Like when Bill Bailey and Rich Hall were doing that Exploration episode on QI, where Stephen asked how native Americans and the founding fathers communicated - it just makes it that much more funnier
@@slowfreq When measured against infinity, ANY finite set of numbers means you have excluded "almost every number". You have excluded an infinite amount of numbers. Regardless of how big the finite set, it will always pale in comparison to the infinite amount that is left.
for those who would like to understand the question: imagine a 2 dimensional cube (also called a square). now try to connect the corners with diagonal lines of two colors(4 corners, two possible diagonal lines), without all the four corners beeing connected with the same color. (quite easy: one red diagonal \ and a blue / diagonal) now, in 3 dimensions you have 8 corners, and 16 diagonals. and still the same task: try to draw all possible diagonals with only two colors without getting a square that has all it's corners in thesame color. (harder than in 2 dimensions, but still possible) now, the question is, at how many dimensions will it become impossible to accomplish.
You didn't state the problem very well, unless perhaps you know a simplification of the problem that I don't. Firstly, the way you said it, it sounds like the corners of the cube somehow receive a unique colour ("without getting a square that has all it's corners in thesame color"). But each edge can be coloured independently, so talking about corners confuses the problem. But more important is that you need to colour the usual edges of the cube too, not just the diagonals between the corners. It has nothing to do with "corners", per se. A better way to say it is that you consider _all_ possible edges between _every_ pair of vertices of the n-dimensional cube (in 3d, the original 12 edges, the 12 diagonal edges on each face, and the 4 edges between diagonally opposite vertices of the cube). You need to colour each such edge with one of two colours so that, whenever you pick four vertices in one plane (e.g., the four vertices of a face of the 3-cube, or 4 along a diagonal slice across the 3-cube), then both colours occur on an edge between some of those vertices. The theorem says that for sufficiently large dimensions you can't do this, but unfortunately the gap between where we know you can do it, and where we know you can't (at dimension Graham's number), is beyond cosmically large.
Calculations estimate the number of photons in the observable universe to be 10^89. Still alot missing to get to G64 Its quite simple. Even if you wrote the digits as small as the plank distance the entire size of the observable universe is not enough to write G64 down.
Something that escaped me the first time I seen this, but now that I'm thinking of it: I'm sure most people who see this assume the problem is "really hard" (and in the sense of solving it, it certainly is). To actually be able to understand what the problem means and to at least begin to understand how one would solve it, however, really is quite trivial once one knows basic algebra/geometry AND what is actually going on in this sort of math. It's simple layers built upon even simpler layers.
I already knew about the Grahams number. What Stephen didn't mention is that this is the biggest number that has ever een used in a calculation. It's also rediculously much bigger than a googolplex.
Graham's number is significant because it was the largest number every actually used, sure you can think of bigger numbers but we haven't found a use for them yet. So there is another large number called a googolplex 10^(10^100). If you filled the observable universe with grains of sand, a googolplex is very roughly equal to the number of ways you could arrange the grains of sand. The relative difference between 1 and a googolplex is basically the same as a googolplex to Grahams number.
@@JaneDoe-ci3gj "The relative difference between 1 and a googolplex is basically the same as a googolplex to Grahams number" Not even close to a comparison. Graham's number is so much bigger than that. Also, if you want an even bigger number, look up tree(3).
Grahams number is insane so i think this is how it goes correct me if i'm wrong The number uses arrow notation so it starts as 3^3 which is 27 then it is 3^^3 which is basically 3^(3^3) this equals 7.6 trillion then you get 3^^^3 which is 3^(3^(3^3) and this is a huge number i can't type out then it goes to 3^^^^3 which is another ridiculously big number now here's where it gets insane the ridiculously big number which is the answer to 3^^^^3 is now the number of arrows and the answer to that is the number of arrows for the next one and you do this 64 times until you get to grahams number
Actually, it's much, much worse than that. Start from 3^^3 which is, yes, about 7.6 trillion. A double arrow expression basically makes a power tower of the first number with a height of the second number. Next, 3^^^3 becomes 3^^(3^^3) or 3^^(~7.6T) which makes 3^3^3^3^3^3^3^....7.6 TRILLION TIMES....3. In other words, 3^^^3 is power tower of 3s which is over 7.6T high. Next, 3^^^^3 becomes 3^^^(3^^^3) which then equals 3^^3^^3^^3^^....3 where the number of threes is 3^3^3^....3 where THAT number of threes is 7.6T. Insanity ensues after only adding a single arrow each time, it's much MUCH worse than just exponentiation. That number, 3^^^^3, is called g1. The number 3^^^^^....^^^^^3, where the number of ARROWS is g1, is called g2. 3^^^^^....^^^^^3 with g2 arrows is g3, and so on. Grahams number (the more popular version at least) is g64, or big G. It's absolutely incomprehensible.
