 10^80. So if you turned the whole universe into some kind of storage device where every atom would store one bit in its spin, you could not even remotely store G in it.

[@sdrtyrtyrtyuty]

+steve1978ger Unless thr universe turned out to be substantially bigger than previously thought? Maybe we could find enough resources for this worthy feat ;) Also which way would we write? He said they dont know the first number but they do know the last one so I guess theyd start from the end and work forward. And also what is the left most known digit of Grahams number?

[@rangarolls6018]

There wouldn't be enough atoms in the universe to do that

[@PC_Simo]

Donald Knuth: ”How many arrows do you want?” Ron Graham: ”Yes.”

[@dash0173]

After a while, numbers just get to be scary...

[@AceInAcademy]

loved the explanation once again, hope to grasp the complete number in one go.

[@regan3873]

2:15 I love this dude’s handwriting

[@as7river]

Between 6 and G64. Matt: we've pretty much nailed it. That's a big nail, Matt.

[@mil3761]

Ok, I get how Graham's number is obtained. But what I don't understand, how did they work out with 100% certainty that Graham's number is the upper bound to this problem? Surely there is mathematical proof but how do you work on such a number to even prove it? And why is Graham's number specifically 64 iterations of the arrow notation? Is there a reason why it stopped there or could it have stopped at 63 or 65 instead?

[@depthsofabjection]

Milan Shah I wish someone would answer this.

[@nameguy101]

***** I think 3 is the smallest number that grows in this context. Because 2 + 2 = 4 2 * 2 = 4 2 ^ 2 = 4 so it doesn't grow.

[@shubhamsengar2558]

Nameguy 2 does grow, as long as its not 2 arrow 2.

[@mil3761]

Shubham Sengar I was just about to reply in the same way. The first step of Graham's number is 3 and then four arrows and then 3 again. If you replaced the 3 with a 2 then it will grow a huge amount (but not as much as the 3).

[@mil3761]

Milan Shah Numberphile could you please answer my question above please. It's a very interesting number but many people are still confused with how you can use Graham's number in a mathematical proof when it is so huge. This would make a really great video addition if you could explain it please.

[@rumoursinspace]

Even yo momma weighs less than Graham's number...

[@bonelessbooks9263]

oof

[@dtripakis]

I always thought the largest meaningful number was the number of atoms/electrons/whatever, in the whole universe. I was so wrong!

[@flixyy]

.

[@subscribefornoreason542]

Here's a bigger number- Behold...G65 Now I just need recognition

[@Xris10BUL]

G(G(G65))

[@sigalig]

g-2=3(g-1 arrows)3.....g-sus.

[@angelinahinkley9730]

sigalig g-2=3 g-1⬆ 3g-sus

[@Artificial-Insanity]

I think it's -1/12. ;)

[@blue9139]

Nope lol

[@youregonnaletityeetyouaway2882]

fun fact: g(64) wasn't the number in grahams original paper, the original upper bound was actually much lower than that but martin gardner used g(64) to make it easier to explain so he could popularise it. the upper bound is now even lower (i think 2^^2^^2^^9?) and the lower bound has also changed to 13

[@finmat95]

from 11 to 13? that's a huge improvement!

[@MABfan11]

the original number is roughly equal to G(7), which is why it has got the nickname Little Graham in the Googology community

[@andrewbloom7694]

5:15 "And all people appear in....I forget" Ah yes. The Parker Graham's Number Analogy

[@eemikun]

8:48 Tony foreshadowing the TREE(3) video that came out five and a half years later!

[@thepersonwhocomentz]

Is Graham's Number larger than the amount of cubed Planck Lengths in the observable universe? and if so, and the observable universe were an equilateral cube of equal size to its current estimated size, how many dimensions would you have to add of equal additional length (equal to the length of the sides of the stated observable universe cube) (thus going from cubed Planck Lengths to Planck Lengths^4, Planck Lengths^5, etc.) in order to eclipse Graham's number?

[@royhe3154]

the number of plank volumes in the observable universe is only about 10^185, no contest there

[@TheSpotify95]

Which is smaller than googolplex, which in turn is smaller than 3^^5. And 3^^^3 (the bit before you get to even G1) = 3^^7.6tn - i.e. a stack of 3's, 7.6 trillion high. Then you've got G1, i.e. 3^^^^3, where you've got an unbelievable amount of power towers to deal with. And that's just G1. Wait until you see what G2 is...

[@rykehuss3435]

3^^^3 is already ridiculously, retardedly larger than that. Let alone G1.

[@mannaggiacristo]

What are the 3 theorems that use bigger number ? o.o

[@logenezell6442]

Tree* theorems, I haven't the slightest clue what they are, but he called them tree theorems.

[@tallestfinder]

Tree theorems

[@tysonprice5058]

I think he said "tree theorems."

[@gyerekakilaci]

It's tree-theorems not three theorems :D

[@TheGiagoskap]

I think he said tree theorems, not three theorems.

[@jamessmith84240]

Can we take a moment to appreciate how lucky we are to have our human brains? I just realised we have the power conceive ideas larger than the universe we live in! Crazy stuff.

[@QuantumBraced]

What if you took every plank space in the observable Universe and expanded it to the size of the observable Universe and counted all the plank spaces in all of those Universes, does that number come anywhere within 1% of Graham's Number?

[@cardogkitchen4106]

10^185^10^185 still nothing compared to G1

[@PeterGeras]

cardog kitchen Not even, it's just (10^185)^2 = 10^370

[@BokanProductions]

cardog kitchen I think G1 (and all the other G's) are googolplexian (or more than likely a few thousand digits more than that) digits. Googolplexian is a number with a googolplex digits. Also I think graham's number is a bit exaggerated. Sure it's beyond human comprehension but I think you COULD fit it in the observable universe. I mean come on when you think about how many galaxies, quasars, stars, planets, and the possible alien civilisations that might live there then there's bound to be a lot of atoms you could right a digit of the number on. Not that you could since not only we don't know how many digits graham's number is exactly but no one could write on an atom.

[@PeterGeras]

BokanProductions I smell a troll.

[@BokanProductions]

How am I a troll? I'm just giving you guys my theory.

[@tommythai1043]

Numberphile I've got a question, how would you even go about working out the final digit of said graham's number? I really wanna know

[@beo3812]

very carefully

[@haddenindustries2922]

are you home between 7 a.m. and Graham's number?

[@murphycooper7557]

Graham Number: Oh, yeah, I'm the king of numbers, I'm sooo unbelievable large... TREE(3): Hold my beer, baby!