Visualising Large Numbers

11 Nov 2013mathematics

I was out walking Winnie the other morning and got to thinking about the storage capacity of neural systems (I’m reading Nils Nilsson’s The Quest for Artificial Intelligence at the moment, which was what triggered this line of thought). I’ll write about that in another article, but I realised as I was figuring through this that I would end up talking about some rather large numbers. Everyone throws around big numbers (millions, billions, trillions, and up and up), but how often do you stop to visualise what these things mean?

There are some nice animations around that help with thinking about relative scales (for instance, this one is really good). What these things don’t do is to give you a sense of numbers. What does a billion of something look like? Can you get a really strong physical feeling for numbers of this magnitude and larger?

Here, I’ll show you what I do. It’s nothing ground-breaking, and it’s kind of obvious once you get started, but it is effective. It’s also quite a nice mental exercise.

Easy numbers

We’re aiming for large numbers, so let’s start ambitious. One. How do you visualise “one of something”? There are lots of ways, but let’s start with a little square:

Being (mostly) bipedal (mostly) pentadactyl lifeforms, we’ll follow tradition and work in powers of ten. Ten of something is a small enough number to capture in a single glance:

But then we need to start getting a little creative. One hundred is pretty easy–it’s 102, so let’s just make a square with ten things along a side:

Nice and easy. We can think in three dimensions as well as two, so one thousand (103) is just as easy:

Now we’re kind of stuck–we can’t really think in four spatial dimensions, so we need to do something else.

In fact, it’s pretty easy to visualise 100 and 1000 things directly. Remember those metre sticks you used in school science classes? One metre of wood or plastic with centimetre and millimetre markings all the way along?

Metre stick

Those 100 centimetres, they’re easy to visualise as a unit. Most people have handled a metre stick enoughEven if only to pretend that it’s a light sabre and make swooshing noises around the classroom while the teacher wasn’t looking... to have a physical intuition about its size, and those centimetre marks can be pinned to something really direct–for me, 1 centimetre is about the width of the fingernail on my index fingerYou can joke all you like about using cubits as a measure leading to all sorts of problems with incommensurability between the Cubit of Rhodes and the Cubit of Alexandria, but I challenge you to find a better and more intuitive unit of measure for personal use than something directly associated with your own body..

Once we’ve got our heads round 100 things in a row, we can try 1000. Focus on the millimetre marks on your mental metre stick. Visualising all 1000 of them in one go isn’t really practical, but that’s not what we want to do–we want to hold in our minds the idea of 1000 things in a row. What do I mean by that? Well, we have a strong physical intuition of what a metre is. From there, we can get a good idea of the relative sizes of a metre and a millimetre, just on the basis of their relationship to our own physical size: a metre is, for most people, not terribly far from the distance from fingertips to shoulder with a straight arm. And for a millimetre, you can think of ten laid end to end across your fingernail.

Metre stick

At this point, we have two ways of thinking about hundreds (as 10 × 10, or as 100 in a row), and two ways of thinking about thousands (as 10 × 10 × 10 or as 1000 things in a row). We can use these representations to work our way up in size.

Big but imaginable

So, what’s next? Let’s try 10,000. Here’s a picture of 10,000 things (admittedly not very practical, and in some ways quite eye-watering, depending on your computer monitor). It’s just a 100 × 100 grid. Think of taking four metre sticks and laying them out to form the boundary of a square. The centimetre markings on each square delineate the edges of a 100 × 100 grid, just like this one.

It would take a long time to count all the squares in this grid one by one, but we can still take in the magnitude of the number of squares more or less “at a glance”.

And here’s an interesting thing. If each square in this grid represents one year, a typical human lifespan is about two-thirds of a row (70 years is filled in black in the diagram above). Human civilisation has been going for about 10,000 years (depending on who you ask and what you mean by “civilisation”). You can get some feeling for the place of an individual human, birth to death, in the whole history of civilisation this way.

Moving on up, it’s going to get harder to draw pictures, so you’re going to have to rely on your imagination more. Let’s think about 106 next, one million. We can go to three dimensions, framing a cube with metre sticks marked in centimetres, since 100 × 100 × 100 = 1,000,000. Or we can measure out a square metre with metre sticks marked with millimetres, since 1000 × 1000 = 1,000,000. Either way, we get a nice physical picture of what a million things looks like.