+lsrwLuke Here's my proof that that claim is false. Consider Graham's number + 1. It is greater than grahams number by one, and is used as the proof of this argument.
It is necessary. The continuous uniform distribution is defined on any finite interval of the real line and such an interval contains an uncountable number of points.
I love Graham’s number and so will pitch in the fun facts and say that if you were to learn every digit of Graham’s number your head would collapse into a black hole
Interesting, the fellow on the far left reminds me of a UK version of Timmy from WKUK. I have to say I find the answer to the question hilarious. It's like saying "It's somewhere between a little bit, and a bit less than infinity".
Yeah, there's often more than one way of proving something, and which is better or more elegant depends on personal taste. I happen to like this proof because most people could understand it, even if it's by contradiction, which one of my maths lecturers once said was the last resort in proofs.
@WakingLife55 I don't think they are too clear here, but what they means is as follows: There is some problem they are trying to solve, related to how many dimensions you need a cube to be so that every colouring of it has some property. Now, Graham (and Rothschild) proved an upperbound for this value - called Grayham's number now. So whatever the answer is, it is less than Grayham's number. The actual number is between 12 and this. It is Grayham's number which ends in a 7, now the answer.
The question is actually really easy to grasp, they're just presenting it in the most obtuse way possible. It's literally drawing squares with diagonal lines. Children can do that. The proof is the hard part.
The not enough ink thing isn't about how big a number is. Stick a decimal point in after the first digit and it's then quite a small number with a lot of kerfuffle. And also in need of ever so slightly more ink for the decimal point.
Graham's number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number.
I read that, but couldn't have got it into 500 characters as well as you have. I'll maintain that Cantor exploited a loophole of reasoning, which is that he assigns infiniteness to types of numbers that are mere subsets of all possible numbers. Those subsets may be infinite by the best mathematical or linguistic definitions that can be expressed, yet they are provably smaller than something else. They are simultaneously both infinite and less than infinite. An infinite recursion, beautiful.
No, he hasn't divided it by infinity. He's calculated the limit of (Graham's number / x) as x approaches infinity, which is 0. You might want to read up on limits of functions.
Can a mathematician imagine a __-dimensional hyper-anything without knowing the number of dimensions (Forgetting the ~difficulty~impossibility of imagining a number of spatial dimensions greater than 3 for a moment) or do they manipulate words? Either way, how do they do it?
You're close :) The corners are all connected to each other, and they are all absolutely randomly coloured (not just alternately). How many dimensions must the hypercube have to always form at least one square with two lines across it which are of the same colour?
I know we can never get to infinity, however your observation is that as you approach infinity the percentage of prime numbers reduces to 0 in the set, surely you need to show that the number is never 0 pre infinity. Limits were never my interest/strong point and I haven't done proper maths in ages.
that people with actual intelligence instead of actors could solve it? now, please don't misunderstand: i don't mean that actors can't be intelligent, but Fry and Mitchell are not as intelligent as they are perceived to be. and what's even more important to this question is, that both of them have next to no idea about natural sciences or mathematics. (especially fry... he said so many extremelly false things on the show)
+andrew7taylor my anwer was to arthur, not you... but: does he? in hi wiki page it's only written, that he studied it. not that he actually got the degree. but yes, this would give him much more chances to understand this question as he actually knows, without having to think too much about it, what an n-dimensional hypercube is.
That is to say, if you cut a section out of this hypercube into the dimension one lower than the total hypercube (so a tesseract becomes a cube, a cube becomes a square) how many dimensions must this hypercube extend into before you have to have a slice that has only one colour.
Stephen really should have explained what he meant by Graham's number. Wikipedia explains it "Graham's number is a very, very big natural number that was defined by a man named Ronald Graham. Graham was solving a problem in an area of mathematics called Ramsey theory. He proved that the answer to his problem was smaller than Graham's number." which makes sense. But here it leaves you thinking "What, it's the largest number ever? What about if Norton is right and you add an 8? It has to end in 7?" etc.
So how is Grahams number expressed? As in X to the power whatever? And how did Graham come about and select this particular number, seeing as the only digit they serm to know is the 7 at the end? There seems A LOT of room for variation.
Although my background is actually in chemistry rather than mathematics, I'll have a stab at explaining what I think the question is: The picture in the background is a clue to the nature of the question although it's inside out. If you take a cube and draw a line from each inside corner to the opposite corner - how many dimensions (n) must the cube (in fact a hypercube) have before (if you alternate black and red lines) you must have a completely red n-1 (in the case of a cube, a square) plane?
