How Big is Graham's Number? (feat Ron Graham)
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- Опубликовано: 21 июл 2014
- See our other Graham's Number videos: bit.ly/G_Number
More Ron Graham Videos: bit.ly/Ron_Graham
More links & stuff in full description below ↓↓↓
The magnitude of Graham's Number is difficult to fathom - Ron Graham himself attempts to explain.
WHAT IS the number: • What is Graham's Numbe...
Extra footage: • Ron Graham and Graham'...
Our original Graham's Number video: • Graham's Number - Numb...
Any errors in this video are mine (Brady's) in editing... For example 3^^^3 is actually even bigger than I say... Graham's Number is just too big for my small brain!
See an improved explanation at: • Graham's Number Escala...
Wikipedia page on Graham's Number: en.wikipedia.org/wiki/Graham's...
And the arrow notation used here: en.wikipedia.org/wiki/Knuth%27...
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Наука
Graham once gave his number to a girl. She never called him.
Untrue, she just hasn't finished dialing. . ...
that's awesome xD
Kelly Jackson hahahaha
33333333333333333333333333333333333333333333333333333333333333333333x3^333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333334
Ivan & Fritz you wouldn't be able to press that many numbers in a googleplex number of lifetimes even if you could press a googleplex keys every billionth of a second. You wouldnt even get to g1
"somewhere between 13 and graham's number" is one of my favourite phrases now.
That and the Parker Square.
"How old are you?"
"How many pieces of chicken would you like?"
"How many times are you going to use that phrase?"
"Somewhere between 13 and Graham's number."
we've pretty much nailed it, as far as I'm concerned.
All of a sudden, its proved grahams number + 1 can be possible without making that pattern....
A card in Cards Against Humanity: Mathematician Edition
B: "What do you want the first digit to be?"
R: "Well in binary it's 1!"
Clever guy, lol
Brian Bethea a very nice understatement I think😊
In base 3, it’s 1 followed by a loooooooooooooooooooong line of zeroes.
Niiiice.
SUPER. UNIMMAGINABLY *SUPER.*
Depends on where you put the MSB
At 3:27 when Graham said "You ain't seen nothing yet.", I knew i was boutta see some THICC numbers.
I think infinity is easier to imagine than this number.
ILikeWafflz i think thats because infinity is more of a concept instead of a number
well, you should read about a number called TREE(3). It is so vast that the numbers of arrows needed to reach it is close of TREE(3) itself. This even holds for much smaller number such as Hydra(100), 3&3&3 (Triakulus) and much larger number like BH(100), SCG(13), Loader's number, BB(1000), Xi(10^6) or Rayo(10^100)
Fluorosulfuric Acid It cannot be that the number of Knuth's arrows needed to reach TREE(3) is close to TREE(3) itself.
George Hamilton its exactly what I said :V
Fluorosulfuric Acid "[TREE(3)] is so vast that the numbers of arrows needed to reach it is close of TREE(3) itself." Unless you are using a rather arbitrary definition of _close_, your statement is not possible because the function of Knuth's arrows increases numbers by so much.
I really liked the part where he said "three"
Iron Dorito three through the three to the three
Your comment is 20 likes away from being appropriate (313 likes when i made this comment
3 replies......................until I came XD
I would like this comment, but it has 333 likes and I don’t want to ruin it :)
Don't worry: it has 424 likes.
Make that 425 likes.
This gave me a lot of existential dread
I didn't expect my favourite terraria youtuber here :o
@@mackan2277 I didn't expect my favorite Terraria youtuber or another Swede (tror jag iaf) here
HeIIo
E
Because this is the top comment, it made me hear music. You know what music😁
You can't even imagine a 4-dimensional world, right. So imagine a world with Graham's number of dimensions
well, we do live in a 4-, not 3-dimensional world! We're just not very able to control our movement in the 4th direction and therefore mostly "float" in positive t-direction with more or less uniform speed. (And there are probably 6 - 20 more "hidden" dimensions which are so tighly wrapped up that we'll never notice them...) Yet I agree, that' not much compared to g64.
Well, I'm pretty sure spheres of Graham's Number of dimensions would be pretty inefficient at stacking!
No.
STOP MAKJING MY HEAD HURT!!
@@mfhasler dimensions in math and not physics are spatial dimensions so time wouldn't count
"You ain't seen nothing yet"
True words from a man who has wrapped his head around things that most everyone can't hope to understand
*YES!* 2🙏🏻 ↑(🙏🏻+ 🙏🏻 +🙏🏻 )↑2(🙏🏻 ²🙏🏻³ +10🙏🏻)↑3 🙏🏻
3^^.........^^3
Graham's wife: "Stop your work and get in the kitchen! I made spaghetti!"
