Tall Numbers

The Universe is too small to store some numbers that lie in plain sight.

One grain of rice isn’t much. It’s so tiny that it’s almost difficult to pick up with your fingers, and its calorie content — 0.07 calories — is so small as to be insignificant to the general human diet. If you were to put it in your mouth, it might get lost; you probably wouldn’t register its passage down your throat. We leave them hanging around on the floor or the bottom of a bowl without a second thought.

But if you take enough of these small grains, it could add up to a lot. Such was the case in the legend of the origin of chess.

The Emperor of the kingdom, it is said, was so impressed that he commanded the inventor to ask for any boon — gold, land, riches — and it would be granted. After refusing, the inventor finally caved in and asked for this:

Take the chessboard from the game I have invented, and, on the first square, put one grain of rice. On the second square, put twice the amount — that is, two grain. The third square must have four grains of rice, and the fourth square eight. In this manner, keep doubling the amount until you reach the end of the chessboard.

The Emperor was astounded. “Is that all?” he asked. “Just rice?”

But when he tried to carry out the chess-maker’s wishes, he realised that the numbers were fast adding up. The fifth square needed a manageable 16 rice grains, but by the twentieth square it had reached 524,288 grains, a substantial lunch for a dozen people.

If you follow along with a calculator, you’ll either find your self up against the small ‘E’ in the corner or realise that the number of rice grains the Emperor was trying to procure came to a whopping

18,446,744,073,709,551,616

which, in terms of rice grains, is about 140 times the volume of Mount Everest. More rice than could be obtained in the entire kingdom!

Numbers, impossibly large on the human scale, can be fascinating. I accidentally stumbled into such numbers in a different way, while thinking about probabilities. It was unexpected and mixed with a little tingling of awe.

I decided to share my venture, maybe it will amuse others, as well. Although there is a little math involved, it is easy to understand.

My so-called stumbling into large numbers came about while reading Carl Sagan’s science fiction novel Contact. Carl Sagan was an astrophysicist; a scientist through and through. He was also active in the sceptical movement confronting all matters of pseudoscience — which is why I was quite surprised that he sprinkled Contact with a small dose of mysticism. What follows is the account of my encounter with the spiritual ending of the story.

The last chapter in the novel, The Artist’s Signature, concerns a familiar number that is universally constant: π (pi), or the ratio of a circle’s circumference to its diameter. If a circle has a certain diameter, its circumference will always be π times that diameter, no matter the size of the circle.

π is a so-called transcendental number, because if you try to write its decimal expansion, it starts as ‘3.141592…’ but the digits go on endlessly. It’s not like a ¼ that stops at ‘0.25’, or even a ⅓ which goes on endlessly but with a neat repeating pattern of 3’s. Statistically, the digits of π behave like a sequence of random numbers: even if you know the previous digits, there’s no way of foretelling what digit will come next.

In the Artist’s Signature chapter, Ellie instructs her computer to look for “anomalies” in the digits of this supposedly random sequence. And! Lo and behold, Ellie’s computer does find something interesting.

The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones […] The program reassembled the digits into a square raster, an equal number across and down. The first line was an uninterrupted file of zeros, left to right. The second line showed a single numeral one, exactly in the middle, with zeros to the borders, left and right. After a few more lines, an unmistakable arc had formed, composed of ones. The simple geometrical figure had been quickly constructed, line by line, self-reflexive, rich with promise. […]

And there you have it. Deep inside the digits of this transcendental number, in a grid arranged in the right way, Ellie saw the ‘0’s and ‘1’s patterning to form the image of a perfect circle!

Ellie has no choice but to conclude: “The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle — another circle, drawn kilometers downstream of the decimal point.”

There is, she concludes, a pre-existing intelligence in the Universe.

Anybody who watched Carl Sagan’s documentary series Cosmos, first broadcasted in 1980, could feel the marvel and awe that he felt for the Universe. In Contact he presented his readers — I’d guess with a wink and a nod — with this hidden intelligence of his beloved Universe.

I began to ponder at this point: what if once we indeed find such a circle in the digits of π? Would we really have received a message from the Universe, or could we have, among the many digits of π, happened to bump into the right sequence simply by chance?

If you have a large playlist on iTunes — or Spotify, or any other music streaming service — chances are you’ve used the “shuffle” function. Perhaps you haven’t thought much about it: after all, it just has to pick random numbers, right?

Actually, it’s not so simple.