@Neylonx Yeah, like all the numbers between 0 and 1 (decimal) and the last number in the whole number series (integer) are both infinities. I guess the infinity in the integer series could be viewed as "large" given the definition "more than average amount", but I did want to stress that there is that infinity [isn't a size]/[doesn't have an actual size]. It synonymous with "boundless" by another - figurative - definition, and that's exactly what all forms of it are: boundless/arealess/sizeless.
Googolplex is actually ten multiplied by itself a googel times, where a googel is ten multiplied by itself 100 times. A consequence of this is that writing a Googolplex (in base 10) takes a googel digits. There are less than a googel particles (I think it is like 10^80 or so in the observable universe. Google seems to agree), and so that holds. However, Grahem's number is so big that a Googelplex is tiny compared to it!
So he read out the question and my mind froze over.. I listened to the words in a fog of 'wtf' and then thought "6 !" I was totally amazed when norton said 6 and even more so when steven said that 6 had been most peoples answer. Lucky guess or passive abilities hidden by stupidity? Wrong nonetheless, but curious now..
It wasn't really what people (graph theorists) thought was the answer. It was a lower bound on the answer - they didn't know the answer, but they knew it couldn't be less than six. Now it's known that it cannot be less than 13.
He may means that, practically speaking, Graham's number might as well be infinity. That is to say, infinity is of course much larger (if you give it the attribute of largeness, I view infinity as a concept rather than a number anyway, but...) So, for example, you had a genie and wished for a building that defied the laws of space and had inside infinite food for all the people of the world, it would be practically no different than having a "Graham's number pounds" of food inside.
@NCaradoc2008 No, not infinity + 7 XD. Anything finite (as Graham's number is) is an infinite distance away from infinity, so it's as far away from infinity as 1 is. It's just incredibly, incredibly large. "Large" isn't applicable to things like infinity, despite some people's notion of it.
OK, I'll play. Infinity is limitless and indefinable, because any assigned value can be endlessly increased. Georg Cantor, who I think you're referencing, reasons that there's an infinite number of natural numbers but also an infinite number of real numbers between each successive natural number, so the tally of real numbers is an infinite multiple of infinity. I suspect the loophole was saying that natural numbers represented an infinite quantity when they were only a subset of a larger group.
That sums it up, yes. That is, there is an infinite list of real numbers between even directly adjacent natural numbers such as one and two. The thing that bothers me is that what is being describe here is infinitessimals which are bounded by finite limits, which are then being directly compared to the whole gamut of limitless infinity, rather like comparing apples and oranges in my view.
You've misunderstood g1. 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3) But what you've taken it as is to be 3↑↑(3↑↑↑3). To understand the quadruple up arrow, you begin with 3, and that describes the height of a power tower, which gives us 3^(3^3) = 7.6 trillion, then that number describes the height of the power tower 3^^^3, and then that number describes the height of the next power tower (this is where you stopped) but then that number describes the height of the next... and again and again... repeat this 3↑↑↑3 times.
Graham's Number couldn't even be a small fraction of infinity, it is simply (according to Martin Gardner as quoted in Wikipedia) "... the largest number ever used in a serious mathematical proof." In a simple thought experiment it's possible to imagine Graham's Number being multiplied by 2 or 10 or any other number without needing to know the value of Graham's Number itself. To me, the true nature of infinity is that its reciprocal is vanishingly close to zero.
@Varoonmg Hey Varoonmg. I just had a list through to the source, and it doesn't appear to be in this one. There is a little bit about IQ though, and David Mitchell mentions about exams a little. If this is still of interest, I can post it though.
The point of Graham's Number isn't that it's simply big; it's the biggest number to have a genuine mathematical use. changing the 7 to an 8 would just take the use away.
The number is larger than the number of elementary particles in the observable universe. It is larger than the double factorial of said number. It is still unimaginably larger even after 10 factorials on the previous number. In fact, the number of Knuth arrows in the 64th layer is larger than the previous number. You can not write it out on notepad; Your computer would not have enough memory. Your computer doesn't just take up space when you save something...
Welll, for a time I believe its sole application was as an upper bound to this exact graph theory problem, sooo... that tells you something. I do believe the upper bound has since been refined, though.
Suppose there are finitely many primes. Multiply them all together. Add 1. The resulting number does not factorise into any of the other primes, therefore it must also be prime, which is a contradiction; hence the original statement must be false, i.e. there are infinitely many primes. QED
@Truthiness231 In the same sense, if you have people in a square formation, each an infinite amount of distance from each other, standing on an infinitely radial disc. They are all relatively at the center of the disc.
