How long is grahams number

For example we don't know what R 5,5 is. We know it's somewhere between 43 and 49 but that's as close as we can get for now. Part of the problem is that numbers in Ramsey theory grow incredibly large very quickly. If we are looking at the relationships between three people, our network has just three edges and there are a reasonable 2 3 possible ways of colouring the network.

Mathematicians are fairly certain that R 5,5 is equal to 43 but haven't found a way to prove it. One option would be to check all the possible colourings for a network of 43 people. But each of these has edges, so you'd have to check through all of the 2 possible colourings — more colourings than there are atoms in the observable Universe!

Big numbers have always been a part of Ramsey theory but in mathematician Ronald Graham came up with a number that dwarfed all before it. He established an upper bound for a problem in the area that was, at the time, the biggest explicitly defined number ever published. Rather than drawing networks of the relationships between people on a flat piece of paper as we have done so far, Graham was interested in networks in which the people were sitting on the corners of a cube.

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.

But not all colourings of a three-dimensional cubes have such a single-coloured slice. Luckily, though, mathematicians also have a way of thinking of higher dimensional cubes.

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. Graham wanted to know how big the dimension of the cube had to be to guarantee that a single-coloured slice exists. Graham managed to find a number that guaranteed such a slice existed for a cube of that dimension. But this number, as we mentioned earlier, was absolutely massive, so big it is too big to write within the observable Universe.

Graham was, however, able to explicitly define this number using an ingenious notation called up-arrow notation that extends our common arithmetic operations of addition, multiplication and exponentiation. We can carry on building new operations by repeating previous ones. The next would be the triple-arrow.

See here to read about the up-arrow notation in more detail. The number that has come to be known as Graham's number not the exact number that appeared in his initial paper, it is a slightly larger and slightly easier to define number that he explained to Martin Gardner shortly afterwards is defined by using this up-arrow notation, in a cumulative process that creates power towers of threes that quickly spiral beyond any magnitudes we can imagine. But the thing that we love about Graham's number is that this unimaginably large quantity isn't some theoretical concept: it's an exact number.

We know it's a whole number, in fact it's easy to see this number is a multiple of three because of the way it is defined as a tower of powers of three.

And mathematicians have learnt a lot about the processes used to define Graham's number, including the fact that once a power tower is tall enough the right-most decimal digits will eventually remain the same, no matter how matter how many more levels you add to your tower of powers. Graham's number may be too big to write, but we know it ends in seven. Mathematics has the power not only to define the unimaginable but to investigate it too. Rachel Thomas and Marianne Freiberger are the editors of Plus.

This article is an edited extract from their new book Numericon: A journey through the hidden lives of numbers. OK now the question is how does the expansion for different numbers work. Googolplexitoll, Googolplexigong is larger than googolplex. Omega is even larger than infinity! They all follow the same pattern.

For up arrows, you just have one less up arrow than in the original problem. Put that back in your other problem. Unimaginably big. And that's only with only 4 up arrows. Thanks for posting the correct definition. People should NOT comment unless they know what they are talking about. If you take a little time and watch Numberphile on YouTube, you can see Graham himself define the number.

Your explanation was the correct one. Thank you for not saying something stupid. Basically the number of arrows tell you the amount of times you repeat the operations that come before it. Long story short the arrow just tells you to take the process used for the previous number of arrows and use a repeated form of that. The difference between the towers is the height. The first tower is 3 high the same as the base number , and once that is multiplied out, that is the HEIGHT of the next one.

This continues forth. It just iterates upwards. Great explanation, but I'm somewhat puzzled by the fact that you never actually told us how Graham's number is defined. It's like there's a missing paragraph after you finish explaining the rapid recursive growth of up-arrows. That's "Step 1". Well beyond numbers anyone can really hope to imagine in any meaningfully representative way without deep mathematical understanding, this completes Step 2.

Anyway, that's what I think an explanation might look like; I'm surprised something like this wasn't included. Otherwise, great explanation,. Graham went one step further. However, that number is called "g1". That is "g2". Are you sitting down? Graham's Number is g64!!! They are not the same thing. Actual up arrows shouldn't be shown in it because like you said they don't mean the same thing.

