How big is a Googol, anyway?
published May 14, 2011
A friend of mine told his son that a Googol is a huge number. My kids caught on, so now I hear googol used in conversation all the time.
You know, like this:
I could eat 100 chocolate Easter eggs!
I could eat 1000!
Oh Yeah? Well, I could eat a googol!
A googol is 10100. That’s 1, followed by 100 zeroes. It’s a ridiculously big number. It’s a number purely designed to awe the imagination. It’s not really useful in real life.
After hearing googol-bombs dropped all the time like this, you sort of get inured to it. A couple of days ago, though, I started thinking about how big a googol really was. My friend (the same one who brought googol in to my life) and I tried to think of something where googol was a reasonable scale, and came up blank.
So, the question I want to ask is: is there anything that has a googol of it in the universe?
Let’s start with some standard “big quantity” things.
The number of grains of sand on the earth.
I don’t really care about accuracy here. So, I’ll just google it. The first answer says
There are seven quintillion five quadrillion grains of sand on all the beaches of the world. That’s a 75 with 17 zeros following!
So, in scientific notation that’s 7.5 x 1018. It’s a big number, but we’re nowhere near close.
How many stars are there?
Okay, let’s up the ante a bit. Another quick google search yields this answer:
With this simple calculation you get something like 1022 to 1024 stars in the Universe. This is only a rough number, as obviously not all galaxies are the same, just like on a beach the depth of sand will not be the same in different places.
Obviously, these numbers are nowhere near big enough. Surely the number of atoms is going to be bigger.
How many atoms are in the universe?
From the Wikipedia article on the Observable Universe (which is a wealth of numbers for making these kinds of calculations):
two approximate calculations give the number of atoms in the observable universe to be close to 1080.
We’re getting closer!
Well, it may feel like it, but we’re still ridiculously far off. We’re off by a factor of 1020. That’s part of the mind-bend of playing around with googol. The biggest number we’ve come up so far with is 1080. That means we’d need 1020 universes just like ours to have a googol of atoms. Yikes!
How many particles are in the universe?
So, if we ask how many particles are in the universe, the number of fermions (things that make up matter, like protons and neutrons and electrons) isn’t going to get much larger than the number of atoms. Most of the universe is made of hydrogen, which has one proton and one electon. Even if we pretended that the atoms in the universe were made of some heavy element (say, Uranium), and then break the protons and neutrons up into their component quarks, we’d only add two or three orders of magniture to the number.
Okay, so what about photons and neutrinos and stuff like that? Well, I found something here:
Of course, besides material particles there are also lots of photons and neutrinos flying around the Universe. It is estimated that there are about 1e9 times as many photons and neutrinos as atoms in the Universe.
So, that gives us ~ 1089. Closer, but still off by a factor of 1011.
It looks to me like our initial goal of finding something tangible that had more than a googol of it has failed.
Just for fun, let’s make one more calculation.
Packing the universe with sand
Okay, let’s go crazy here. Remember the grains of sand in the earth question we started with? Let’s pretend we pack the whole universe with grains of sand. Yeah, the whole thing. Ignore questions of gravitational collapse and turning the whole universe into a giant black hole of what used to be sand and all that.
So, from that same Wikipedia page on the observable universe:
The observable universe is thus a sphere with a diameter of about 28 billion parsecs (93 billion, or 9.3 × 1010, light years). Assuming that space is roughly flat, this size corresponds to a comoving volume of about 3 × 1080 cubic meters.
Cool! So that means, based on our number of atoms in the universe number above, there’s 1 atom for every three cubic meters in the observable universe. That’s not enough! Let’s see what happens if we pack the whole thing with sand.
This page here gives a reasonable estimate for the size of a grain of sand. The number they come up with is 8 per cubic millimeter. Let’s make life easier and round that up to 10. That means we’ll get 10 grains/mm3 X 109 mm3/m3 = 1010 grains/m3. Multiplying that by the volume of the universe (and if you’re feeling bored, think about how crazy that phrase is…) gives us 1090 grains of sand.
That’s right, if we fill the entire universe with grains of sand we’d still need 1010 universes to get a googol grains of sand.
Hmph. Okay, but how many atoms would that be?
To make things easier, let’s just make the universe one giant silicon crystal. Silicon has a density of 2.3 g/cm3 (again, Wikipedia is our friend), or 2.3 x 106 g/m3. Silicon has an atomic mass of 28 g/mol (the stable isotope of Silicon has 14 protons and 14 neutrons).
So, a cubic meter of silicon has
2.3 x 106 g/m3 / 28 g/mol X 6.022 1023 atoms/mol = 4.95 X 1028 atoms/m3
Ooo, we may be on to something here.
Multiply that by the volume of the universe:
4.95 X 1028 atoms/m3 X 3 × 1080 m3 = 1.48 X 10109 atoms = 1.48 X 109 googol atoms!
We did it!
We had to make a ridiculous assumption that could never happen, but in the end managed to get a number that was bigger than a googol.
What did I miss?
Thanks for reading this far.
If you’re interested, upvote it on hacker news.
What did you think? Is there any reasonable number of real-live countable things in our current universe that has more than a googol? Perhaps if we include gravitons, or dark matter, or something like that?
Also, this post was written in about an hour in a coffee shop, so I’m sure I’ve made errors. Let me know if you find any!