There isn’t a googol of anything

Before Google, there was googol, the number 10^100, written as a 1 followed by 100 zeroes.

There are about 4 × 10^79 atoms in the universe. (Here’s a derivation of that number.) You could bump that number up a little by counting particles rather than atoms, but not by much. There’s not a googol of anything physical in the universe.

On the other hand, numbers larger than a googol routinely arise in application. When you’re counting potential things rather than physical things you can run into numbers much larger than a googol. This happens all the time in probability calculations. Inconceivably large numbers pop up in intermediate steps on the way to moderate-sized results.

Related posts:

How to calculate binomial probabilities
Floating point numbers are a leaky abstraction

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16 comments on “There isn’t a googol of anything
  1. Fun. 70! is the first factorial over a googol, yes? The number of unique shuffles of two decks of cards is over 10^166. These are “inconveniently large numbers” indeed!

  2. Michael says:

    Example for a number of potential things larger than a googol: The number of possible chess games that can be played out is about 10^123. At least according to New Scientist (see Link above).

  3. Scott says:

    The number of possible go games of length 400 or less (a reasonable limit for go games) is on the order of 10^800, according to http://senseis.xmp.net/?NumberOfPossibleGoGames or about 10^768 according to Wikipedia. (There are more go games partly because the board is so much larger. There are 361 opening moves in go, compared to 20 opening moves for chess.)

  4. John S. says:

    There isn’t -1 of anything physical in the universe either, is there?

  5. duggi says:

    i would like to kindly disagree. i personally believe, that matter, space, and time are infinite. in the infinite sense, not in the bounded surface sense. in this view, there is definitely a googol of atoms.

  6. Duh says:

    Wrong, you say Universe instead of observable Universe.

    The total Universe is probably a googol times bigger that the observable Universe.

  7. Jonathan says:

    hogwash
    hogwash
    hogwash

    Physicists have no coherent story to explain the edge of the known universe, if there are regions of the universe beyond our horizon, if there are multiple universes, or a sundry of other questions. We don’t know quarks are the smallest things. Can we count strings? We don’t know what dark energy or dark matter really are, or what kind of particles they are made of – there maybe lots of countable physical things in there. Photons are real things too, something I’d put into “anything”. Lastly, virtual particle pairs are absolutely real, and counting those would take you well over anything countable in atoms.

    I thing you’ll need a lot more caveats to this story.

  8. The number of ways of splitting a 1000 instance data set into 80% training and 20% test is greater then a googol
    http://www.wolframalpha.com/input/?i=1000!+/+(800!+*+200!)

  9. Vulcan Eager says:

    What if we count events? For example the number of times an electron has passed from atom to atom since the beginning of the universe or something like that?

  10. Duh says:

    Photons?

  11. vonjd says:

    Well, I won’t be so sure – first of all we don’t really know how many things there are *really* because we don’t even know if the universe is infinite (said above) – but anyway: even if it is finite: if you count all the different positions of all the different particles since the big bang a googol won’t suffice. And that is *not* potential, but *real* stuff.

  12. John says:

    It’s interesting that googol makes a sort of universal upper bound. I’ve had people ask whether there could be a googol words in all the world’s books. Nope. Not even if there were sentient beings all throughout the universe writing books. If they need at least an atom to write a word, there aren’t a googol words.

    You can’t have a googol kittens, or cells, or grains of sand, or bits of data storage in all the world’s disks. I used this as an example in this blog post to argue that you could never store a table of 80-digit primes.

    And yet it’s easy to come up with situations with over a googol theoretical possibilities. Several great examples in the comments here. I particularly like Will Fitzgerald’s example of shuffling two decks of cards because it’s so tangible.

    I remember as child being surprised to find out you could never write down a list of all permutations of the alphabet. 26! is far less than a googol, but it’s still quite large, on the order of 10^26.

  13. Dan P says:

    And of course numbers have many more uses than merely counting. For example Godel numbering, which can quickly lead to vast numbers.

  14. vonjd says:

    I think much more interesting would be to know if there is something like “infinity” in the “real” world…

  15. As another example, the number of possible amino acid sequences of length 62 (counting only the 20 naturally occurring amino acids, ignoring additional ones that occur by covalent modification) is 20^62, which is more than the number of atoms in the universe.

    Until you do a calculation like this, it’s hard to appreciate just how huge the sequence-space that proteins occupy really is.

  16. In my previous comment, I meant to add that there are more than a googol possible sequences that are 77 amino acid sequences long. Typical proteins are much longer than this.

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