I don't know if someone already suggested this to you, but check out Numberphiles video about Graham's number. That will give you an idea how huge this number really is.
If you watched the video... It provided an upper bound to the problem described in the video. And the lower bound was guessed by one of the contestants.
I just invented a number twice as big as Graham's number, which is the largest number of dimensions required to make TWO of what Stephen said. It ends with a 4. It shall henceforth be called Maximilian's Number.
I remembered this from a documentary I watched about the problem of infinity. Sadly, I only remembered when he gave the answer, so that doesn't really count.
No, limits are a part of "standard" mathematics (and in fact, calculus wouldn't work without them). Again, I encourage you to look up "limits of functions".
i love so much when they give a joke answer and it's almost/mostly/completely correct.
Like when Bill Bailey and Rich Hall were doing that Exploration episode on QI, where Stephen asked how native Americans and the founding fathers communicated - it just makes it that much more funnier
yaseen reza that’s one of my favourite QI bits, Sean Lock especially
Or Victoria and the jack-rabbit.
@@richardlloyd2589 "WORSHIP IT WORSHIP IT"
that's... that's numberwang???
"Numberwank!"
"It's numberwang!"
"F**k!!"
genius
What is this? A crossover episode?
Thats wangernum, rotate those boards...
Yes, that is a number!
"Try and think of a really big number."
"17."
Oh Alan! I love you.
In mathematical terms, having your bounds be between 12 and Graham's number is actually not the worst bounds we have.
Yeah. We'll get there eventually :p
Saying that it's a natural number between 11 and grahams number instead of "we don't know" means that you exclude almost every number
@@TheLuuuuuc A finite number of numbers is not "almost every number"
@@slowfreq Depends on the field of mathematics you're in, but I would say in this context it makes sense (we excluded 100% of numbers)
@@slowfreq When measured against infinity, ANY finite set of numbers means you have excluded "almost every number". You have excluded an infinite amount of numbers. Regardless of how big the finite set, it will always pale in comparison to the infinite amount that is left.
Personally, I don't believe it would take very much ink to write "Grahams Number" at all.
for those who would like to understand the question:
imagine a 2 dimensional cube (also called a square). now try to connect the corners with diagonal lines of two colors(4 corners, two possible diagonal lines), without all the four corners beeing connected with the same color. (quite easy: one red diagonal \ and a blue / diagonal)
now, in 3 dimensions you have 8 corners, and 16 diagonals. and still the same task: try to draw all possible diagonals with only two colors without getting a square that has all it's corners in thesame color. (harder than in 2 dimensions, but still possible)
now, the question is, at how many dimensions will it become impossible to accomplish.
RKBock what the........??!
You didn't state the problem very well, unless perhaps you know a simplification of the problem that I don't. Firstly, the way you said it, it sounds like the corners of the cube somehow receive a unique colour ("without getting a square that has all it's corners in thesame color"). But each edge can be coloured independently, so talking about corners confuses the problem.
But more important is that you need to colour the usual edges of the cube too, not just the diagonals between the corners. It has nothing to do with "corners", per se.
A better way to say it is that you consider _all_ possible edges between _every_ pair of vertices of the n-dimensional cube (in 3d, the original 12 edges, the 12 diagonal edges on each face, and the 4 edges between diagonally opposite vertices of the cube). You need to colour each such edge with one of two colours so that, whenever you pick four vertices in one plane (e.g., the four vertices of a face of the 3-cube, or 4 along a diagonal slice across the 3-cube), then both colours occur on an edge between some of those vertices.
The theorem says that for sufficiently large dimensions you can't do this, but unfortunately the gap between where we know you can do it, and where we know you can't (at dimension Graham's number), is beyond cosmically large.
@@jamma246 ok i pick all vertices and i color them blue. blam, theorem proved wrong.
th-cam.com/video/HX8bihEe3nA/w-d-xo.html
@@loveboat Not a theorem, and the question asks when is it impossible, not when someone stupid can't manage it.
"You couldn't write down Graham's Number using ink with all the material in the universe". It's SCARILY bigger than that. SO MUCH bigger.
If you used every single sub atomic particle as a digit you would not be able to describe the number. Thats how large it is. Actually its bigger.
aren't photons subatomic particles? and i don't think there's anything that limits their total number.
Calculations estimate the number of photons in the observable universe to be 10^89. Still alot missing to get to G64
Its quite simple. Even if you wrote the digits as small as the plank distance the entire size of the observable universe is not enough to write G64 down.
You wouldn't have any universe left to write it on anyway so I wouldn't worry about it.
Walter Kingstone and then some. Times forty.
"Six."
"That is exactly what people used to think."