Look at the picture of the 100 × 100 square above, and think about 100 copies of the square stacked up to make a cube. If each cube is a single year, we have one million years stacked up in front of us. A million years isn’t such a long time geologically speaking–the dinosaurs became extinct about 65 million years ago, and the earth has been around for about 4.5 billion years, 4,500 times one million years. And yet, an individual human life rarely makes it to the end of the first of one hundred rows in the first of one hundred layers of our million year cubeNo, I’m not trying to be depressing! One of the things that thinking about some AI topics brought home to me were some of the problems of scale involved in trying to simulate real neural systems. To understand these scale issues, you need to get a feeling for how big these big numbers are. If a little existential angst helps with that, it’s all to the good..

Going ahead, I tend to prefer the “cubic metre” standard over the “square metre”. A cubic metre of water weighs a ton, 1000 kilograms, which is 1000 litres (or about 2000 pints, if you’re American). As I stand at my desk, I can look to one side and visualise a cubic metre of water sitting there confined by invisible walls, and I can use that cubic metre as a framework for imagining numbers.

So, if we’re going to go on up to a billion, 109, let’s think of a cubic metre of water, framed by metre sticks graduated in millimetres: 1000 × 1000 × 1000 = 109. If you can hold this in your head, you can see what a billion of something looks like. You can look at the whole of the cubic metre of water, then you can zoom in and look at the millimetre graduations on the metre sticks framing the cube and you can see what a single cubic millimetre of water looks like. Zoom in, zoom out, zoom in, zoom out. You can think of the relative sizes of your cubic metre and the cubic millimetres from which it’s composed: a cubic millimetre of water is a tiny amount (one microlitre)–a very small raindrop that you might just be able to feel on your skin. A cubic metre of water, on the other hand, is a quite considerable drenching.

Duplication takes us further

Mental zooming in and out is the key to getting a handle on bigger numbers. It’s hard to hold more than one level of zooming in your head at a time (at least for me!), so before we take the next step, you should do a bit of zooming around your billion cube.

All zoomed out? OK, let’s continue.

Suppose we imagine zooming in to one of the little cubes of our 1000 × 1000 × 1000 cube. Then zoom in some more, and imagine that each of those little cubes is a copy of the big cube, i.e. that each of the billion millimetre cubes is itself a 1000 × 1000 × 1000 cube (with each side being 1µm, 10-6m).

How many of the smallest cubes are there? Well, there are 109 millimetre cubes in our original metre cube, and we’re now saying that each of those millimetre cubes is itself composed of 109 sub-cubes. That means that there are 109 × 109 = 1018 of the tiny cubes.

Now, 1018 really is quite a large number, so let’s stop to think about it carefully. Although we’re really dealing with three levels of sizes here (our one metre cube, the billion millimetre cubes and the billion billion micrometre cubes), the self-similarity between the embedding of the millimetre cubes in the metre cube and the embedding of the micrometre cubes in each millimetre cube allows us to use the same mental template to represent both scales of zooming. I can form a clear image in my mind of what the one billion cube looks like (one cubic metre broken up as cubic millimetres), and I can then hold that image in my mind alongside a zoomed in version of one of those millimetre cubes.

It takes a little practice, but if you sit staring into the middle distance visualising your cubes of cubes of cubes long enough, you can start to get an inkling of what 1018 of something looks like.

Buoyed by our success, we might think about replacing each of our 1018 micrometre cubes with copies of our billion cube. That would give us 1018 × 109 = 1027 really small cubes (each 10-9m = 1 nanometre) on a side. Now that’s a BIG number! However, I have to confess that this doesn’t work for me. The “duplication scaling” trick works for only one level for me–I can’t keep a mental grasp on the original one metre cube, the billion millimetre cubes, one of the 1018 micrometre cubes and its constituent nanometre cubes all at once.

