The “power tower feeding frenzy” and hyper-large numbers
What’s a large number? A billion? A billion times a billion? A billion to the billionth power? A googol? A googolplex? A googolplex is 10^googol, BTW:
So a googol is 1 with just 100 zeros after it, which is a number 10 billion times bigger than the grains of sand that would fill the universe. Can you possibly imagine what kind of number is produced when you put a googol zeros after the 1?
That’s pretty big, right? Not. Even. It turns out you can construct numbers that are so much larger than a googolplex, that it’s gonna light your head on fire just to read about them. Put on your asbestos hat and feast your eyes on Graham’s Number.
Moving up another level, exponentiation is iterated multiplication. Instead of saying 3 x 3 x 3 x 3, exponentiation allows me to bundle that string into the more concise 3^4.
Now, the thing is, this is where most people stop. In the real world, exponentiation is the highest operation we tend to ever use in the hyperoperation sequence. And when I was envisioning my huge googolplex^googolplex number, I was doing the very best I could using the highest level I knew — exponentiation. On Level 3, the way to go as huge as possible is to make the base number massive and the exponent number massive. Once I had done that, I had maxed out.
The key to breaking through the ceiling to the really big numbers is understanding that you can go up more levels of operations — you can keep iterating up infinitely. That’s the way numbers get truly huge.
You might get lost around the “power tower feeding frenzy” bit or the “power tower feeding frenzies psycho festival” bit, but persist…the end result is really just beyond superlatives. (via @daveg)
Update: In this video, you can listen to the inventor of Graham’s number, Ron Graham, explain all about it.
(via @eightohnine)
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