Graham: "No! Just ONE MORE ARROW!"
StarTrek123456 LOL
StarTrek123456 xDD
+StarTrek123456 They eat in the kitchen? Where do they cook the food, the bathroom? :D
Kelly Jackson I have a big table in the kitchen, so i eat there. The only other table i have is in my living room, but i am too lazy to take my food there :D
+StarTrek123456 Gawd you made my day xD
Graham's number in base Graham's number: 10
I love it. :) Base 11???
It is 1
@@fakefury1198 grahams number plus g64^0 or grahams number plus 1
@@vighnesh153 No. x in base x is always 10.
2 in base 2 is 10.
16 in base 16 is 10 (A).
@Graham's Video World
What about 10^10bk^(10^bk)?
“You ain’t seen nothing yet”
Said after the number is too large to be on the screen or even imagine.
Well, that escalated quickly.
Zekzram Reshirom i mean that really got out of hand
Then do that 64 times, even dubble 64 times is huge!
"Could just be 13, though." >_>
Got to love the huge interval between the two.
huge doesn't even *BEGIN* to cut it. xD
Daniel Cannata Indeed.
Let's you question the sanity of this guy.
How did he calculate that Graham's number is the limit?
3:27 mathematecian goes beast mode
Here's the prime factorisation of Graham's Number:
3x3x3x3x3...x3
But 5 can go into (G64 - 2) because the last digit is 7 and 7 - 2 is 5, so (G64 - 2) isn't prime.
Ohhhhh. Nice.
@@AC-fl1le he said G64 not (G64 - 2)
Onyuhhno
Huygcduyhrifdrdfjgfkngkrjjgibghehfrjukrfji(jijmjihdr
New use for Graham's number - counting the number of times in his life Graham has said the word 'three'
He talks like a bot tbh
+The True Fizz nothing can ever be compared to grahams number realistically that's how phenomenally enormous it is :)))))))
+The True Fizz 3 to the 3 to the 3...
cause 3 is a word :|
+Daniel Cannata I said the word "three", which is a word. "3" is a number, but when he speaks it he is saying the _word_.
whats the leading number?
"in binary, its 1"
LOLLED
it has to be lead by a one in binary
Ariel Sproul That's the joke
last digits in base 2 is 1
Ariel Sproul r/whoosh
My bank account: 3arrow3
My spending: 3arrowarrow3
I wonder what you bought that cost 3 trillion dollars
@@bunbunnbunnybun* > 7.6
My credit scores 3arrow0
@@BlokenArrow Glad you still got 1 dollar in your pockets, eh?
You mean 3↑3 and 3↑↑3?
What a legend
Ron graham you will be missed
He died?
Yes I think he did
@@kevinnguyen552 yes. Sadly, he passed away on the 6th of July 2020
I didn't even know he died until recently.
"It's a number so big, we had to use Comic Sans." - editor
Ralph Anthony Espos yes
Legend has it he's never stopped saying "3 to the 3 to the 3 to the 3 to the 3 to the 3..."
Justin Ly i wonder how many lifetimes of our universe it would take if you could say 3 to the 3 every second
@@oz_jones a number close to Graham's number
Osmosis Jones: :-| (!!!!!!!!!!). As much as He would live.
@@oz_jones well I know the no of lifetimes you would take is 3 to the 3 to the 3 to the three to the...
I want to see 10h version of this
The first time i watched this, a few years ago, i thought there was only 64 arrows. Now understanding it better actually hurts my brain
Same.
And if I'm not mistaken, they still understated the size of even 3↑↑↑3 at 03:07 .
3↑↑↑3 is a power tower of ~7.6 trillion 3's.
Well, 3^3^3 (powers are evaluated top to bottom if there are no parentheses) is 3^27, about 7.6 trillion. So, 3 to that power is a bit less than the square root of 10 to that power, which would have 7.6 trillion digits. THAT number has 3.6 trillion digits already, and it's only a power tower of height 4. 3↑↑↑3, power tower of height 3^3^3 is unimaginably huge, and you need another arrow before even starting _Grahamization_ , the process of using G(n) arrows to define G(n+1). 3↑↑↑↑3 is far, _far_ below 3↑↑↑...↑↑↑3 (with 64 arrows), which is in turn tiny compared to the second step, G(2).
And THEN, there are another 62 (or 63?) steps to come.
