Too big to write but not too big for Graham

Rachel Thomas and Marianne Freiberger

Recently, when we were writing our book Numericon, we came across what has now become one of our very favourite numbers: Graham's number. One of the reasons we love it is that this number is big. Actually, that's an understatement. Graham's number is mind-bendingly huge.

Observable universe

The observable Universe is big, but Graham's number is bigger. Image: ESA and the Planck Collaboration.

Our new favourite number is bigger than the age of the Universe, whether measured in years (approximately 14 billion years) or seconds (4.343x1017 seconds). It's bigger than Avogadro's number, a sizeable 6.02214129 x 1023. This is the number of hydrogen atoms in 1 gram of hydrogen, which is called a mole and is the standard unit for measuring an amount of a substance in chemistry or physics.

Graham's number is bigger the number of atoms in the observable Universe, which is thought to be between 1078 and 1082. It's bigger than the 48th Mersenne prime,

257,885,161-1,

the biggest prime number we know, which has an impressive 17,425,170 digits. 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. (This might sound familiar, as Google was named after this number, though they got the spelling wrong.)

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 10googol=10(10100). A googleplex is significantly larger than the 48th Mersenne prime. You, or rather a computer, can write out the 48th Mersenne prime in its entirety, all 17,425,170 digits of it. But, despite the fact that I can tell you what any digit in the googolplex is (the first is a 1, the rest are all 0), no person, no computer, no civilisation will ever be able to write it out in full. This is because there is not enough room in the Universe to write down all googol+1 digits of a googolplex. 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."

Graham's number is bigger than the googolplex. It's so big, the Universe does not contain enough stuff on which to write its digits: it's literally too big to write. But this number is finite, it's also an whole number, and despite it being so mind-bogglingly huge we know it is divisible by 3 and ends in a 7.

A big party

The origins of Graham's number go back to 1928 when a brilliant young mathematician, Frank Ramsey, noticed a surprising thing when he was working on a paper about logic: complete disorder seemed to be impossible. No matter how complicated your system looks, pockets of order of any size are always guaranteed to be there if the system is large enough.

This result, which was just a small part of the particular paper he was working on, was the start of a whole new field of mathematics called Ramsey Theory. This area of maths is often explained with the example of a party. Suppose you're having a party and you want to make sure you invite a good mix of people and decide to keep track of who knows who. Suppose you draw a map of the relationships of all your friends, linking two people with a blue edge if they are friends and with a red edge if they are strangers. Then you might end up with something like this:

friends network

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. But if you zoom in on just Ann, Bryan and David, they are all joined by red edges. This red triangle is an example of order hiding in the messy overall network. The more ordered a system is, the simpler its description. The most ordered friendship network is one that has all the edges the same colour: that is, everyone is friends or everyone is strangers.

Ramsey discovered that no matter how much order you were looking for – whether it was three people who were all friends and strangers or twenty people who were all friends and strangers – you were guaranteed to find it as long as the system you were looking in was large enough. To guarantee yourself a group of three people who are all friends or all strangers you need a friendship network of six people: five people isn't enough as this counterexample shows.

friends network

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). We know a few Ramsey numbers: we've seen that R(3,3)=6, and it has been proved that R(4,4), the number of people you need to guarantee you'll find four friends or four strangers, is 18. But we hit a wall very quickly. 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 23 possible ways of colouring the network. For four people there are six edges and 26=64 possible colourings. But for the relationships between six people there are fifteen edges and we already have to consider an unwieldy 215=32,768 possible colourings. 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 903 edges, so you'd have to check through all of the 2903 possible colourings – more colourings than there are atoms in the observable Universe!

Too big to write but not too big for Graham

Big numbers have always been a part of Ramsey theory but in 1971 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.

graham's 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.

Ronald Graham

Ronald Graham who gave us his beautiful number. Image: Cheryl Graham.

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 think of multiplication as repeated addition:

3 x 3 = 3+3+3

and exponentiation as repeated multiplication:

33 = 3 x 3 x 3.

If we define the single arrow operation, ↑, to be exponentiation, so:

3↑3 = 33 = 3 x 3 x 3 = 27,

then we can define the double-arrow operation ↑↑ to be

3↑↑3 = 3↑3↑3 = 333 = 327 = 7,625,597,484,987.

We can carry on building new operations by repeating previous ones. 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! (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.


About this article

Rachel and Marianne

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.


Comments

Yes it is bigger it makes googolplexian look like the smallest thing smaller than the smallest thing and makes Grahams number look as big as the BOX that is the largest thing POSSIBLE!

.... But why that big, why does we know that the solution will be that number or smaller?

Is it a guess from Grahams part or is there a reasoning to go that far? I never seen that explained. 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?

Not sure if true, but IIRC: perhaps the best thing about Graham's number is that this absurdly massive number is an upper bound to the solution, and it's believed that the actual answer to the question is six.

You're forgetting about the fact that if you consider our universe as a good example of space then you have forgotten that our universe is much smaller than, THE INFINITE AMOUNT of universes in space. Space is believed to be just a vacuum of stars and planets that are grouped together, and that these groups of groups of planets go on seemingly forever. So why are you fighting about fitting it in our universe. When if you tried, you wouldn't actually know when you've reached the outer reaches of our universe. You'd die before you could ever even attempt to record it. So let me put it to you like this, you're fighting about fitting that number into a place that could quite possibly have more universes in it than any number in the known universe could ever come close to being. Think about it like this, it's pretty much like fighting to put a number that can be used to explain how many universes our in existence. There's probably more than a g^g amount of universes in existence so why are you fighting about it, when you couldn't tell if you were in our universe or the one that's two universes over. It's a pointless argument. You're fighting about fitting something that is pretty much infinite into a space that is pretty much infinite. It's like fighting to fit infinity into infinity.

