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.

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33 thoughts 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. 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. 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 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.)

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  5. 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. Wrong, you say Universe instead of observable Universe.

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

  7. 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. 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?

  9. 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.

  10. 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.

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

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  13. 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.

  14. 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.

  15. It’s easy to find things with a googol of possibilites, but not a googol physical things.

  16. Hello. If we are looking at atoms within a cell, within organisms, shouldn’t there be more atoms in the universe than one googol? Atoms are not imaginary and are possible. If you’re referring to tangible physical items such as the example before of kittens, salt, grass, then no, there does not seem to be one googol of anything in that sense that it is physical and visible. But there are more than a googol of atoms. Using a human being as an example, if it is true that there are about 10^14 atoms in one cell, and there are 10^14 cells in one human, considering there are about 7.2 billion humans, this does not equal a googol. But looking into other organisms on this planet alone, albeit organisms that may not have as many cells as humans, there should still be more total atoms on Earth, surpassing the one googol number.

  17. Libby: No, there are far fewer than a googol atoms comprising the earth.

    The mass of the earth is less than 10^25 kg. The mass of an electron is about 10^-30 kg. So if the earth were made up of only electrons, much lighter than atoms, there would be 10^55 electrons in the earth. That’s 45 orders of magnitude less than a googol.

  18. In regards to the number of atoms, not the mass. And of things that are on Earth, anything from water, humans, fig trees, carp. Not the mass of them but the number.

  19. Libby: I’m using mass to get an upper bound on the number of things. If you have a box of things that weight at least 10^-30 kg each, and the box weighs 10^25 kg, then there cannot be more than 10^55 things in the box.

  20. What about things in space-time. You can represent all integers up to a googol with 333 bits. The age of the universe is ~10^66 planck times. So if there are 10^79 atoms in the universe then it’s conceivable that a googol integers are representable if you relax the constraint about them existing at the same time.

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