BTW, literature about Grraham's number is highly contradictory, often with itself. Some say that 3↑↑↑3 is the starting point G(0), others say it's 3↑↑↑↑3 (with another arrow, like Graham himself did). Then, some treat one of the above as G(0), others as G(1) (and G(0) would be either 3 or 4). Nevertheless, the last number in the sequence is huge beyond comprehension either way.
@@achtsekundenfurz7876 I got that too.7 trillion digits which they have written down for it is nothing compared to a number with trillions of towers of 3. You get to trillions of digits using just maybe 15 towers of 3.
@@Parasmunt you get above trillions of digits using just 4 Towers of 3
@@achtsekundenfurz7876 Yeah, when they say about 3↑↑↑3 in the video, what they say is the result is actually just 3↑↑4. 3↑↑4 = 3↑(3↑↑3) = a 3.6 trillion digit number (bigger than Googol). 3↑↑5 = 3↑(3.6 trillion digit number) = bigger than Googolplex.
3↑↑↑3 = 3↑↑(7.6 trillion) which, if each 3 is written as 2cm tall, the tower will stretch to the Sun. And remember that the top 10cm is already bigger than Googolplex.
So the result of even 3↑↑↑3 cannot be written down, and remember, 3↑↑↑↑3 has that many iterations in it!
July 8, 2020, RIP Ron Graham, the big number man...
Wait what
He died?
I used to be a mathematician, then I took some arrows to the knee.
lol nice
+philphy101 ..... Graham's Number of arrows?
+philphy101 an arrow to the three?
some arrows to the three
+Marco Roque i was just thinking that 😂
"So, there you go. Graham's number. And if you think you understand it, you probably don't." 😂
Bit patronizing I thought
+Ynse Schaap I think he means no one understands it.
Christopher King Well Graham obviously does ;-)
+Ron Volkovinsky mind being precise when quoting the professor, please?
+Ynse Schaap - I'm pretty sure he doesn't either. Nobody can even conceptualize small numbers like a googol. Write them down, yes. Calculate them, yes. Understand them, no.
I think its outstanding to have various mathematicians who have decades of experience talk about their work on this channel. Great!
This made me have a thought: infinity is exactly that, _infinite._ Graham's number, no matter how abstract it has to be in order to even be measurable, written, or put into language anything can understand, is still finite. Bigger numbers, still finite numbers, are being made all the time. Infinity never ends and goes beyond all of those.
Just think of the absurdity of that, though. You can be more precise and make smoother lines with Graham's number (let alone infinity) than you could be making a line with individual strings.
Grahams number is closer to 0 than it is to infinity.
Yep, Grahams number is nearly zero compared to infinity.
@@M-F-H Would it not be practically 0?
Here's what people don't understand.
Technically, no number is near 0.
People are like n is near 0 than infinity.
But think about it
1 is near 0, so are
0.1
0.01
0.001
0.0001
0.00001
.
.
0.{infinite 0's}1
The fact is the only way you get a number to 0 is to subtract that number by itself or multiply it by 0.
@@asagiai4965 You’re right. Any number is equally between zero and infinity.
I'm going to have nightmares where I just see seas of 3s and hear 'three to the three to three three three th-..' until i wake up in Graham's dimension.
*"In binary it's 1"*
Brilliant!
Oh i see
Extraordinary.
I bet if I jump scared him while he was sleeping he would just wake up saying “3 to the 3 to the 3 to the 3 to the 3”
Also my name is graham
game: you take a shot every time Ron says 3
Brb mopping my brain bits off the floor and walls.
Need help?
hi thiojoe
What does brb mean?
Oh, hey there ThioJoe, didn't expect a computer expert to pop up on a math video's comment section.
@@addieperkins2599 Be Right Back.
333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333
This isn't spam, it's a quote from this video.
haha
that is am impossibly big overstatement.@~@
Daniel Cannata No it isn't you heard it yourself.
696969696969696969696969696969696969 is a better quote
So emotional. So inspiring. Almost cried.
3:09 makes no sense
you said the 3↑↑↑3 has 3.6 trillion digits.
the number itself has a tower of 7 625 597 484 987 threes.
At the top (level 1): it's 3
level 2: 3^3 = 27
level 3 = 3^27 = 7 625 597 484 987
level 4: 3^ 7 625 597 484 987 (if 3^300 has approx 140 digits, 3^7 trill must have at least thousands)
so level 5: 3^(number with thousands of digits) must have millions of digits. And we still have 7 625 597 484 982 threes left in this tower.