What you're missing here is that a lot of the time (as in this case) when people say "the Universe" what they really mean is "The OBSERVABLE Universe", which is a VERY large but definitely finite size (about 93 billion light years). It is defined by the distance that light has had time to reach us since the beginning of the universe. https://en.wikipedia.org/wiki/Observable_universe
So you may be right about how large the universe outside of that area could be, but it's not really relevant.

I'm not a mathematician - far from it. Every time I come back to Graham's number I can't get my head around Knuth's up-arrow notation. I interpreted it in my own way, but I have a feeling I've got it wrong. The way I thought of it is that:

Step 1: 3^3 = 27. Take that result (27) and make it the new power:

Step 2: 3^27 = 7.6 trillion. Take that result (7.6T) and make it the new power:

Step 3: 3^7.6T = ? (An "Impossibly Huge Number.") Take that result (I.H.N) and make it the new power:

Step 4: 3^I.H.N = ? (A "Savagely Huge Number.") Take that result (S.H.N) and make it the new power:

Step 5: 3^S.H.N = ? And so on, and so on. (I forget which one would be "G1", step 4 or step 5 maybe)

- What I'd like to know is whether this way of looking at it is right or not. Obviously you can't write anything down without coming up with "Magnificently Ridiculous Names" for these numbers so it defeats the object, but I'm trying to get my head around the process.

Thanks in advance for any help, I really appreciate it.

You actually are interpreting the arrows quasi-correctly.
Step 1: 3^3
Step 2: 3^^3 = 3^27 = 7.6T
Step 3: 3^(3^^3) = 3^3^3^3 = 3^^4
Step 4: 3^(3^^4) = 3^^5
The tricky part to wrap your head around is the addition of a third arrow.
New Step 3: 3^^^3 = 3^^(7.6T) which means repeating your original version of Steps 4, 5, 6, etc. all the way to Step 7.6T! Then you would have to add yet *another* arrow, which means you have to take the result of that Step #7.6T and continue repeating your steps to reach *that* step number. Then, FINALLY, you would have G1.

Brain broken yet?

Numbers come from counting. If you can't physically count to the number than it doesn't exist. Counting is just a short hand for grouping things into a location. For example I pick apples from a tree and place them in a basket. The number of apples remains the same but I am defining my counting by only the apples in the basket. So the short hand of multiplication and exponents etc. represent the physical act of counting things one at a time but is just a short hand. If you can't actually count to a number than it doesn't exist in the physical universe. You need things to count and there are only so many atoms or particle to count and then you have to have a method of grouping the things you are counting. So it involves physics. Quantum physics, time , gravity etc. Why is this number considered to be the largest ?? You have a way of defining Grahams number but carrying out the definition is in itself impossible. This is why infinity is impossible and infinity + 1 is impossible also. When you count and attempt to count higher than anyone or anything has ever counted to before you will eventually reach a number and if you try to add 1 to this number you will fail. Something physical will prevent you from counting higher.

The great thing about mathematics is that it's unbound by reality. We don't need physical things to count. It may have started that way but it's evolved way past that. Mathematics is not defined by physics, mathematics is a study in and of itself.

"Why is this number considered to be the largest ??"
This is the largest NAMED number. Obviously, there are an infinite amount of numbers larger.

Even if our use of the concept of numbers began with counting, mathematics (which encompasses not only numbers but the relationships between abstract quantities like time and gravity) is a language. It's the language of our reality. In English, we can certainly use words to express concepts that don't physically exist in our Universe, and we can describe imaginary objects that don't (or even can't) exist physically. By your logic, the following words wouldn't exist for me to write: the orange-colored Beaver Queen of donuts.
Math is no different. It's a language fully equipped to describe any numerical concept, whether or not it could exist in our Universe.

It's fun to toy with big numbers, but all this iterated power stuff is primitive recursive. All primitive recursive functions are dwarfed by Ackermann's function for relatively small arguments, whereas the Ackermann function itself is in turn dwarfed by the Busy Beaver function.

I was wondering when someone would mention Ackermann's function. Imagine Ack(Graham's number)!

g(Graham’s number)

Graham’s Number plus 1?

I imagine that an encryption scheme incorporating g64 would be well-nigh unbreakable.

When it comes to encryption u should be aware that as of now (june 2020) advances in quantum computing have made encryption a thing of the past, in fact a 512 qubit computer could anylise every known atom in the universe in less than a second!. But when it comes to maths we are approaching an entirely new and unknown area, the more we learn the more we are starting to realise that numbers are not a unit of measurement or way of defining amounts but more like a system of programming that exist everywhere and are undeniably fundemantal to not just the field of mathematics but life itself..you have golden ratios and fibonacci sequences...etc...etc... which the more i have researched the more weird things start to get...grahams number, googleplexians are all well and good but the current field of quantum physics is on the cusp of rendering everything we know and believe to be absolutely incorrect....watch this space...

grahams number is g64
g's will be 1 layer of grahams
p will be one layer of googol/googolplex/googolplexian
starrting from a level playing field (0)
how long will the exponential increase of grahams take to catch up to p's expansion
one layer of g increases it by a power of 3
one layer of p increases it by a power of 10
meaning that 3 layers of g is equal to 1 layer of p
googolplexian = ~g9

Graham’s number plus one. I win.

My own maths thing I(x) is B I G
I(x) = x^x^x and so on x+1 times
I(0)=0
I(1)=1^1=1
I(2)=2^2^2=2^4=16
I(3)=3^3^3^3=3^3^27=3^7625597484982=a number with over 36 TRILLION DIGITS
I(x) where x is Bigger than 3 is in computable
I(0.5)=0.25
I(1.5)=1.5^0.75≈1.3554
Bye!