Sometimes I think about numbers. Big numbers. Numbers so big that you couldn't write them down if you wrote a digit on every particle in the universe.
At the low end of the big number scale is the googol. A googol is pretty big - a one followed by 100 zeros, or 10100, or 10^100. (This is much larger than numbers like a million, billion, trillion, or decillion, which have 6, 9, 12, and 33 zeros, respectively.) A google is about a thousand times larger than at least one estimate of the number of particles in the universe. In other words, if you were to write a thousand numbers on every particle of the universe, you could write out all of the numbers up to a googol
The next step up is the googolplex. This is a one followed by a googol of zeros, or 10^(10100), or 10^(10^100). Just to write out this number would require one particle to have a one written on it, and every other particle in the universe to have a thousand zeros on it. Forget about writing all of the numbers below that in the sequence. And yet, this is still relatively small potatoes in the big number realm.
To start talking about really big numbers, you need a new, more powerful notation. Knuth's generalized up-arrow notation works well up to a point. The basic idea is to extend the power of the exponentation operator. Let x^y be x * x * ... * x, where there are a total of y x's. So 10^3 = 10 * 10 * 10 = 1000.
Now we take it to the next level. Let x^^y = x ^ (x ^ (x ^ ... ^ x)), where there are a total of y x's. This number grows quite rapidly. For instance, 2 ^^ 4 = 2 ^ (2 ^ (2 ^ 2))) = 2 ^ (2 ^ (2 * 2)) = 2 ^ (2 ^ 4) = 2 ^ (2 * 2 * 2 * 2) = 2 ^ 16 = 65536.
2 ^^ 5 is very large, roughly 2 followed by 19728 digits (bigger than a googol). 2 ^^ 6 dwarfs a googolplex - it has upwards of 10^19727 digits (recall that a googolplex has only 10^100 digits.)
We've just started down the path of big numbers. The next step is to increase the number of up-arrows, which allows numbers to grow much faster. But that is a topic for another day.
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