SO how is the bottom of this tower 3.6 trillion digits long
It's true. The error has the origin in this way of thinking: 3^^^3 = 3^^(3^^3) = 3^^(7.625.597.484.987) = 3 to the power of 7 thrillions. And so, how many digits this operation has? Log 3 * 7.625.597.484.987 = 3.6 thrillions of digits. This is a wrong answer, because the right answer of 3^^(7.625.597.484.987) is 3 to the power of 3 to the power of 3 to the power....and so on 7.6 thrillions of times!!! The real answer to the question "How many digits this operation has?" is "Only the devil, maybe, knows the answer".
@@ivantchakoff4067 if you keep multiplying by 3 you will see a pattern.
for example if you start at 1 x 3 you get 3, and keep count of how many times you multiply, so that's One, again 3 x 3 = 9 that's Two, and we didn't increase in a digit,
of course again will give you 27 and that's Three,
so it took three times but actually it's a 1/21 chance this will happen every other time it'll take just Two
you will continue until you get an extra digit and the number lies between 1 x 10^n and 1.1111111111111111111..11 x 10^n
like 10,460,353,203 (21st multiple of 3) than it's 3 more multiples
So 3^7,625,597,484,987 = [1- (1/21)] x 7,625,597,484,987 = 7,262,473,795,218 and / 2 for the difference between 10^n and 3^n
to get that extra digit = 3,631,236,897,613 amount of digits..............
3^3^3^3 or 3^7,625,597,484,987 will have about 3,631,236,897,613 digits..........
The 1/21 is for a little more accuracy,
Just know 3^n will have about half as many digits as 10^n.
I know right
yeah the number he showed was 3^3^3^3, not 3^^^3
And in video about tree(3) they say "Graham number is effectively zero compared to tree(3)".
Given that Graham's number is g(64)..
The size of TREE(3) is probably bigger than g^(g^(g^(g^(....
)(64))(64))(64))(64) with g(64) storey power tower high
@@r.a.6459It's worse than that.. the G function just doesn't grow fast enough to be relevant to TREE(3). Writing universes full of stacked G functions still doesn't get anywhere close. You need pretty advanced mathematics to even describe how fast the TREE function grows.
@@vincentvandergoes444 you know functions can go beyond exponentiation (i.e. repeated iteration).
Functions can be _tetrated onto itself_ right, not just integers. Like (g↑↑g)(n). It works the same way as the up arrow notations.
Now imagine (g↑↑↑↑g)(63) which is:
(g↑↑↑g↑↑↑g...g↑↑↑g)(63) with g(63) 'g's.
...but comparing (g↑↑↑↑g)(63) to TREE(3) is like comparing the size of 11D Universe to 3D planck volume!!
Now, how big is TREE(4) is, compared to TREE(3)??? It's beyond human logic, it involves dimensions alien to us.
@@r.a.6459≈3{{3}}g64
"Three four-arrows three. That's a big number" - Graham, 2014.
Yeah, "quite big".
+Ricardo Pieper When Ron Graham says it's a big number, you know it's a big number.
"For you"
Maybe this is why Valve can't count to 3?
I think that Valve ran out of paper.
*Paper change*
Now do that 3↑↑3 times.
3↑3 = 3^3 = 27
3↑↑3 = 3↑3↑3 = 3^3^3 = 3^27 = 7'625'597'484'987
3↑↑↑3 = 3↑↑3↑↑3 = 3↑↑(7'625'597'484'987) = ~1.258014298121 * 10 ^ 3'638'334'640'024
3↑↑↑↑3 = 3↑↑↑3↑↑↑3 = 3↑↑↑(~1.258014298121 * 10 ^ 3'638'334'640'024) = *INSANE NUMBER*
After G64 years of development, we cancelled half life 3
Tristan Jacquel they canceled half life at 3, but apple is still eventually going to make iphone g64? How is that fair?
What I love about these kinds of numbers is that if you were to physically write the single digits you would need several universes worth of matter just to make the ink.
I believe we would run out of Quarks, not Atoms, Quarks, in the entire universe (if it isnt infinite) if we were to assign each quark a digit in Grahams Number
@@alexanderzippel8809you’ve not really understood in a conceptual sense how big Graham’s number is if you’re trying to talk about its number of digits.
Even 3^^^3 has far too many digits to represent with any physical thing in the universe, be it quarks, atoms or Planck volumes.
It depends what you mean by physically. But it you just mean by ink no.
People under rate the universe so much.