Avogadro’s number

Let’s try something else. Let’s try to visualise Avogadro’s number. This is about 6 × 1023, but let’s approximate it to an order of magnitude as 1024. Why think about Avogadro’s number? It’s really determined by an arbitrary choice of units: it’s the number of particles in one mole of a given substance, so it’s not really anything more than a way of defining what a mole is. So why bother thinking about it? Well, it’s a number that makes a very clear connection between the macroscopic world of “everyday things” and the microscopic world of atoms and molecules. For instance, one mole of water is 18 grammes, which would more or less fill a medium sized test tube. That means that a test tube of water contains about 6 × 1023 (or for our purposes, 1024) molecules of water. So if we could build a mental picture of how big Avogadro’s number is, we might be able to develop some intuition on how big a water molecule is.

Now, 1024 = 1018 × 106, so we’re somehow going to need to visualise one million of our nested 1018 cubes. As I said before, I have trouble doing more than two levels of zooming in–the nested billions to give the 1018 cube is about all my poor little brain can handle. However, I find that something interesting happens if I try zooming out instead of in. I can keep a level of “out” along with my two levels of “in”.

Let’s think about making a million copies of our cubic metre of water and stacking them into a single big cube. That means a cube 100 metres along each side. To get some perspective on what that means, let’s take a typical outdoor area of around 100 metres in dimension. An athletics stadium works really well–the 100 metre straight that sprinters run along is pretty clear, and we can get some idea of what our big cube of water looks like in a setting with people and some other objects for scale:


To get 1024 things, we need to imagine that each cubic metre of water in this big 100 metre cube is one of our 1018 cubes. To see one of the 1024 individual “things” (which are tiny tiny cubes of water one micrometre on a side), we zoom into one of the one metre cubes that make up our big 100 metre cube, then zoom in by a billion (thinking about the millimetre gradations along the edge of the metre cube to guide us), then zoom in by a billion again (we can mentally “magnify” the millimetre cube we pick out by our first “billion” zoom and think about it being graduated by thousands along each edge as well).

So why does this way of thinking about things work, but trying to zoom in three levels from a metre cube leaves me sad and confused? I was thinking about this earlier today, and I think that the reason for this is that the “middle” level, the one metre cube, is a human order of magnitude. In some way, it’s our “reference” scale, the one compared to which we measure everything. People often talk about “the width of a human hair” or some such visual stand-in for a small length scale, but that doesn’t really grab your visual and proprioceptive sense in the way that “as long as your arm” does. This engagement of our “internal” proprioceptive sense is what makes us “feel” the size of things. The span of your hand, the width of a finger, the length of your arm, these are all measures for which you have both a visual model and an “internal” tactile/proprioceptive model. That means that it’s easy to mentally zoom in or out from these scales.

Anyway, let’s spend a bit of time zooming in and out of our stadium-sized cube of water. Once you get the hang of that, think about shrinking that cue of water down so that you could pour it into a test tube. At that scale, the micrometre cubes we’ve been thinking of as our bottom level are the size of water molecules!


Avogadro’s number seems like a good place to stop. Visualising Avogadro’s number is actually quite useful, because of its role in linking everyday scales to atomic scales. Bigger numbers do turn upA quick back of the envelope calculation revealed that I might need to talk about numbers as big as 1042 if I want to talk about storage capacities. This number comes from an absolute limit for information storage capacity based on the Bekenstein bound, which is way larger than the information storage capacity of any realistic neural system–it’s just an interesting theoretical upper limit for these things. but it’s not clear how useful it is to try to visualise these numbers–billions, trillions and Avogadro’s number have a lot of practical applications for understanding, but these bigger numbers really are getting on for “unimaginably large”. Also “uselessly large” in some cases, except for some pure mathematical issues.

Here’s an example to show how unwieldy things get. A googol is 10100. It doesn’t really have any practical application. It’s just a big number. If we start with our 1018 cube, and progressively zoom in, replacing every one of the 1018 small cubes with a copy of the whole 1018 cube, we can make the following numbers:

1018 → 1036 → 1054 → 1072 → 1090

Four levels of zooming in and we’re still not there! It seems as though some other means than our duplication approach would be needed to build intuition about numbers of this magnitude.

Despite that, for “everyday” large numbers–millions, billions, trillions, even up to 1024–the methods we’ve looked at really can help to give a direct feeling for these numbers. That intuition can then be applied to thinking about relative sizes (molecules versus test tubes, for instance) in a way that’s not accessible via by step-wise scaling.