The thing about random sequences is, even the unlikeliest pattern will emerge if you wait long enough. Think about repeatedly tossing a coin: you would expect to get more or less the same number of heads and tails — but you can get a sequence of five consecutive heads if you wait long enough.

On the other hand, if you ask someone to imagine tossing coins and come up with a random sequence, then getting those five heads will be harder, because that someone will mentally think “oh, I’ve done three heads already; it’s probably time for a tails”. You can even try this experiment with your friends: ask one of them to toss a real coin while the other one tosses an imaginary coin. When you see both their sequences, the one with seemingly less-random strings is probably the one involving a real coin.

Similarly, in a long enough playlist, you will eventually get songs from the same album playing three times in a row. Because humans are quick to notice patterns, we normally think it’s too much of a coincidence; that the music player must be playing tricks on us.

Or, in the words of Apple CEO Steve Jobs himself, “We’re making it less random to make it seem more random”.

If random playlists can seem so non-random, can the same be said of the digits of π? Would we really have received a message from the Universe, or would we have happened to bump into the right sequence simply by chance?

The book says the circle was “kilometres downstream of the decimal point”. Well, kilometres are plenty of space to write down a lot of digits! Is there a reasonable chance of finding the string of our circle, within those kilometres of random digits, without the intervention of the Universe?

If the chance of finding a sequence is one in a billion — why, then if you try ten billion times you’ll probably see the sequence pop up quite a few times. And if you keep trying, forever? Of course that very rare event will keep on cropping up forever as well.

π has infinitely many digits — in other words, you can keep trying forever — so any string you search for will show up, eventually, at some point. The only question is: when?

Okay, leaving ‘forever’s aside for a moment, let’s figure out the likelihood of running into a circle, depicted by the symbols ‘0’ and ‘1’, within a random sequence of digits.

First, what sort of a circle are we dealing with? Small number-grids will make the circle very pixelated, so we need to have one large enough. I tried a few and concluded that a frame of 31 digits per side was adequate to yield at least a decent looking circle, even if not a nice smooth one.

This circle is of course formed by a string of digits: in this case, a 31×31=961-digit-long string that looked something like this:

000000000000000000000000000000000000000000000010000000000000000000000000010000000100000000000000000000…

…which we then arranged into a square to get the picture.

That is the string we have to look for in the digits of π. What are the chances of finding one?

Let’s put the question more precisely. We know this string exists somewhere in the digits of π, because π has an infinite number of digits. It will surely pop up if we wait long enough.

But though the digits of π may be infinite, our patience certainly isn’t. That begs the question: how many digits would we need to calculate to have, say, at least a 50% chance of finding our pattern within those calculated digits?

The novel has one last instruction for us: “The anomaly showed up most starkly in Base 11 arithmetic….”

In our everyday Base 10 mathematics we have ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. But if we were eleven-fingered aliens, we might well have used a system that works just the same but has eleven symbols instead. What matters for our purpose is that, in Base 11, any of π’s digits have an equal chance to turn out to be any one of the eleven possible symbols — the chance is 1/11 rather than 1/10.

Why Base 11 arithmetic? Here’s my theory: Contact was published in 1985. Superstring theories came to the fore of theoretical physics just in that period, and, according to them, the universe is ten- or eleven-dimensional. Wouldn’t it make sense that an eleven-dimensional Universe communicates in Base 11 arithmetic?

With that out of the way, let us begin.

First question: what is the chance that — in Base 11 — our 961 digits long string would appear after, say, the 100th digit?

Well, the 101st digit would have to be a ‘0’. The probability of that is 1/11. The 102nd again would have to be 0, the probability of that is also 1/11.

That means the probability that the 101st and the 102nd digits are both 0s is (1/11)².

The probability that the 101st, 102nd, and 103rd digit are ‘0’ is (1/11)³, because the same thing has to happen three times in a row.

And so on, until we get the chance of finding the correct 961-digit-sequence at position 100: (1/11)⁹⁶¹, because each of the digits have to be a specific number — either ‘0’ or ‘1’ — which has a 1/11 chance of happening.

Now, there was no reason we had to start this calculation at the 100th digit: there is nothing special about it. Finding our string after any particular digit, be that 100, 1000, or 123456, has always the same probability: (1/11)⁹⁶¹. This fact allows us to think of finding our string as an event. The probability of this event occurring is (1/11)⁹⁶¹, which is approximately (1/10)¹⁰⁰¹.

That’s 0.0…followed by another nine-hundred and ninety-nine zeros and then a 1!

In human terms, this is beyond small. Compare it to the probability of winning the jackpot in one of the national lotteries, typically in the range of a few times (1/10)⁹. A jackpot is also an event and upon many trials the event happens and the jackpot is won.

In the lottery’s case, each (non-duplicate) ticket is a trial. All it takes is a few hundred million tickets and the jackpot is conquered.

Same with our string: here, each successive digit of π is a new trial. Each digit might turn out to be the “magic” one, the first in our sought-after string. Except that finding the string is an event with an incomparably smaller chance than winning the lottery. On the other hand, although our chances may be tiny, we do have many trials at our disposal. We only have to keep churning out the digits of π, until bingo! it crops up at some point.

Ellie found the string after a few miles worth of digits. How far can we expect to go on calculating π’s digits in Base 11 arithmetic to have a 50% chance of finding our sting somewhere in there?

The number of trials needed for a random event to happen at least once with a 50% chance is about 0.7 divided by the probability of the event. In our case, how we got to 0.7 is irrelevant: the emphasis is on “divided by the probability”. Since our probability is ≈ (1/10)¹⁰⁰¹ we have to calculate out to 10¹⁰⁰¹ digits—that’s 1 followed by a thousand and one zeroes!

10¹⁰⁰¹ is not only more than a few kilometres worth of digits; it is  beyond any human scale — and more: it’s way beyond anything relating to our Universe. To let this sink in, let’s try to see what it would mean to even note down such a vast number of digits.

The Kingston DataTraveler Max 1TB pendrive is about 8×2 centimetres in size, with a thickness of about a centimetre — making for a volume of 16 cubic centimetres. But because we’re dealing with exponents here, a meagre 10³⁹ digits would take 1.6×10²⁴ litres, more than the volume of planet Earth.

To truly get a sense of the scale, let’s assume we have an ingenious technology, way better than our pendrives, that can store 10¹⁸ digits in a cubic millimetre. That’s way beyond anything we’d ever be able to do in reality.

Now let’s see how many of those we can fit into the observable Universe.

A characteristic measure of the observable Universe is about 25 billion light-years, which gives us a volume of ~ 10⁸⁸ mm³. So if we stuff every cubic millimetre of our observable Universe with super-pendrive, we get to store ~ 10¹⁰⁶ of them…only getting us to 10¹²⁴. If we are truly ambitious, we could corral the part of our Universe that we can’t observe. (Due to the expansion of the Universe, light cannot reach us from there.) This unobservable part may be good for at most another factor of about 100, thus giving us ~ 10¹⁰⁸ storage places.

No matter which way you look at it, we are short of space to store our number. If we were to store all of it, we’d need, oh about 10⁸⁹³ extra Universes.

Such estimates are the ones I referred to in the beginning as capable of filling one with awe. Numbers like these cast a shadow. We really did not ask for much: we only wanted to find a specific string among random numbers — and look what it got us into.

What chance did Ellie’s computer have to find the string if there was no intervention by the Universe itself? You guessed it: absolutely none. Even if she had an impossibly powerful computer able to calculate 10⁴⁰ digits per second, and had it running since the Big Bang, she’d still need 10⁹⁴³ more computers to do the job.

But let’s get off our high horse. What if Ellie’s circle was barely distinguishable from a square and had no frame of ‘0’s? If she had imagination — and in the story she certainly had — she could have gleaned a circle in a 7×7 square of properly placed 0s and 1s. Then, all it would take is to calculate about 11⁴⁹ digits which is about 10⁵¹. Even that, though, would take up enough pendrives to fill up…the entire Solar System.

I think we can conclude that yes, if Ellie’s computer came up with any kind of recognizable circle of ‘0’s and ‘1’s in Base 11 arithmetic it definitely was a message from the Universe. Professor Sagan did allow a little play with wonder. Kudos for giving such an imaginative twist to his book!

Not everything is within the realm of supercomputers though. If we use Base 2 instead of Base 11, we could get our ugly circle in less than 2⁴⁹ digits. That’s less than 10¹⁵, a number so small that it even has a name: one quadrillion.

Things get better with even smaller images. For instance, with a little goodwill, one can discern a human figure in the 3×6 rectangle shown here.

This gets us within the range of “things you can do on your home laptop”.

One can easily spend an afternoon in front of a computer playing with such things and observe that finding various figures in random sequences is not a miracle — but can also  turn into one pretty soon, if we are too ambitious with the length of our strings.

In the meantime, π smiles. I am not fazed by your 961-digit-long predefined string any more than by a single digit, it seems to say. I have room to spare for them all.

I have an infinite number of digits.