What Does 1,000,000 Look Like?

This post is one of a series of articles adapted from my university dissertation on data visualisation — see the rest of the series.

School children are taught the order of the powers of ten the same way they are taught to count to ten, say the alphabet or recite their times tables. There is barely a distinction between taking steps of one in sequences such as counting from 1 to 10 and counting in steps of powers of ten (1, 10, 100, 1000...); in fact a lot of children tend to miss out the tens of thousands and hundreds of thousands and seem to think that the hundreds become thousands and the thousands become millions — all because they are seen to be the major milestones.

When talking about extremely large numbers, like the age of the universe (given by Wikipedia as 13.7 billion years, plus or minus 0.11 billion years), it's incredibly difficult to keep a sense of scale, because the number is so large.

Somebody reciting a large number from vague memory could yield massive error ranges. "He won 32 or 3.2 million or something on the lottery" — a 28.8 million margin! Or: "Dinosaurs haven't been on Earth for a million years." "Actually it's over 160 million years." A million years seems an eternity to most people, so multiplying it by 160 doesn't have a significant impact — only scientifically is this number actually important in its accuracy.

Rubik's Cubes (standard 3x3 cubes) are advertised as having "billions of combinations" when in actual fact the number is significantly larger. It's calculated by:

8! × 3^7 × (12! / 2) × 2^11

This equals 43,252,003,274,489,856,000 (forty-three quintillion), which is 43 billion billion. If I said a number was x billion you'd think it had to be big anyway — but the number of billions the number actually is is over 43 billion times as big as 1 billion. That's how many billions it is. It's an astonishingly huge number, especially for a small puzzle with 26 pieces! The next size cube up from this, the 4x4, has 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (7.4 quattuordecillion) permutations, whereas the smaller 2x2 cube has 3,674,160.

A MATLAB plot of these three values is not an effective visualisation of the scale of comparison, except that the 4x4 number is vastly larger than the other two — but that much is clear from the sheer length in digits of the number!

This plot shows that the numbers of permutations on the 2x2 and 3x3 cubes are completely insignificant in comparison to the 4x4. Since the y-axis has to go up to 8×10⁴⁵, the other numbers can't be represented as they are smaller than can be seen on this scale.

However, if rather than plotting the actual values we plot what power of ten the number has to be raised to, this is a much more measurable scale for vastly accelerating numbers. This practice is called the log plot. There are a few variations: the log-log plot (log of x against log of y) for when both sets of data are rising on the same scale; and the semi-log plot (log of one variable against the actual number of the other). Here is a MATLAB semilog-y plot of the same data as the previous plot:

Getting the decimal point wrong when entering a customer's payment or transfer amount could cause huge problems. A similar erroneous occurrence took place at an Asda petrol station recently, where customers were granted cheap petrol (12.9p per litre) from a particular card payment pump. The fault was said to have been due to the decimal place being put in the wrong place.

So, what does 1 million look like?

A way I like explaining the magnitude of large numbers is to show the difference between each power of ten. To go from the 0th power of 10 to the 1st you must add 9, and so on:

On each iteration the new number is 10 times as big, obviously, but it is also 90% larger than the previous number. For example, if a child has been saving up pennies for a month and at one point has enough to say he has a whole pound, and is then given £10, this makes all the pennies he put towards the pound somewhat insignificant because the £10 is so much more in comparison.

The way I explain the magnitude of increasing powers of ten is by starting with a single object — I'll refer to it as a unit. To help visualise this, try to think of this unit as a real object, something rather small, like a pound coin or a tennis ball.

In dimensional terms this unit — considered to have no width, height or depth, just a single unifying point in space — has no dimension. No x, no y, no z.

If we want to see 10 of these, we simply make a line of 10. We've had to obtain 9 additional units to get here. This is a 1-dimensional line of points.

If we want to see 100 of these we must obtain an additional 90 (9 more lines of 10 units), which makes a 10x10 square containing 100 units. This represents two dimensions of space, as it now exists in the x and y plane.

If we want to see what 10 of these squares looks like, we must obtain 9 more squares of squares like this one and lay them on top of the original square in the third dimension (the z plane):

This 3-dimensional array of arrays of units contains 1,000 of the original units — I'll call this the cube. Note that we had to obtain a further 9 sets of squares of 100 to get this. As seen previously, to get from 100 to 1,000 we multiplied by ten, but another way of looking at it is that we added 900. Now if we treat our cube as a single point in a fourth, conceptual dimension, which has the same properties as the dimension in which we started — to get from that dimension to the first, we created a line of 10 single points. Likewise, if we line ten 3D cubes up to make a "4D line", we get a line of 10 stacks of cubes of squares of rows of units:

Next, we make 10 of these rows of cubes to get a square of 100 cubes, each one worth 1,000 units, giving us 100,000 in total:

Going along with the idea that each level of stacking units, rows, squares and cubes enters a new special dimension, we are now in the fifth. To get to the big 1 million, we must enter the sixth. Following the pattern from previous steps, we must turn this square into a cube by stacking 9 duplicate squares of cubes on top of this one. We currently have 100,000 units, and to get to 1 million we are multiplying by 10, which is equivalent to adding 900,000. When we started this process, multiplying by ten was equivalent to adding 9 — and now it's adding 900,000.

Remember, these units are representing a small object like a pound coin or a tennis ball. We can easily comprehend 10 tennis balls, imagine 100 in a basket, maybe 1,000 in a large container, but any more and it seems too much to picture, so this method of building up sets of smaller sets helps us visualise it. If we can imagine 100 tennis balls in a basket, it's easy to imagine 10 baskets, then 10 rows of 10 baskets, maybe several of these rows of baskets ready to be loaded onto a truck, and so on. Or with pound coins — anyone who has ever counted money for a shop, charity or other organisation will be able to picture a row of stacks of ten £1 coins — that's £100! And 10 rows of these stacks is £1,000. That's a lot of money, and yet we just imagined it sitting on a table in front of us.

So, what does 1 million look like?

It looks like this: