What is the largest number?

Sometimes, we talk about what the biggest number is and then arrive at the answer: INFINITY. But, infinity is not a number in a real sense. So, the largest number closest to infinity is ‘Graham’s number’. This number is so huge that it is said that even the entire universe will not be enough to write on!



Graham’s number is really big, but of course, it's precisely 0% as big as infinity. On the other hand, everything we touch is finite, so in some sense, Graham's number is probably "close enough" to infinity for doing most math. 

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. But, Graham’s number is ridiculously larger than googol or googolplex. It is so huge that it cannot even be mentioned in scientific notation of raise to the power of 10.
The only way to describe Graham's number involves some dense "up-arrow" notation. 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. The Wikipedia page on the notation is a good resource if you'd like to read more.

We can describe Graham's number with an enormous stack of these up-arrows.


 
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.

Here, we can’t even describe how large g1 number is! It blows up the brain. The final number is totally beyond comprehension.


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