Googol and Googolplex - Numberphile

ANTONIO PADILLA: Imagine you had a universe which was a
googolplex meters across, OK, so a big universe.
Then, that's very big.
And if you actually traveled far enough in that universe,
you would start to see repetitions.
You would start to see exact copies of yourself.

RIA SYMONDS: Today, we've got a googol.
ANTONIO PADILLA: A Google.
RIA SYMONDS: A googol.
ANTONIO PADILLA: As in the website.
RIA SYMONDS: Kind of, not spelt the same.
It's spelt G-O-O-G-O-L, 10 to the power of 100.
So imagine that's a 1 with 100 0's after it.
ANTONIO PADILLA: And it came about because it was--
a guy called Kasner, who was a mathematician.
He was also very famous in cosmology.
RIA SYMONDS: And it wasn't him that came up with the name.
He just thought of a really big number.
He thought, what's a big number, 1 with 100 0's.
ANTONIO PADILLA: And he wanted to sort of explain infinity.
And he wanted to come up with a very big number--
that wasn't really big at all compared to infinity.
RIA SYMONDS: And he actually asked his nephew, his
nine-year-old nephew, what's a good name to call this number?
ANTONIO PADILLA: His nephew came up with the word
"googol." But it's quite a big number.
I don't think it's that big a number.
But it's quite a big number.
I mean to put it into perspective, you might ask
yourself well, OK, how many grains of sand could you fit
inside this room?
You think that's a lot, right?
That's going to be a lot of grains of sand.
It's only about 50 billion, not that much.
So you might ask well, let's have something more dramatic.
Well, how many particles are there in the universe?
Now, that's 10 to the 80, a 1 with 80 0's after it.
That's how many particles there are in the universe.
So a googol is bigger than that.
And you could ask yourself OK, well how many grains of sand
could I fit in the universe?
Say the universe was full of just grains of sand, how many
grains of sand can I fit in the universe?
And that would be about 10 to the 90, so a 1 with
90 0's after it.
It's quite a big number.
But it's not that big.
There's certainly stuff that's a lot bigger that that.
BRADY HARAN: You're saying it's not big a number, and yet
all these things that we think of as big, like the size of
the universe, and atoms in the universe, and particles,
aren't even touching it.
So it must be big.
ANTONIO PADILLA: Well yeah, OK, but I can think of
something that is bigger than it, that you can think of
physically.
So if you take the smallest volume in the universe, which
is a Planck volume, so it's basically a little cube that
its radius is a Planck length, which is 10 to the minus 35
meters, and you see how many of them you can fit in the
observable universe.
And you get about 10 to the 183, so that's a 1 with 183
0's after it.
So that is bigger than a googol.
So googols aren't that impressive.
We can think of things that are bigger.
RIA SYMONDS: He then thought, well what's a bigger number
than a googol?
And for that he took 10 and raised it to this power, the
power of a googol, 10 to the 100.
So this is taking 10 and timesing it by 10 and 10 and
10 and 10 and 10, and timesing it googol amount of times.
And this number is called a googolplex.
ANTONIO PADILLA: We start writing down a googolplex.
And let's suppose-- well, we're writing on a piece of
paper here.
But let's suppose we actually tried to
write it down on particles.
So for each particle, we're going to write down a 0.
So first particle, we write down a 0, second particle we
write down a 0.
And we carry on doing that on every
particle in the universe.
And we still would not be able to write down a googolplex.
It's too big a number.
And there's something else remarkable about the
googolplex.
And I find this astonishing.
This should emphasize how big a googolplex is.
Imagine you had a universe which was a googolplex meters
across, OK, a big universe.
And some models of eternal inflation and things like
that, you do get universes which could be as big as this.
So we have a universe which is a googolplex meters across.
Then, that's very big.
And if you actually traveled far enough in that universe,
you would start to see repetitions.
You would start to see exact copies of yourself.
Now, you might think, what the hell am I
talking about here, Brady?
But let's take you, Brady.
Let's ask how many quantum states can describe the volume
that you occupy, OK?
The number of quantum states that can occupy that volume,
the total number is, and you can work this out.
It's something to do with entropy and black holes and
stuff like that.
But you can work it out.
And it is 10 to the 10 to the 70.
That's the number of different quantum states, different ways
that you could actually put together that volume of space.
So that number is less than a googolplex.
BRADY HARAN: So the space I'm in could be me, it could be
you, it could be a dog, it could be a vacuum.
ANTONIO PADILLA: Yes.
BRADY HARAN: It could be anything.
ANTONIO PADILLA: All those particles are arranged in any
number of different ways.
Particles might not even be there.
But all those possible states, given that volume that you
occupy, the number of possible states you could have is 10 to
the 10 to the 70.
A big number, but not as big as a googolplex.
But 10 to the googol is equal to 10 to the 10 to the 100.
So this number is clearly less than this number.
So you occupy roughly about a meter cubed of space.
There's that many different possible states that
you could be in.
So if you go this many meters away from yourself, you would
start to expect to see repetitions.
So that means if you're in a universe which is so big, as
big as a googolplex meters across, that eventually you
would start to see repetitions of those
volumes of 1 meter cubed.
BRADY HARAN: So just by chance, I would run into
another arrangement of atoms that matches me.
ANTONIO PADILLA: Exactly.
Exactly.
Exactly.
It matches you exactly.
And then you'd go even bigger, and you would start to see
sort of the entire observable universe is repeated.
So this is truly remarkable.
I think the fact that you would go there and see your
doppelganger if you went far enough away in a googolplexing
universe, I think it just sort of emphasizes how big that
number really is.
The size of the universe is 10 to the 26 meters sort of all
cubed, that's the size of the universe.
So this number is tiny compared to that.
BRADY HARAN: So there's probably not another me in
this universe.
ANTONIO PADILLA: Probably not.
But if we live in a universe which is a googolplex across,
then there probably is.