Big Numbers: Part 1
I used to have a poster that had a million dots on it, and I can assure you that you can’t really appreciate how big a million really is until you actually see a million dots, all at once. I used to look at it and, noticing that each tiny little square was really made up of 100 even tinier little dots, think in amazement that if each dot represented a year, then each hundred-dot-square represented a (long) human lifetime.
Here’s what a small section of that poster looked like. This image has 100 dots per square x 10 squares per row x 10 rows: That is, 100 X 10 X 10 = 10,000 dots.
You’d need to print out a hundred of these squares to see those million dots for yourself.
A million really is a big number for humans to conceptualize; we have 10 fingers and 10 toes, live (maybe) a hundred years, and have (maybe) a few hundred (maybe even a thousand) friends on Facebook.
But to talk about big numbers, it makes things much easier to use exponents. For a while anyway. You remember exponents:
106 = 1,000,000 (a million, or 1,000 thousand)
109 = 1,000,000,000 (a billion, or 1,000 million)
1012 = 1,000,000,000,000 (a trillion, or 1,000 billion)
Etc. This notation will work well for Part #1. Wikipedia has a list of the names of some large numbers, which you can check out here.
Anyway, here we go…
Bremermann’s Limit = 2.56 × 1047
This is a pretty cool “big” number I just learned about a month or so ago. Bremermann’s Limit is “…the maximum computational speed of a self-contained system in the material universe. It is derived from Einstein’s mass-energy equivalency and the Heisenberg uncertainty principle, and is approximately 2.56 × 1047 bits per second per gram”
David Foster Wallace, in “Everything & More: A Compact History of Infinity“, puts Bremermann’s number into context when he says,
“…a hypothetical supercomputer the size of earth (about 6 x 1027 grams) grinding away for as long as the earth has existed (about 1010 years) can have processed at most 2.56 x 2092 bits… Calculations involving numbers larger than 2.56 x 2092 are called transcomputational problems, meaning they’re not even theoretically doable; and there are plenty of such problems in statistical physics, complexity theory, fractals, etc.”
I just thought it was kinda cool.
Atoms in the entire universe [approx] 1080
This isn’t really an official, “named” number, but it’s a good place to start. Primarily because every number you will read about after this is bigger than the estimated number of atoms in the universe.
Think about that for a second; 1080 represents a number that is not just greater than every grain of sand on the planet, but all the atoms in every grain of sand on the planet, as well as all the atoms in all the planets, and all the stars, in the entire universe.
Now we’re ready to start talking about some big numbers!
Googol = 10100
Everyone’s probably heard of a Googol. A googol is the name of the number that is expressed as 10100; the digit 1 followed by 100 zeros in decimal representation:
Note: The search-engine giant Google derived their company name from the number googol, saying that
“The name reflects the immense volume of information that exists, and the scope of Google’s mission: to organize the world’s information and make it universally accessible and useful.”
As I mentioned previously, notice that a googol is bigger than the estimated number of atoms in the universe:
10100 > 1080 atoms in the universe
On one hand, it’s a big number. On the other, it really isn’t. Not compared to the next number on our list…
Shannon Number: Somewhere between 10111 and 10123
I know; you were thinking the Googolplex would be next, but I like chess, and therefore, wanted to bring the Shannon Number to your attention. The Shannon Number refers to the number of legal positions in chess, and is estimated to be between 1043 and 1050, with a game-tree complexity of approximately 10123. That is, there are more ways a game of chess could unfold than there are particles in the universe, so it’s virtually certain that the next game of chess you play will be unlike any other game of chess that’s ever been played, or ever will be played, in the entire, cumulative history of human civilization.
Skewes Number = 1.397 × 10316
Skewes Number has something to do with prime numbers, but I’m not quite sure what. I just know it’s big, and that in 1933 Skewes showed that
Converted to base 10, the value is normally approximated as 101010^34
A more accurate approximation is 10108.852142×1033 or 10108852142197543270606106100452735038.55
Since then, others have improved the estimate dramatically. In 1966, Conway and Guy gave an upper bound of about 101167. Most recently, in 2005 Patrick Demichel found a smaller crossover point near 1.397162914×10316.
Whatever. It’s big.
If you’re still with me, you deserve a reward, and I now bring you, with gratuitous gusto, the next big number on our list, the Googolplex! A googolplex is the number 10googol, which can be written conceptualized as the number 10 raised to the power of a googol of zeros (i.e., 10100 zeros).
However, although a Googolplex can be conceptualized as the number 1 followed by a googol of zeros; it cannot be written down in decimal form like you can with a Googol. This may not be obvious at first, but go ahead and try. First, write down a Google; a 1 followed by 100 zeros:
Finish yet? That was pretty fun wasn’t it. Yeah…OK…
Next, add one zero; just one. The new number I challenge you to write is a 1 followed by 1,000 zeros:
If you bothered to write down 100 zeros, you probably didn’t even begin to write 101000 down, and I don’t blame you. It was a pain in the ass to write down 100 zeros, so the act of writing down 1,000 zeros would be a mind-numbingly insane waste of time. However, I think I’ve probably made my point, and you should now be in a position to understand why writing down a Googolplex is, quite literally, impossible; there isn’t enough material in the universe, not even if every single molecule could represent a number.
Red Turtlenecks: They’re coming back in style, and when they do, I’ll be ready…
That wraps up Part #1 of “Big Numbers”, but stay tuned; the heavy-hitter in Part #2 is so friggin…big…that it needs and deserves it’s own post!