I’ve seen this video at least 5x..it’s still mind blowing each time I see it
"If you think you understand it, you probably don't".
this sums up my life
And also sums up quantum mechanics lol
The funny thing about Graham's number is that it's impossible to describe how big it is in simple, understandable terms.
For comparison, if you were explaining for example how big the largest prime number found so far is, you can say "it has over 17 million digits", and that gives you a simple picture of how large it is.
However, you can't do that with Graham's number. It's so large that no description is sufficient to explain how large it is. You can't say "it has x digits" because x itself is unexplainable in simple terms. You can't say "the number of digits in GN is so large that this number itself has x digits" either because here, too, x is way too large. In fact, the amount of "recursions" you would have to make in this way to make x small enough to be explainable is too large to be explained in simple terms.
It quickly becomes so complicated that there just is no way of doing it.
The way you describe this "inexplainability" is fantastic, by the way.
Or you can just imagine the size of my.... as a comparison
That's right. The number of digits in Graham's Number (in any number, by the way) is it's log (rounded downwards) + 1. So even like that you can't imagine how many digits Graham's Number has...
It's interesting because that exact same explanation you used would apply as early as g1 = 3^^^^3.
Actually, Conway chained arrow notation helps. If you understand how fast arrow chains grow as the terms (and length) increase, you can get an understanding of how much bigger one chain is than another.
Eg 3 -> 3 - > 64 -> 2 is less than Grahm's number, but 3 -> 3 -> 65 -> 2 is bigger, and 3 -> 3 -> 3 -> 3 is much, MUCH bigger than Grahm's number.
It's amazing that I just saw Ron Graham, a man who met Godfrey Hardy, a man who met Ramanujan.
Ramanujan number when?
I can't get enough of this "Big Number" stuff on Numberphile.... who knew it could be so intoxicating??? Blows my mind anew everytime I watch it.
"You ain't seen nothing yet." - Ron Graham
I was going to tell my girlfriend about my favorite number, Graham's Number, so I asked her what her favorite number was. She said it was two, and when she explained why, I couldn't stop laughing. "I like two because it's one more than one, and it's easy to understand."
Taylor Foulkrod love it!!!
Wowwwww
Where do you get your paper
+Taylor Foulkrod Take a drink everytime the word 3 is said. You won't regret it.
+Aqua Man Until the next morning...
2:04
"...3 or 3 to the 3 this is 3 3 to the 3..."
his voice and handwriting are so soothing, i could listen to this guy explain stuff all day.
"It's your number! What do you want it to - first digit - to be?"
"Uh... Well, in binary, it's one!"
Cracks me up.
I was following along with all of this just fine!
But then, I took an insane triple arrow to the three.
How does this comment not have more likes
The best adaptation of a meme I've ever seen.
I actually fell out of my chair I was laughing so hard at this. Well done sir, well done.
Very well, since nobody else will do it, I volunteer to be the one who admits that I to not get this joke.
Jack Sainthill Look up Arrow to the Knee meme
seeing as 3^^3 is 3^(3^(3) ), does that mean 2^^2 equals 2^2? if it's number of exponents on the tower than it doesn't matter how many arrows there are, using 2s always results in 4.
Yes, you are correct. For example 2^^^^^^2 = 2^^^^^2 = 2^^^^2 = 2^^^2 = 2^^2 = 2^2 = 2*2 = 2+2 = 4
(The pattern even continues to the lower operations before the arrows)
2↑^∞ 2= ∞ or 4
I never would have thought that trying to understanding the mere number of digits in a number would be such a bamboozle.
Terrific video, I enjoyed it very much!
I lost him at "3".
Nice. Had to smile at that comment.
After that is another 3, I believe.
Lol
😂😂😂
Hey look, the average global word usage list has updated, and three has risen >90%
Ron was the coolest prof I TAed for when I was in Uni. A legend.
Would like to submit an erratum here: at 3:16 in the video it says that 3↑↑↑3 = 1.258 * 10^3638334640024. Actually, 3^(3^(3^3)) = 1.258 * 10^3638334640024 and 3^(3^(3^3)) is just the top 4 levels of 3↑↑↑3 exponential tower, which is a total of 3^(3^3) levels high.
Now factorialize it...
+Pixelater4 I'm pretty sure that number wouldn't even be close to g65. 27 Factorial is nowhere near the 7.6 Trillion monstrosity.
+gredangeo He means factorialise Grahams number, not 27.
+gredangeo Actually, 27! has 29 digits, far bigger than 7.6 trillion which has 13 digits.