O_o
The lamb sauce is somewhere between 11 and Graham's number
USS Trekky Gordons Number :p
Underrated.
(Undercooked?)
Considering there are infinitely more numbers that aren't between 11 and Graham's number, I'd say they pretty much nailed it.
Watching this 6 years after your comment, but that was my first thought too.
(I am not a mathematician.)
1:31
The funny thing about that is that 17 is actually the only known value for multicoloured (three or more colours used) examples of the theorem.
Something that escaped me the first time I seen this, but now that I'm thinking of it:
I'm sure most people who see this assume the problem is "really hard" (and in the sense of solving it, it certainly is). To actually be able to understand what the problem means and to at least begin to understand how one would solve it, however, really is quite trivial once one knows basic algebra/geometry AND what is actually going on in this sort of math. It's simple layers built upon even simpler layers.
And now it's 13. We'll get it eventually.
tossabaddle This is Correct
+tossabaddle My guess is that it is 17, it is what it is for every other version of the question.
+The Major That was Alan's guess as well! Better watch out for that one....
That's numberwang
It's totally going to end up being 42.
Is it wrong that I got hugely excited to see two of my favorite internet obsessions (QI and Graham's number) intersect? God, I need a life.
You are not alone
I'm in the same boat as David, so I'll just say 42.
BNSFandSP Ah yes but do you know what the question is? And how is it even true in this universe?
Good call 😂👍
11 and Graham's number seems like an absurd range, but that still narrows it down from an infinite number of potential answers.
I already knew about the Grahams number. What Stephen didn't mention is that this is the biggest number that has ever een used in a calculation. It's also rediculously much bigger than a googolplex.
I thought this was going to be the Aaron Ramsey theory that when he scores a famous person dies
IT FUCKIN RAAAAAWWW.... oh... not THAT Ramsey, well *ahem*... I shall be going then!
HELLO, MY NAME'S NIIIIIIINO!!
Graham's number is significant because it was the largest number every actually used, sure you can think of bigger numbers but we haven't found a use for them yet.
So there is another large number called a googolplex 10^(10^100). If you filled the observable universe with grains of sand, a googolplex is very roughly equal to the number of ways you could arrange the grains of sand.
The relative difference between 1 and a googolplex is basically the same as a googolplex to Grahams number.
Cool😊👍. You actually explained it so that I understood. Perhaps you should be a teacher or be on QI👏👏! Thanks🌹
@@JaneDoe-ci3gj "The relative difference between 1 and a googolplex is basically the same as a googolplex to Grahams number"
Not even close to a comparison. Graham's number is so much bigger than that.
Also, if you want an even bigger number, look up tree(3).
Grahams number is insane so i think this is how it goes correct me if i'm wrong
The number uses arrow notation so it starts as 3^3 which is 27 then it is 3^^3 which is basically 3^(3^3) this equals 7.6 trillion then you get 3^^^3 which is 3^(3^(3^3) and this is a huge number i can't type out then it goes to 3^^^^3 which is another ridiculously big number now here's where it gets insane the ridiculously big number which is the answer to 3^^^^3 is now the number of arrows and the answer to that is the number of arrows for the next one and you do this 64 times until you get to grahams number
Actually, it's much, much worse than that. Start from 3^^3 which is, yes, about 7.6 trillion. A double arrow expression basically makes a power tower of the first number with a height of the second number. Next, 3^^^3 becomes 3^^(3^^3) or 3^^(~7.6T) which makes 3^3^3^3^3^3^3^....7.6 TRILLION TIMES....3. In other words, 3^^^3 is power tower of 3s which is over 7.6T high. Next, 3^^^^3 becomes 3^^^(3^^^3) which then equals 3^^3^^3^^3^^....3 where the number of threes is 3^3^3^....3 where THAT number of threes is 7.6T. Insanity ensues after only adding a single arrow each time, it's much MUCH worse than just exponentiation. That number, 3^^^^3, is called g1. The number 3^^^^^....^^^^^3, where the number of ARROWS is g1, is called g2. 3^^^^^....^^^^^3 with g2 arrows is g3, and so on. Grahams number (the more popular version at least) is g64, or big G. It's absolutely incomprehensible.
best youtube comment thread ever.
snuff meister : Well, that's a low bar, but yes.
true,true
It's the biggest number used in a proof
+lsrwLuke Here's my proof that that claim is false. Consider Graham's number + 1. It is greater than grahams number by one, and is used as the proof of this argument.
+lsrwLuke Not true actually, larger integers were used in Harvey Freidman's proofs of the various finite forms of Kruskal's theorem.
the biggest number is actually i
+James Thurlow true. And then there are also numbers like TREE(3) and SCG(13) which are also very closely related to many mathematical applications.
