Need to read: A Series
My definition of a large number, is a number you will never ever need to use in your daily life. For example, I would say 12 is not a large number, as you use it with eggs and donuts and other items. But a number such as a billion (10^9), would never be used in the average persons day to day life. But why do we even care about big numbers. Why should we care?
Because why not? Thinking about such large numbers can be a great creative exercise. And also, large numbers are commonly used in pure mathematics (such as the famous Shannon's Number (10^120)).
The first truely large number in my eyes is a million. This is 10^6, and is one of the most well done big numbers. Another big number is a billion. 10^9. So, we can see that they both share a -illion suffix. We can define this suffix to mean that n-illion will be equal to 10^(3n+3). So for 4-illion (also known as a quadrillion), will equal 10^15. Here is a short list of -illion numbers.
Now, we are already to the point that these numbers have no real world meaning. The amount of atoms in the universe is equal to about 10^80. But, we are only getting started. We can define a Googol (not the website) as 10^100, which can also be written as 10^10^2. But, we could go one step further. We could take 10, to the power of a googol! This number is known as a googolplex (for a reason we shall soon get into) and is written as either 10^10^100 or 10^10^10^2. We can continue, so that a googolplexplex (or googolduplex) is equal to 10^10^10^100. We can keep on going on forever.
Before, we used the -plex suffix. This suffix was created by John Conway and Richard guy in their book, The Book of Numbers. n-plex is defined as 10 raised to the power of n. So 2-plex is 10^2 which is just a hundred, and a 6-plex is 10^6 (or a million). So now, we can see where googolplex gets its name (10 raised to the power of a googol). But we can extend this suffix. An example is n-duplex, which is defined as 10^10^n, and n-triplex, which is defined as 10^10^10^n and so on and so forth. So, we can define n-m-plex as 10^10^10^...^n where 10 is repeated m times.
To advance to even larger numbers, we must (once again) define a new notation. I know this may feel a bit overwhelming, but don’t worry, I’ll introduce an even newer notation in the next section >:). Before that, though, we need to talk about operations that come after exponentiation. Just as multiplication is repeated addition, and exponentiation is repeated multiplication, we can extend this pattern to create new operations (called hyperoperations) that repeat these already powerful processes.
We will define the first hyperoperation before we define generally. Tetration of n and m (n^^m) equals n^n^n^...n^n m times. So for example, 3^^3 will equal 3^3^3, which is 3^27, which is around 7 trillion. So now, we can define all hyperoperations.
So now, we can define Knuth arrows. n↑m, equals n^m. And each arrow we add one increases the operation by one. So n↑↑m is tetration, and n↑↑↑m is pentation and so on. But more formally, we can define a ↑^n b where n is greater than 1 to be (a ↑^n-1 (a ↑^n-1 (a ↑^n-1.....↑^n-1 a)...)) where a is repeated b times (this is where ↑^n means ↑ repeated n times). So here, we can see how the numbers can increase quite quickly.
Now, we only have one finally push before we can describe one the largest numbers ever used in a mathematical proof, Graham's Number! Conway's Notation can be defined as a→b→c is equal to a ↑^c b (so 3→3→2 is the same as 3↑↑3, which is 7 trillion). So now, we can start to write down Graham's Number. Starting with g1, which is equal to 3→3→4, which is already a MASSIVE number. Now, we go, for g2 3→3→g1. So 3↑↑↑↑...↑↑↑↑3 with g1 arrows! Graham's number is g64. So this number is WAY bigger than every number we have mentioned so far.