I am going to talk about some cool math tonight. Basically I am just going to be explaining some ways to talk about big numbers that most people have never heard of. It might be kinda short because I need to get to sleep fairly soon, but I hope it will be interesting.

To start, lets establish a pattern. One of the simplest things you can do with a number is to count up by one. To go to the next whole number bigger. Its how we first learn to add numbers, by counting up a number of times equal to the amount we are adding. We want to add 4 to 3, we count 3, 4, 5, 6, 7. When you add a number, it means you are counting up that many times. So we can use counting in order to understand how to add.

Next we can use adding to learn to multiply. If we want to multiply a number, we are just adding that number a number of times equal to what we are multiplying by. So if we multiply 3 by 4, we are saying 3+3+3+3. We are adding 3, 4 times. So we are using addition in order to understand how to multiply.

After multiplication is exponents, or powers. If you want to take a number to the power of another number, that just means you are multiplying by that number a number of times equal to the second number. So if we take 3 to the fourth power, we are just multiplying 3*3*3*3. We are multiplying by 3, 4 times. So we are using multiplication in order to understand how to take powers.

Now most people know how to do all of these things, even if they might be a bit hazy on powers, and if they stop to think about it, it is pretty obvious they are all built on top of one another. When they are all presented in an ordered fashion like this however, it becomes easy to see that the pattern can continue. Let me show you the next step then.

After powers, is an operation called tetration. It is so called because it is the 4th level of operation, if you start with addition as the basic operation instead of counting. If you want to tetrate a number by another number, you simply take that number to the power of itself a number of times equal to the second number. So if you want to tetrate 3 by 4, then you would have 3^3^3^3, where ^ is the symbol commonly used on computers for powers due to the difficulty of using the normal superscript notation for powers. We are going to the power of 3, 4 times here. Thus we are using powers in order to understand tetration.

A couple of interesting things to note in terms of tetration. First, the representation. It is usually looks like taking to the power, except the small number is in front of the number instead of after it. So the tetration of 3 by 4 discussed above would look like a tiny 4 hanging in the air followed by a normal sized 3. If you want to write this out on a computer, you use two ^s, so the above 3 tetrated by 4 would be 3^^4. Another important thing to understand is that when you are taking the powers in order to find the answer, you start at the highest point and work down instead of working up. As an example, if we had 3 tetrated to the third, it would look at first like 3^3^3, then 3^27, because we solve the furthest out operation 3 cubed equals 27, so we replace the 3^3 at the end with 27. Then we would take 3 to the 27th power, which works out to be 7,625,597,484,987. If we worked the other direction instead, and changed 3^3^3 into 27^3, we would only get an answer of 19,683, which is a fair bit smaller. So its important that you go from the top down if you are using normal notation, or from right to left, if you are using the computer notation. The next notable thing is how fast tetration grows. Lets look at some examples.

2 tetrated by 2 = 4

2 tetrated by 3 = 16

2 tetrated by 4 = 65,536

2 tetrated by 5 = 2.00353 × 10^{19,728} This is a number with almost 20,000 digits. If I wrote it out longhand it would be longer than the rest of this article all together by a significant amount.

3 tetrated by 2 = 27

3 tetrated by 3 = 7,625,597,484,987

3 tetrated by 4 = A number with 3 trillion digits.

3 tetrated by 5 = A number who’s number of digits is not even expressible in standard notation.

4 tetrated by 2 = 256

4 tetrated by 3 = 1.34078 × 10^{154} A number with 154 digits

4 tetrated by 4 = A number who’s number of digits is a number with 153 digits.

4 tetrated by 5 = A number who’s number of digits has a number of digits not expressible by standard notation.

As you can see, these numbers get pretty absurd, pretty fast. Still, tetration is obviously not the end of what you can do. You can do something called pentation which is the next step, taking a number and tetrating it a number of times equal to the second number. This can extend indefinitely, and it does not make sense to keep coming up with unique notation for each potential operation. Because of this, a guy called Donald Knuth, famous in the programming world fro writing a series of books on the fundamentals of algorithm creation, created a notation called Knuth’s Up Arrow Notation. Basically it involved putting a number of upwards facing arrows between the two numbers, with the number of arrows indicating which operation you are using. One up arrow is powers, two up arrows is tetration, three is pentation, four is the next one after that. While the ideal depiction of the arrows includes a full arrow, online we simply use the ^ symbol for the arrow. So ^ is powers, ^^ is tetration, ^^^ is pentation and so on. As you can see, we have already been using Knuth’s Up Arrow Notation with the computer depictions.

One problem with Knuth’s up arrow notation is that depending on what you are doing, the number of arrows can sometimes make the notation unweildy. If you are doing the 10th up arrow operation, you don’t want to be writing ^^^^^^^^^^ each time between your numbers. As such, once you get past five or six arrows, it is conventional to simply write the number of arrows in a superscript next to your arrow. Online you put the number of arrows in parenthesis after the initial carrot symbol. So 3^^^^^^4 could alternately be represented as 3 ^(6) 4. Using this notation we can now talk about one of the biggest numbers ever actually useful for anything, a number called Graham’s Number.

Graham’s Number was a number used as an upper bound for a mathematical proof in the field of combinatorics, the mathematical study of combinations and permutations of things. The number is generated in 64 steps. The first step is to take 3 ^^^^ 3, ie 3 taken to the operation one above pentation by 3. This is a number so incomprehensibly large, that you could not fit it into the universe if you subdivided the entire thing into plank volumes, a unit of volume that is the smallest it is possible to measure. We call this number G1. So already we have a number generally beyond comprehension and conventional notation in any sense. Then to get G2, we take 3 and we put G1 up arrows in between it and another 3. So using our notation, it is 3 ^(G1) 3. So this is a number generated by increasing the rate of size increase by an already incomprehensible number. For the next 62 steps we repeat the procedure, with G3 having G2 up arrows and so on. At the highest level G64, we have Graham’s Number. I am not even going to try and explain how big this is. Each step is increasing the size of this by increasingly incredible incomprehensible increments, and their are 64 of these steps.

So with that explanation of Graham’s Number, that’s where I am going to stop today. A few things to note before I sign off though. First, the number that Graham’s Number is an upper bound to has a lower bound of 13. So that number is somewhere between 13 and Graham’s Number. Actually, a smaller number was found later, which puts the upper bound at 2^^^6, a still incredibly large number, not expressible in normal numerical notation, but much much smaller than Graham’s Number. So the answer to the problem is somewhere between 13 and this new number. Additionally, while often considered the largest number ever useful for anything, it has been supplanted by even larger numbers as time has moved on. Some of these numbers are so large that they are not even expressible using Knuth’s Up Arrow Notation in a reasonably concise manner, and their are other notations for talking about even bigger numbers. One of the most famous is called Conway’s Chain Arrow Notation, which I may at some point right another post about, but currently am leaving for another day. If you are interested, look it up. Anyways, all this is just a math thing that I think is really fun and interesting. I might try to do another math related post next week if I can find something equally fun. So long, and happy imagining of numbers beyond imagination.