+didanyonethinkofthis very fitting to your username haha
"i'm worried about what might come out when i pull it" missed opportunity there!
if you zoom out enough on the numberline, eventually 11 and G64 will look like the same point!
Damn son
There is a video by numberphile where graham describes his number.
When the clip first started I thought Alan was having a nosebleed.
It is necessary. The continuous uniform distribution is defined on any finite interval of the real line and such an interval contains an uncountable number of points.
I saw the title and I thought, well, I know this is QI and all, but they can't REALLY mean Ramsey theory, but … yes, they do!
At first glance I thought this thumbnail was the Scenes We'd Like To See screen going terribly wrong.
I love Graham’s number and so will pitch in the fun facts and say that if you were to learn every digit of Graham’s number your head would collapse into a black hole
Geez, these job interview questions just get curiouser and curiouser
It's necessary. There's a uniform distribution on the real numbers between 0 and 1, for example.
Interesting, the fellow on the far left reminds me of a UK version of Timmy from WKUK.
I have to say I find the answer to the question hilarious. It's like saying "It's somewhere between a little bit, and a bit less than infinity".
I thought Ramsey theory meant whenever Aaron Ramsey scores a goal a worldwide catastrophe happens.
Yeah, there's often more than one way of proving something, and which is better or more elegant depends on personal taste. I happen to like this proof because most people could understand it, even if it's by contradiction, which one of my maths lecturers once said was the last resort in proofs.
@WakingLife55 I don't think they are too clear here, but what they means is as follows: There is some problem they are trying to solve, related to how many dimensions you need a cube to be so that every colouring of it has some property. Now, Graham (and Rothschild) proved an upperbound for this value - called Grayham's number now. So whatever the answer is, it is less than Grayham's number.
The actual number is between 12 and this. It is Grayham's number which ends in a 7, now the answer.
You're correct: there is no uniform distribution on any countably infinite set.
it's very reassuring that they don't understand the question either
Jordan Davies yeah
The question is actually really easy to grasp, they're just presenting it in the most obtuse way possible.
It's literally drawing squares with diagonal lines. Children can do that. The proof is the hard part.
The not enough ink thing isn't about how big a number is. Stick a decimal point in after the first digit and it's then quite a small number with a lot of kerfuffle. And also in need of ever so slightly more ink for the decimal point.
Graham's number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number.
I read that, but couldn't have got it into 500 characters as well as you have. I'll maintain that Cantor exploited a loophole of reasoning, which is that he assigns infiniteness to types of numbers that are mere subsets of all possible numbers. Those subsets may be infinite by the best mathematical or linguistic definitions that can be expressed, yet they are provably smaller than something else. They are simultaneously both infinite and less than infinite. An infinite recursion, beautiful.
To give you an idea, Graham's number is so big Graham had to create a new notation in order to show the algorithm to calculate it.
No, he hasn't divided it by infinity. He's calculated the limit of (Graham's number / x) as x approaches infinity, which is 0.
You might want to read up on limits of functions.
It does sat at the end of the video. series G, episode 6 ("Genius")
Can a mathematician imagine a __-dimensional hyper-anything without knowing the number of dimensions (Forgetting the ~difficulty~impossibility of imagining a number of spatial dimensions greater than 3 for a moment) or do they manipulate words? Either way, how do they do it?
You're close :) The corners are all connected to each other, and they are all absolutely randomly coloured (not just alternately). How many dimensions must the hypercube have to always form at least one square with two lines across it which are of the same colour?
I know we can never get to infinity, however your observation is that as you approach infinity the percentage of prime numbers reduces to 0 in the set, surely you need to show that the number is never 0 pre infinity. Limits were never my interest/strong point and I haven't done proper maths in ages.
Years ago, "I've got your/his/her number" meant you knew what they were up to or what their intentions were. Seems relevant in Grahams' regard.
When David Mitchell says he doesn't understand the question, you can safely assume no one else will, and that it's time to move on.
Graham's number: The only number ever to approach Stephen Fry's IQ...
I think you’ve surpassed being able to write it down by G2, most certainly by G3.
Six! That numberwang!!!
numberphile also did a video on this if anyones interested. They explain it much better than on here and also talk more about grahams number.
Just saying thank you for putting the name of the episode in the end :)
If Stephen Fry and David Mitchell don't understand a question, what hope is there?
Dara had more chance to understand this one, actuaIIy ;->
that people with actual intelligence instead of actors could solve it?
now, please don't misunderstand: i don't mean that actors can't be intelligent, but Fry and Mitchell are not as intelligent as they are perceived to be. and what's even more important to this question is, that both of them have next to no idea about natural sciences or mathematics. (especially fry... he said so many extremelly false things on the show)
That's why I made that remark about Dara. He has a degree in mathematics and theoreticaI physics.