W by the way you just made a argument waiting to happen by saying "They are not the same thing. Yeah graham's number is huge, but not as big as a googolplexian, which is a one followed by a googolplex number of zeros.

In fact, the googolplexian is the largest number that has a name. Man I was curious about googol then I stumbled on googolplex then I noticed it was only the 2nd biggest number according to google then I found out about grahams number bloody hell i f I ever get cocky about my intelligence I'll just think about grahams number and think about how confusing it is man I don't think I'll be able to sleep at night I wish I had just stuct to studying pi multiplied by the radius squared.

Actually, Graham's number is now considered pretty small by mathematicians. For example, TREE 3 is so big that it makes Graham's number look like pretty much like zero in comparison.

The interesting thing about the TREE function is that it grows so rapidly eg. I don't think so, but I didn't count the arrows. Also in this range are the odds of winning the really big lotteries. A recent Mega Millions lottery had 1-in,, odds of winning. Both of those are dwarfed by the number of cells in the human body 37 trillion. People say the words million, billion, and trillion a lot. No one says quadrillion. Either way, there are about a quadrillion ants on Earth.

People would be mad at you. Also the number of references to Kim Kardashian that entered my soundscape in the last week. Please stop. No one who has social skills ever says the word quintillion. The number of grains of sand on every beach on Earth is about 7. So many podcasts. And heard of a Planck volume? More on Planck volumes later. Oh, and our dot image? By the time we get to quintillion dots, the image would cover the surface of the Earth.

You also had to deal with this number in high school— sextillion, or 6. The Earth weighs about six septillion kilograms. No dot posters being sold for this number. So why did I stop here at this number? The name googol came about when American mathematician Edward Kasner got cute one day in and asked his 9-year-old nephew Milton to come up with a name for 10 —1 with zeros. So picture the universe jam-packed with small grains of sand—for tens of billions of light years above the Earth, below it, in front of it, behind it, just sand.

Endless sand. Lots and lots and lots of sand. If that were the case for every single grain of sand in this hypothetical—if each were actually a bundle of 10 billion tinier grains—the total number of those microscopic grains would be a googol.

How many of these smallest things could you fit in the very biggest thing, the observable universe? A Googolplex — 10 googol. After popularizing the newly-named googol , Krasner could barely keep his pants on with this adorable new schtick and asked his nephew to coin another term.

With its full written-out exponent, a googolplex looks like this:. So a googol is 1 with just zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe.

Can you possibly imagine what kind of number is produced when you put a googol zeros after the 1? What I wrote above is just the exponent— actually writing a googolplex out involves writing a googol zeros. Well I just tested how fast a human can reasonably write zeros, and I wrote 36 zeros in 10 seconds. About billion human beings have ever lived in the history of the species. Now to get a glimpse at how big the actual number is—as the Numberphilers explain , the total possible quantum states that could occur in the space occupied by a human i.

What this means is that if there were a universe with a volume of a googolplex cubic meters an extraordinarily large space , random probability suggests that there would be exact copies of you in that universe. Because every possible arrangement of matter in a human-sized space would likely occur many, many times in a space that vast, meaning everything that could possibly exist would exist—including you.

Including you with cat whiskers but normal otherwise. Including you but a one-foot tall version. Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2 n vertices.

Color each of the edges of this graph either red or blue. What is the smallest value of n for which every such coloring contains at least one single-colored complete subgraph on four coplanar vertices?

I told you it was boring and confusing. So anyway, I said above that I had been limited in the kind of number I could even imagine because I lacked the tools—so what are the tools we need to do this? The hyperoperation sequence is a series of mathematical operations e. If I have 3 and I want to go up from there, I go 3, 4, 5, 6, 7, and so on until I get where I want to be.

Not a high-powered operation. For numbers as big as this, Mathematician Donald Knuth came up with a new type of notation that you have probably never heard of. This means that there are g1 arrows between the two 3s. There are now g 2 arrows between the 3s. Why would anyone need a number like this you ask? Mathematician Ronald Graham came up with it when talking to another mathematician named Martin Gardner.