I’m trying to get my head around some math here. The scientific consensus is that there are 10 to the power of 86 subatomic particles in the observable universe. Let’s call that number “L.” Now take L to its own power, L to the power of L. Now let’s say I print out the sum.
If printed, would the resulting sum take up all of the space in a library? All of the space in every library on Earth? Somewhere in between?
Your problem can be approximated by looking at a googolplex.
A googol is 10[sup]100[/sup]. A googolplex is 10[sup]googol[/sup].
You couldn’t fit that into the known universe if you tried to print it out. You couldn’t write it out if you started at the beginning of the universe and kept going for a trillion trillion years.
Such is the power of exponents.
The number on the bottom hardly matters-- It’s the one in the exponent that does all of the heavy lifting. Whatever (10^86)^(10^86) is, it’s surely larger than 10^(10^86), right? So let’s just look at that number for a moment.
It’d be a 1 with a whole bunch of zeroes after it. How many zeroes? Well, 10^86 of them. In other words, as many digits as there are subatomic particles in the observable Universe. So if you wrote one digit on every subatomic particle, it’d still take you the entire observable Universe to write it in.
And how many more zeros would it take to write the number you actually asked about? Well, 86 times that many. Which kind of pales in comparison when we’re talking about numbers that big.
And since the first number was unprintable because there are not enough particles, adding one more level of exponentiation to the tower obviously doesn’t help matters.
As a large power of 10, it would trail a simple string of zeros.
ETA never mind, ninjaed above
Everybody likes to quote Douglas Adams on how huge the universe is. Well, the universe ain’t a bump on a fly fart compared to math. Math’ll gobble up universes by the googol and won’t even notice.
There are lots of ways of writing really, really huge numbers that are incomprehensible. But we can stick to plain old exponentiation and boogle your mind.
Googolplex[sup]Googolplex[sup]Googolplex[/sup][/sup]
I believe we had a thread like that, and the conclusion was that to get some really hugely huge numbers you want some (mathematically rigorous) variant of “the biggest number that can be described in 10^100 symbols”. By their nature, such numbers are divorced from any physical or computational model you might imagine, including the universe.
I had Mathematica software years ago and was playing with it, exploring a function I made up that looked like this:
f(1) = 1
f(2) = 2^2
f(3) = 3^3^3
f(4) = 4^4^4^4
That was about as far as Mathematica could handle it, IIRC.
I don’t even remember if I did it the gentle way or the aggressive way – that is, did I actually say
f(3) = 3^(3^3)
et cetera.
You’ve reinvented tetration.
f(n) = [sup]n[/sup]n
The standard order of operations on exponents is what you’re calling “the aggressive way”: a^b^c is interpreted to mean a^(b^c), because (a^b)^c can be rewritten as a^(b*c).
Here is an on-line calculator intended to work with such large numbers.
Type (10^86)^(10^86) into the data entry box near the bottom of that page and press Calculate! to see 10[sup]8.600000000000 × 10[sup]87[/sup][/sup], or about 10^10^87. This is what Munafo calls a “class 3” number. (Note that, as Chronos mentioned, the number would be almost as big if you replaced the 10^86 at the bottom of the tower with just 10.)
Instead of L^L do you want the much bigger number L!^L! ? Type (10^86)!^(10^86)! into the data entry box and press Calculate! to see about 10^10^10^87 — a “class 4” number. Raise that number to its own self should be preposterously large! Let’s try it:
(10^10^10^87)^(10^10^10^87), press Calculate: to see about 10^10^10^10^87 — just a “class 5” number.
The Calculator allows 100P1 to input a class 100 number. If you enter (100P1)^(100P1)^(100P1)^(100P1)^(100P1) you only get a class 104 number. Add factorial operators (100P1)!^(100P1)!^(100P1)!^(100P1)!^(100P1)! and you’re still just in class 105. Instead of a class 100 number what about a class X number where X is itself a class 100 number? AFAICT that Javascript cannot handle such things, let alone really big numbers like Conway’s or Graham’s.
So basically, if you use scientific notation, it will fit in a tweet.
If you use a cubic mile of outer space to represent a zero, is there enough space to represent 86*10^86 zeroes?
I think you’re asking if the total volume of the universe is enough to contain 8610[sup]86[/sup] cubic miles. The concept “volume of the universe” is problematic, but for simplicity let’s take it as a constant-volume sphere 13.8 billion light-years in radius. Then the volume is about 2 x 10[sup]69[/sup] cubic miles, so not even close. Dividing that volume into 8610[sup]86[/sup] cubes would have each cube being about 1 mm[sup]3[/sup].