+andrew7taylor my anwer was to arthur, not you...
but: does he? in hi wiki page it's only written, that he studied it. not that he actually got the degree. but yes, this would give him much more chances to understand this question as he actually knows, without having to think too much about it, what an n-dimensional hypercube is.
If you think Stephen Fry and David Mitchell are incomprehensibly intelligent then you are an Alan Davies type ;)
That is to say, if you cut a section out of this hypercube into the dimension one lower than the total hypercube (so a tesseract becomes a cube, a cube becomes a square) how many dimensions must this hypercube extend into before you have to have a slice that has only one colour.
Stephen really should have explained what he meant by Graham's number. Wikipedia explains it "Graham's number is a very, very big natural number that was defined by a man named Ronald Graham. Graham was solving a problem in an area of mathematics called Ramsey theory. He proved that the answer to his problem was smaller than Graham's number." which makes sense. But here it leaves you thinking "What, it's the largest number ever? What about if Norton is right and you add an 8? It has to end in 7?" etc.
But surely you just add 10 to Graham's number making it 9 larger than the 8 you changed the last digit to. 😆
So how is Grahams number expressed? As in X to the power whatever? And how did Graham come about and select this particular number, seeing as the only digit they serm to know is the 7 at the end? There seems A LOT of room for variation.
Although my background is actually in chemistry rather than mathematics, I'll have a stab at explaining what I think the question is: The picture in the background is a clue to the nature of the question although it's inside out. If you take a cube and draw a line from each inside corner to the opposite corner - how many dimensions (n) must the cube (in fact a hypercube) have before (if you alternate black and red lines) you must have a completely red n-1 (in the case of a cube, a square) plane?
"Im worred about what might come out when i pull it"
Thats what she said
@Neylonx Yeah, like all the numbers between 0 and 1 (decimal) and the last number in the whole number series (integer) are both infinities. I guess the infinity in the integer series could be viewed as "large" given the definition "more than average amount", but I did want to stress that there is that infinity [isn't a size]/[doesn't have an actual size]. It synonymous with "boundless" by another - figurative - definition, and that's exactly what all forms of it are: boundless/arealess/sizeless.
Googolplex is actually ten multiplied by itself a googel times, where a googel is ten multiplied by itself 100 times. A consequence of this is that writing a Googolplex (in base 10) takes a googel digits. There are less than a googel particles (I think it is like 10^80 or so in the observable universe. Google seems to agree), and so that holds.
However, Grahem's number is so big that a Googelplex is tiny compared to it!
Google can probably be said to be a bit biased on that issue 😉.
So he read out the question and my mind froze over.. I listened to the words in a fog of 'wtf' and then thought "6 !" I was totally amazed when norton said 6 and even more so when steven said that 6 had been most peoples answer. Lucky guess or passive abilities hidden by stupidity? Wrong nonetheless, but curious now..
It wasn't really what people (graph theorists) thought was the answer. It was a lower bound on the answer - they didn't know the answer, but they knew it couldn't be less than six. Now it's known that it cannot be less than 13.
@patrickgpking No, you misunderstand. We know Grayham's number ends with a 7. It is unknown what the last digit of the actual solution is.
1:24 another brilliant Alan-esque response. gotta love him. :D
He may means that, practically speaking, Graham's number might as well be infinity. That is to say, infinity is of course much larger (if you give it the attribute of largeness, I view infinity as a concept rather than a number anyway, but...) So, for example, you had a genie and wished for a building that defied the laws of space and had inside infinite food for all the people of the world, it would be practically no different than having a "Graham's number pounds" of food inside.
@NCaradoc2008 No, not infinity + 7 XD. Anything finite (as Graham's number is) is an infinite distance away from infinity, so it's as far away from infinity as 1 is.
It's just incredibly, incredibly large. "Large" isn't applicable to things like infinity, despite some people's notion of it.
OK, I'll play. Infinity is limitless and indefinable, because any assigned value can be endlessly increased. Georg Cantor, who I think you're referencing, reasons that there's an infinite number of natural numbers but also an infinite number of real numbers between each successive natural number, so the tally of real numbers is an infinite multiple of infinity.
I suspect the loophole was saying that natural numbers represented an infinite quantity when they were only a subset of a larger group.
Fun fact: it's now known that it's anywhere from 13 to 2^^^6 (up arrows not exponentiation) google "Graham's Number is Less Than 2^^^6"
That sums it up, yes. That is, there is an infinite list of real numbers between even directly adjacent natural numbers such as one and two. The thing that bothers me is that what is being describe here is infinitessimals which are bounded by finite limits, which are then being directly compared to the whole gamut of limitless infinity, rather like comparing apples and oranges in my view.
