Numers! Graham numbber and googleplex and few more
========================== Sometimes when we talk about big numbers, we talk about how many digits they have or, basically equivalently, write them in scientific notation involving some power of 10. A googol, for example, is 10^100, which has 101 digits. The largest known prime number has over 17 million digits. A googolplex is 10 raised to the googol power, so it has approximately a googol digits. It takes a while to write down a 1 followed by 100 zeroes, but we can do it. ---------------------- A couple years ago, Numberphile made a nice video on Graham's number. The good folks in the video run into some of the same problems I did comprehending how big the number is. ====================
The only way to describe Graham's number involves some dense "up-arrow" notation. The Wikipedia page on the notation is a good resource if you'd like to read more.
Up-arrow notation is an extension of exponentiation. In the same way that exponentiation is iterated multiplication (and is denoted by one up-arrow), two up-arrows denotes iterated exponentiation, three up-arrows denotes iterated iterated exponentiation, and so on.
More concretely, a↑b is the usual exponential notation, ab=a×a×a.., for a total of b copies of a multiplied together. Adding a second arrow, so we have a↑↑b, means we take b copies of a in a tower of exponentials. An example is worth a thousand words: 4↑↑3=444. When we add yet another arrow, a↑↑↑b means "make b copies of a↑↑." So 4↑↑↑3 means we do 4↑↑(4↑↑4). (There are three copies of the number 4, separated by the ↑↑ symbol.) The thing inside the parentheses is 4444, and then we raise 4 to the 4th power that many times. I can't say much about that number because 4444 has 8×10153 digits, and we have to make a tower with that many powers of 4 in it.* The important thing about up arrows is that they grow mind-bogglingly quickly. Exponentiation grows much more quickly than multiplication. 2×10 is just 20, but 2^10=1024. In the same way, each level of up-arrows grows much more quickly than the level before.
We can describe Graham's number with an enormous stack of these up-arrows.
It's easiest to think of this as an iterative process. We start at the bottom with g1=3↑↑↑↑3 and then creating the second row (call it g2) by having two threes separated by g1 up-arrows. Then g3 is two threes separated by g2up-arrows, and so on, until g64 is Graham's number. The only one of these numbers that I can hope to describe a little bit is g1=3↑↑↑↑3. Peeling the onion of up-arrow notation, we have 3↑↑↑↑3=3↑↑↑(3↑↑↑3). So far so good. The thing in parentheses, 3↑↑↑3, is 3↑↑(3↑↑3), which is a stack of over 7.5 trillion 3s.* So to get g1, we have to do "↑↑" to 3 over 7.5 trillion times. It boggles the mind! Check out Wikipedia if you'd like to see some more details about the magnitude of g1. (Warning: your brain might explode.) I can't even describe how large the first layer is, and each layer grows much more quickly than the last. The final number is totally beyond comprehension.
The origin of Graham's number is one of those slightly legendary math stories. In 1977, Martin Gardner wrote, "In an unpublished proof, [mathematician Ronald] Graham has recently established ... a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof." Graham had worked on a fairly complicated question about combinatorics. But Graham's number doesn't actually appear in the published proof of his result. The number he used was quite a bit smaller. In a Google+ post, John Baez describes both the combinatorics problem Graham was working on and where the number came from. He writes,
"I asked Graham. And the answer was interesting. He said he made up Graham's number when talking to Martin Gardner! Why? Because it was simpler to explain than his actual upper bound - and bigger, so it's still an upper bound!"
*These sentences were changed after publication to correct numerical errors. Thanks to Mike for pointing it out!
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