Ok I had the same problem for a while too understanding how arrows worked but it’s actually a rather simple pattern just different from how we’re used to thinking about operations. Therefore, "^" is the natural and obvious substitute for the up-arrow, if the computer keyboard doesn't have the up-arrow. There aren't enough of them to write it. ...with a long enough string of "trillion" to write Graham's number you'd still need 2/3 as many characters, and so it would still be impossible to write all that string of "trillion", by writing one character in of the Planck volumes in the observable universe. Graham's number may be too big to write, but we know it ends in seven. Rachel Thomas and Marianne Freiberger are the editors of Plus. Thank you for not saying something stupid. 4 ^ 3 = 4 x 4 x 4 The two sites erroneously claim that googolplexian is "the world's largest number with a name" and that googolplex comes in second place. So you are counting apples that you pick from a tree and put in a basket. But this number, as we mentioned earlier, was absolutely massive, so big it is too big to write within the observable Universe. 3 raised to the third power a number of times equal to (3^3^3^3^3^3^3^3...) from above. A Googolplex is defined as $10^{\text{Googol}}$. What is larger? Now, compare this with just the first layer of Graham's number,i.e., 3 ↑↑↑↑ 3. 3↑↑3 = 3↑3↑3 = 333 = 327 = 7,625,597,484,987. 3^^4 = 3 ^ ( 3 ^ ( 3 ^ 3 ) ) Given that we are talking about filling the entire universe with numbers, extremely small numbers, it's not unreasonable, at least not more unreasonable, that we can consider that this 'font' we are talking about has zero thickness and as such there is more than enough space to write any number on the head of a pin. i.e. They all follow the same pattern. 2. As Kasner, and his colleague James Newman, said of the googolplex (in their wonderful 1940s book Mathematics and the imagination which introduced the world to these numbers): "You will get some idea of the size of this very large but finite number from the fact that there would not be enough room to write it, if you went to the farthest star, touring all the nebulae and putting down zeros every inch of the way.". The higher the dimension, the more corners there are: a three-dimensional cube has 8 corners, a four-dimensional cube has 16 corners, a five-dimensional cube has 32 corners and so on. In this picture we can see that for a particular flat diagonal slice through the cube, one that contains four of the corners, all of the edges are red. We can carry on building new operations by repeating previous ones. That's the only use for Graham's number that seems practical anyway. You're fighting about fitting something that is pretty much infinite into a space that is pretty much infinite. Graham's number is also bigger than a googolplex, which Milton initially defined as a 1, followed by writing zeroes until you get tired, but is now commonly accepted to be 10 googol =10 (10 100). We can think of multiplication as repeated addition: If we define the single arrow operation, ↑, to be exponentiation, so: then we can define the double-arrow operation ↑↑ to be. And it's bigger than the famous googol, 10100 (a 1 followed by 100 zeroes), which was defined in 1929 by American mathematician Edward Kasner and named by his nine-year-old nephew, Milton Sirotta. Instead, something called "Knuth's up-arrow notation" is used. You're right but you're forgetting something... https://en.wikipedia.org/wiki/Observable_universe, Physics in a minute: The double slit experiment, Clearing the air: Making indoor spaces COVID safe. Hi. A good reason why "^" is a good substitute is the fact that one up-arrow means the same as "^": Exponentiation. The number of people you need to guarantee that you'll find three friends or three strangers is called the Ramsey number R(3,3). 4 ^^^ 3 = 4 ^^ 4 ^^ 4 F(1)=1+1=2 How many orders could that list be arranged in? To get to G (Graham's Number), one starts with 3^^^^3 (a clearly demonstratedly STOOPID BIG number. 