You've misunderstood g1. 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3) But what you've taken it as is to be 3↑↑(3↑↑↑3). To understand the quadruple up arrow, you begin with 3, and that describes the height of a power tower, which gives us 3^(3^3) = 7.6 trillion, then that number describes the height of the power tower 3^^^3, and then that number describes the height of the next power tower (this is where you stopped) but then that number describes the height of the next... and again and again... repeat this 3↑↑↑3 times.
Graham's Number couldn't even be a small fraction of infinity, it is simply (according to Martin Gardner as quoted in Wikipedia) "... the largest number ever used in a serious mathematical proof."
In a simple thought experiment it's possible to imagine Graham's Number being multiplied by 2 or 10 or any other number without needing to know the value of Graham's Number itself.
To me, the true nature of infinity is that its reciprocal is vanishingly close to zero.
@Varoonmg Hey Varoonmg. I just had a list through to the source, and it doesn't appear to be in this one. There is a little bit about IQ though, and David Mitchell mentions about exams a little.
If this is still of interest, I can post it though.
I'd say that the reciprocal of infinity IS zero to an accuracy of an infinite number of decimal places.
I'd ask how they came up with the upper limit but I don't think I really want to know how.
I may or may not be able to comprehend the problem they're trying to solve, but I very definitely can't do it when it's rattled off at me verbally.
The point of Graham's Number isn't that it's simply big; it's the biggest number to have a genuine mathematical use. changing the 7 to an 8 would just take the use away.
Graham's Number doesn't even come close to the largest numbers used.
Look up TREE(3).
(which is still tiny to some other concepts)
Thank you Numberphile.
Dara has used this Handkerchief trick since he started in Showbiz
@JaxWeb Graham's number is just the upper limit, not the solution, so we do not know that it ends in 7.
I got a nosebleed halfway through this video and now I look like I got into this way too much.
@Alexbrainbox Graham's number is the maximum value for this problem to work. It's the largest number ever used in an equation.
Possibly my favourite ever youtube comment. A thank you.
The number is larger than the number of elementary particles in the observable universe. It is larger than the double factorial of said number. It is still unimaginably larger even after 10 factorials on the previous number. In fact, the number of Knuth arrows in the 64th layer is larger than the previous number. You can not write it out on notepad; Your computer would not have enough memory. Your computer doesn't just take up space when you save something...
Welll, for a time I believe its sole application was as an upper bound to this exact graph theory problem, sooo... that tells you something. I do believe the upper bound has since been refined, though.
Suppose there are finitely many primes. Multiply them all together. Add 1. The resulting number does not factorise into any of the other primes, therefore it must also be prime, which is a contradiction; hence the original statement must be false, i.e. there are infinitely many primes.
QED
As soon as he said "6" I knew this was going to involve Graham's number.
Holy shit, when Graham Norton said 6 i was thinking the exact same thing
Added to description. 1st Jan 2010 according to Wikipedia.
@Varoonmg Sure. But which video was it you wanted me to post? I've forgotten now, sorry!
@Truthiness231 In the same sense, if you have people in a square formation, each an infinite amount of distance from each other, standing on an infinitely radial disc. They are all relatively at the center of the disc.
I don't know if someone already suggested this to you, but check out Numberphiles video about Graham's number. That will give you an idea how huge this number really is.
Regarding the grahams number, i was wondering what is the point of it ? Is it used as a measurement or just someone having a boring day
It is one of the largest numbers ever used in a serious mathematical proof.
Alcesmire I'm guessing it's purpose is to understand maths better.
If you watched the video... It provided an upper bound to the problem described in the video. And the lower bound was guessed by one of the contestants.
I got that at 0:11 just because I remember graham's number being related in some way to hypercubes
I just invented a number twice as big as Graham's number, which is the largest number of dimensions required to make TWO of what Stephen said. It ends with a 4. It shall henceforth be called Maximilian's Number.
@patrickgpking Indeed, you can work out the right hand digits of it. It ends, "...4195387", in fact =]
I remembered this from a documentary I watched about the problem of infinity. Sadly, I only remembered when he gave the answer, so that doesn't really count.
Excluding this one, these are the best comments to a video ever!
Finally, a place for math nerds to feel part of popular culture. Try not to sap all the humour out of it too soon.
Oy it's a nerd! Kick him!
No, limits are a part of "standard" mathematics (and in fact, calculus wouldn't work without them). Again, I encourage you to look up "limits of functions".