Big Numbers

Think about all the grains of sand in all the deserts and beaches on the planet. That’s a ridiculously big number—seven and a half quintillion (7.5 x 10¹⁸), at least according to one estimate.

As many grains of sand as there are, there are far more stars out there in the observable universe. This kind of weird comparison can be useful when you need to understand something like deep time, where the numbers that measure our daily experiences don’t amount to a hill of beans.

It’s useful to us because we can get an idea of how many grains of sand there are in the world. Well, we can’t really get a sense of this number—it’s still well beyond our normal human comprehension to try to imagine what a million is, much less a quintillion.

A quintillion is a million million million.

This analogy works because we’ve taken something from the world of our regular experience—the amount of grains of sand in the world—and asked us to think about that unimaginably large number for a second. We’re asked to turn this number around in our minds, to consider all those tiny grains that get stuck in your flip flops, and the tens of thousands of grains that somehow end up in your pockets whenever you visit the beach.

We realize that this is a number we can’t grasp, and we’re forced to sit in awe of the number for a moment. Then, we’re told there’s a number that’s even bigger—much bigger, even—than the number we can’t really grasp.

The number of stars is greater than a number we can’t really grasp, which is a useful idea in and of itself. Suddenly, something we had no hope of understanding can at least be discussed in relation to something else.

This phenomenon can work in reverse, too. Physicists often become very good at thinking in non-intuitive ways about the universe, and that includes the way ordinary matter like atoms works.

It also includes weird things like knowing (roughly) how many atoms would be in a given space. Remember thinking about grains of sand? Well, let’s zoom in on one grain for a moment, and consider that there are 100,000,000,000,000,000,000 (10²⁰) atoms in a grain of sand.

Now, let’s zoom out and think about the universe. We know that there are way, way more stars out there in the observable universe than there are grains of sand on Earth. We also know that each grain of sand has a mind-numbing number of atoms in it.

How many atoms must there be in the observable universe? As difficult as this number is to calculate or estimate with any real degree of accuracy, it’s still useful as a concept. Each of those stars is made up of an unimaginable amount of atoms—more atoms than are on the Earth, never mind sand in particular.

One rough estimate places the number of atoms in the observable universe at 10⁸⁰. That’s a ten followed by eighty zeroes. All you can really do to grasp how enormous that is, is to think about how many atoms are in a grain of sand, then how many grains of sand would be in a star, and how many stars there are out there.

We can compare these numbers to one another and get a slightly better sense of the truly enormous scale of the universe, but we can’t think of them in isolation.

This brings me to Claude Shannon. In Programming a Computer for Playing Chess, Shannon pointed out that there are around 10¹²⁰ game configurations—a figure now known as the Shannon number. When engineers and scientists were working on a machine that would beat a human at chess, they knew (thanks to Shannon) that a brute-force approach was hopeless.

Decades later, scientists pointed out just how much bigger Shannon’s number is than the number of atoms in the universe, and we’ve been using that as a comparison point ever since. We can’t understand astronomical numbers, but we can grasp that there’s a much bigger number elsewhere, and that there are many times more of X than there are of Y.

By comparing different categories of unimaginably huge numbers, we can start to work with the numbers, even if we don’t understand them.

Claude Shannon was one of a kind, and his observations about chess were the tip of the iceberg in his life. If you want to read more about him, I wrote a bit here:

Have you ever thought about very big numbers like these? What’s your favorite way to do it? Do you compare two unimaginably huge things?