+Pixelater4
here's a large number
g128!^(g128!^(g128!^...(g128!))))
the tower is g64! layers -i think thats what the amount of powers is called, im no expert at maths- high, and the trend continues
Now take the square root of it!
Let's make a new number, but instead of repeating that 64 times, we repeat Graham's Number times.
+David Andrei Norgren the mind blowing thing is, no matter what you do with it or how many times you multiply exponentially by graham;s number or anything, it would still be infinitely smaller than infinity
Let’s actually make a number where you repeat that process Graham’s number of times.
hoi te doi
grahamplex
Stuff by David why? What does it solve/help/prove/convey?
Rip the legend ❤️❤️
Rest in peace Ron Graham😔
Could Graham just say: "my number"?
😂he's sooo humble
"The so-called 'Graham's number.'" - Ron Graham
He probably calls it "so-called" because this was the publicized much bigger version, the actual number he used in the proof had G1 equal to 2↑↑↑↑↑↑↑↑↑↑↑↑3 and he defined the number at G7.
It's not really his number. It's named after him.
That's the name of the number. He doesn't own the number.
@@KalOrtPor Isnt it 2↑↑↑↑6?
Three.
MrTristan [TREE]3
sit down, son.
Daniel Cannata
Loader's Number: Shut up, TREE(3).
FOUR!!!!!
SSGC(3)
3^^3^^3^^3^^3^^^^^^^^
R.I.P. Ron Graham 🙏🏻
RIP Ron Graham
You will be forever missed.
Covid 19 got him?
@@spiderjump IDK but probably not
You can always use the back of the paper...saves you money and trees.
The marker usually shows through.... Also, the papers are sometimes sold to raise money for various reasons (extra production costs, charity, etc) so they do not go to waste.
+MegaHayzer When you deplete the tree farm, what do they do? They plant more, but if they need bigger production, they have to increase the area, which is very bad for the environment because it's not at all like a forest, it's like agricultural ground, which is really really detrimental. You're deluded if you think it's 100% solved...
+Numberphile if you guys wrote down Graham's number, how many sheets of paper could you give away?
+Christopher King It's actually literally impossible to write down Graham's number. There are 10^82 atoms in the (observable) universe, which is just a laughable fraction of the number of digits in the number ^^
박수연 I think you mean in the *observable* universe.
i begun feeling sick at 5:45 ... i'll talk to my cat about it and see his opinion
here I am imagining you talking to your cat and the cat replying to you his honest opinions regarding on that topic which seemingly somehow makes sense to you and agreed while nodding.
This is the greatest comment ever. I will lol 3 to triple arrow 3.
Schrodinger's cat?
"Hey Bastet come to see that sht"
Math teacher: do you understand
Me: uhh yeah...
My brain: wtf
Grate vid, thank you to Ron Graham what a legend!
I'd really like to know, why his proof works for G64 but not for G63.
Wouldn't that make a nice video??
Yeah I can't seem to find how this was actually used in a proof, I don't understand how a number bigger than we can understand was used to prove anything :(
or G69
Actually, Graham's original proof was for a "much" smaller number, though still huge. However, the pop math author writing about it (Martin Gardiner) found this version to be easier to explain.
The current upper bound is 2↑↑2↑↑2↑↑9, where even 2↑↑9 is equal to 2^2^2^2^2^65536. There's already no point in talking about how many digits this number has - your next reduction is to ~ 2^2^2^2^2.0035e19,728. 5 layers up is a number with almost 20,000 digits. Next reduction is to ~ 2^2^2^Xe(6.0312e19,727) - there's no point in figuring out what that X is though, as multiplying Xe(6.0312e19,727) by 10 gives you Xe(1+6.0312e19,727). But we would have to add 1e19,723 inside the parenthesis to even get to Xe(6.0313e19,727), so we can just discard the 1+. So at this point we can just flip over all the instances of 2^ into 1e, and get 1e(1e(1e(1e(6.0312e19,727)))). The 1s genuinely don't matter anymore. It doesn't matter how many we add either, from 2^^5 onward, you just tack on one more "1e(" at the beginning. Try to picture the jump from the number X to some number with X digits as you basic operation - then 2^^n means making that jump n-5 times starting with 6.0312e19,727. [Note that Numberphile actually got this wrong - 3^^(3^^3) is not a seven trillion digit number, its a number where you make the jump from X to a number with X digits seven trillion times. Seven trillion cases of "1e(" to write the number down with nested scientific notation.]
