eli5, Graham’s Number?

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Multiplication is just defined as doing addition over and over. 4*3 is the same as 4+4+4 as you are asked to add 4 to itself 3 times. Similarly, 4^3 is asking you to do 4*4*4, so multiplying 4 to itself 3 times. You can use similar logic to make a notation that makes you do 4^4 to itself 3 times. These numbers get very big, very fast. I only went to three degrees of that logic but Graham’s Number makes you do it to the 64th degree, which is absurd to the point of being incomprehensible.

There is no largest number. As others have said Graham’s number is incomprehensibly large. There just isn’t any real intuitive way to make sense of its scale. It’s notable because it was for some time the largest number used in a serious mathematical proof. It has since been supplanted by some other ridiculously large numbers.

It gained notoriety because of its largest in a proof status, but also because of the fact that just explaining how to construct it to a layman was very difficult to do. It requires repeated iteration of a type of mathematical operation that is not something non-mathematicians would learn. Once you begin to understand how those operations work, and explore the process of forming the number, you start to see that even after one or two of the first iterations you’ve already constructed an enormously large value, and that’s only the beginning.

It’s an upper limit, there is a problem in combinatorics which asks “suppose you are combining things, what is the minimum number of things you need to guarantee a certain result” Now nobody know what the minimum number is but we do know it is less than grahams number. There is absolutely no way to explain how huge Grahams number is to anyone let alone a five year old. If all the material in the universe were turned into paper you don’t have enough paper to write how many digits it contains. Before you could store it in your head your brain would collapse into a black hole. There is simply no physical description that comes close. Even mathematics professionals don’t understand how big it is. They only know how to construct it.