3^^(3^3^3^3^3^3^3...) where "..." continues on for 3^3^3 iterations of powers of three, i.e. Sorry. The next would be the triple-arrow, 3↑↑↑3 = 3↑↑3↑↑3 = 3↑↑(3↑↑3)=3↑↑7,625,597,484,987, a tower of powers of 3 that is 7,625,597,484,987 levels high! Could we write the number in digital form, which from my understanding would incredibly expand the number of digits that could possibly be written, or am I mistaken? Only that G1 is 3 4*arrow 3 and that is the number of arrows between the 3's in G2 and so forth until G64 why not just stop at G1 or why not continue to G1.000.000? It is very low but not zero. That's not even close to Graham's Number. After the first equality sign It should be $3^{3\times 10^{100}}$ instead of $3^{3\times 10^{10^{100}}}$. You can never pin down the location of anything with 100% precision. Graham's Number = G And that's only with only 4 up arrows. This area of maths is often explained with the example of a party. For example we don't know what R(5,5) is. if 4^^3=4^(4^4), then is 3^^4=3^((3^3)^3)? How many peas you could put in a bottle depends on the size of the bottle and the size of the pea. Think the number to the right of the arrow is the exponent: 4^^3 = 4^(3^3) = 4^9 etc. Unless we find new physics. That is "g2". 1. no problems but A down arrow would make more sense for roots. University of Cambridge. So you may be right about how large the universe outside of that area could be (which is unknowable, since it can literally never be observed) but it's not really relevant in this context. Have any other US presidents used that tiny table? 3^^3=3^(3^3)=3^27. Doing so today I see I don’t quite understand the Arrow notation. The name of this number is derived from googol, but there is no logic to the definition otherwise. But for the relationships between six people there are fifteen edges and we already have to consider an unwieldy 215=32,768 possible colourings. even tree[treeG1000] is puny to my number, SSCG.SSCG(0)=2 SSCG(1)=5 SSCG(2)>3x(2^(3x2^95))=10^(3.5775x10^28) SSCG(3)>>>TREE(3) SSCG(3)>>TREE(TREE(TREE(...(TREE(3))..) TREE(3) times, i have created a number bigger than the puny G64, No you are wrong a googleplex^of a googleplex is smaller than graham but larger than googleplex. F(2)=2•(2+2) Absolutely mind dissolvingly, unfathomably, ludicrously large. Then you might end up with something like this: Now this looks pretty complicated and it would take quite a lot of information to describe who is connected by red edges and who is connected by blue edges. Answer: Writing Graham's number as a 1 followed by zeros, suppose you could write one of the zeros in every Planck volume in the observable universe. Answer: As someone mentioned, here is what Graham's notation means: (I'm using ^ to mean up-arrow. Basically F(n) whatever n is equal to tells you the number of operations used and the level of the last operation you’ll use that step. How can you trust that there is no backdoor in your hardware? That g(n) function increases _very_ rapidly, as n increases. Since all subsequent terms will end with a (-x) - (-x) in a denominator, they are ALL undefined. When it has been specified that "^" is being used to stand for Knuth's up-arrow, then yes, it becomes the same thing. Basically the number of arrows tell you the amount of times you repeat the operations that come before it. Even worse, googolplex.com contains a hidden 8, and googolplexian.com contains a hidden 6. This article, though clarifying Knuth's "Arrow Notation", doesn't get you to Graham's Number. Or you could use a operation of my own design it’s kinda like a hyperoperation but I haven’t seen it officially listed anywhere. One adjective for two singular nouns (same gender). In how many orders could you arrange a number of objects equal to 10 to the 72 billionth power? But if you zoom in on just Ann, Bryan and David, they are all joined by red edges. That number of orders is about the 400 millionth power of the number of Planck volumes in the observable universe. Ex, F(0)=0 See Numberphile -- Tree(G64) for a really fun and technical explanation of where they fall in the infinity system. It's said that you couldn't even write the number of digits in the number of digits. 3^^^^3 = 3 ^^^ ( 3 ^^^ 3 ). Copyright © 1997 - 2020. You'd die before you could ever even attempt to record it. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Graham wanted to know how big the dimension of the cube had to be to guarantee that a single-coloured slice exists. GET A QUOTE