2↑↑2↑↑9 then is a number with 2↑↑9 cases of "1e(" in it. You no longer care even whats at the top of the tower, because the exact height of the tower has five *layers* of "put what digit you want here, it doesn't matter" to describe it. 2↑↑2↑↑2↑↑9 then has so many cases of "1e(" that we don't even care precisely how many cases of "1e(" it takes to describe how many there are. Silly big, yet not even the tip of the iceberg for large numbers. That analogy falls short though. Every analogy other than "literally nothing by comparison" sells the difference short, and even that is both just barely appropriate and at the same time too extreme to be factually accurate. Think about that - to try to communicate the scale of these numbers in succinct English, YOU HAVE TO LIE.
I prefer to explain the size of Graham's number in terms of scientific notation.
We all know scientific notation, right? Something like 4*10^15. It's used to write big numbers, and it's helpful because it tells us how many digits are in the number in the exponent. In that example, it's 16 digits, which is 15+1.
Now, lets start looking at just the exponent when we start putting arrows between 3's. (Also, because there's no easy arrow character, I'm going to use ! for an arrow)
3!3 is 27, which is 2.7*10^1. So 2 digits.
3!!3 is about 7.63*10^13. So 14 digits.
3!!!3 is about 1.26*10^3638334640024. That's a big enough number that I would tell you how many digits are in the number of digits in the original number. (13) This number is so big, that even if you had a typical 1 TB hard drive, you could not store this number in regular form. You would need 37 TB of storage (which of course, does exist, but it's not for mass-consumer use).
Now, just look at how insanely fast the digits are piling up when you add just one arrow.
For 3!!!!3, I'd probably need to tell you how many digits are in the number of digits of the number of digits of the original number. No computer anywhere could store this number. Even if you built a universe-computer in which every subatomic particle in the observable universe was its own bit, you would not be able to store this number. Such a computer could easily store every program, every song, every game, every youtube video, every file of every type, millions upon millions of times over, but would still not be able to comprehend the true form of 3!!!!3.
And 3!!!!3 is only g1.
g2 is 3(insert g1 arrows)3.
g3 is 3(insert g2 arrows)3.
Continue this pattern to g64. That's Graham's number.
Long story short? Nothing in the universe can comprehend the true form of even g1. And g1 may as well be infinitely smaller than g64.
On a side note, Graham said that it's unlikely that anyone will know the leading digits of g64. That's true, because it's impossible. Knowing the leading digit implies you know all the digits, and as I just demonstrated, that is impossible.
3!!!3 is a lot bigger than the value you gave; it's already far too big to represent with scientific notation or anything remotely like it.
How did you arrive at that value?
GMann43 I pulled it from the video, at 3:13. It could be wrong, I dunno.
EDIT: It's correct. I used logarithmic logic to determine so. Since 3!!!3 is 3!!(about 7 trillion), and 3!!(that number) is 3 times 3 times 3 times 3...repeated that number of times, I reasoned the the exponent of the 10^n part would increase by log(3) (base 10) for each multiplication by 3. In other words, the exponent would be equal to that number times log(3).
I chucked that into Wolfram Alpha and indeed got the same number.
Manabender Nope - the problem that is that 3!!(7 trillion) isn't 3x3x3x3.....
It's actually 3^3^3^3..... (exponents, not multipliers.)
I just noticed that in the description of the video, they acknowledge the error.
Just to give you an idea, 3^3^3^3 is already bigger than the number that you and the video gave. And that stack is only four-high; 3!!!3 is a stack 7 trillion-high.
***** I was referring to its natural, unabbreviated form. Of course you can store it in that form.
I'd wager that, given your icon however, you already knew that...
Manabender For future usage, you can press shift + 6 for ^ for arrow notation.
R.I.P Ron Graham 🙏
„Mr. Graham - what is your last wish?“ -
„Just write my number on my tombstone“
Not that it matters much to the size of Graham's Number, but there is a mistake in the video :-)
You wrote that 3↑↑↑3 is 1.25*10^3638334640024, but that number is actually "only" 3^(3^27) or 3^3^3^3.
The actual 3↑↑↑3 is a tower of 3's 7625597484987 high (3^27), as was also written in the video and that is MUCH MUCH bigger than 1.25*10^3638334640024 which is only the 4 first 3's.
So if I understood that correctly: Graham's Number is the number of dimensions after which every dimensions HAS to have that configuration at least once in it?
The upper bound to the problem has since been lowered significantly, in 2019 it was established to be 2^^5138*((2^^5140)^^(2*2^^5137)), which for comparison is much less than the closest tetration of 2^^(2^^5138)
Someone call in Brainiac. I think my calculator is running on fire...
"you ain't seen nothing yet." you Just put 3^27 trillionth power, what next?!
Graham's numbers last 500 digit's frequency:
0 - 56
2 - 56
9 - 56
5 - 55
6 - 54
1 - 49
3 - 47
4 - 46
6 - 46
7 - 35
No eights? :_:
How do you know this?
Sammy I got the last numbers from wikipedia and feed them into a character frequency program.
+Autodidactus Communitati Thats an interesting observation.
+Autodidactus Communitati actually look at that chart he just accidentally listed 6 and 8 as 6
3:01: Someone make a remix out of this.
Hi @SuperWindows78
I don't stand anything but the paper and his voice are satisfaying
So to summarise .... Pretty big
Nice video Brady!
You should ask Ron if he will be disappointed if the answer ends up simply being 13!
(Also, ask if him Graham's number +2 is prime!)
The more I think about Graham's number, particularly the construction, the more frighteningly incomprehensible it becomes. But then TREE(3) is incomprehensibly larger than G64. And yet they are/were answers to relatively comprehensible mathematical constructions.
And even more crazy numbers are coming up nowadays. Such as SSCG(3) which even makes Tree (3) look like nothing.
@@vedantsridhar8378WHAAAAT 🤯🤯🤯🤯😱😱😱😱😱
What a genius Graham was! RIP
I still think 20 is like the biggest number ever
21
20^Graham's number
:)
42!!!!
But 24 is still the funniest.
Yea but what happens when you use Grahams number instead of the 3...
Pretty sure you get nothing. At least, nothing to see, until someone finds a way to mathematicaly express this kind of ridiculous abomination ^^'
g(64)^g(64)
:)
Or what if there were 5 arrows hmmmmmmmmmmmmmmmmmmmm
@@johnmarcpandino3043 or what if you read my comment
it could even contain a G63 continuous zeros next to each other some where in the number
rest in peace graham, we will miss you.
Did you know? If you were to take every atom in the known universe, and expand them to all be the size of the universe, then turn them all into solid lead, the weight of all that lead in pounds would still be less than Graham's number!
In fact, the only thing that weighs MORE than Graham's Number of pounds... is your mother!
To be fair, osmium is denser than lead.
+Xhinope I, too, watch day9
+Xhinope Lol thanks dayj
+Xhinope The number you came up with I am fairly certain isn't even 3(3 arrows)3
+Xhinope Correction.. The observable universe. The universe could be infinite.
I love that paper change transition! XD
Hahahaha best ever!
RIP to a legend.
If I am not mistaken, 3↑↑↑3 is MUCH larger than a 3 trillion digit number as the video states. 3^3^3^3 (= 3^(3^(3^3)) ) already has trillions of digits, and that's just a tower four high. We're talking about a tower trillions of 3s high.
Here's a new number:
G64↑↑↑↑G64 = A1
A1↑↑↑↑(A1 times)A1 = A2
A2↑↑↑↑(A2 times)A2 = A3
...
A64↑↑↑↑(A64 times)A64 = B1
repeat until Z64.
No. Just no.
Yeah okay
themanwiththepan or TREE(4) that would kick any of the "i cam eup with a big number" comments.
theoretical physicist What about TREE(G)?? or TREE(TREE(3)) ?? or TREE(Fish number 7). And yet, I still don't know how actually TREE(3) does actually work...
MaxRideWizardLord yeah neither do i... would be cool so see (and hopefully understand the mathematics of it.
So... how many sides in Graham's Number-dimensional "cube"?
6:58
That’s mathematically quite savage lol
Here's the formula for Knuth's up arrow notation:
a↑ⁿ b = a↑ⁿ⁻¹a↑ⁿ⁻¹a↑ⁿ⁻¹a↑ⁿ⁻¹ … a↑ⁿ⁻¹a
The number of arrows is n.
The number of a's is b.
a↑b = aᵇ
a↑⁰b = ab
n ≥ 0
n *_must_* be an integer.
"If you think you understand it, you probably don't"
Just like quantum theory
Now this went from interresting to silly. This number is so ridiculous large that the total amount of integers in it exceeds the theoretical total amount of particles and waves in the universe.
"In binary it's 1.". Spot on.
3:06 Wrong. 3^3^3^3 contains 3.6 trillion digits. Not 3^3^3^3^3^3^3^3^3^3^3^3^3^3^3^3^3^3.... with over 7 trillion 3's. I assume they made the mistake of evaluating the power tower from the bottom up, which is incorrect, exponentiation is right-associative.
THEY DID UNDERESTIMATE THE ->s