12 July 2011

Did you know that there was an 19-digit number that has never been spoken by any human ever?

Well, it’s true. I want to show you why it is true. Before I do that though, I have to explain why I am writing this. During my time studying maths at university, I was never particularly impressed by most “existence theorems.” On the whole, I don’t find it particularly exciting or helpful to set about proving the existence of something without any prospect of actually calculating it (Examples of these include the Ham Sandwich Theorem and the Hairy Ball Theorem). The exception to this is the pigeonhole principle, which I first came across on an open day at Leicester University. The day sticks in my mind because it was pouring down with rain and one of the first guys we spoke to said that he didn’t expect many of the applicants present to go on there, on the basis of the poor weather. His idea was the first impressions were lasting and that bad weather left an overall bad impression, in spite of whatever efforts made by the university staff on the day, the applications would likely think of Leicester as a rainy place, compared to another university they may have visited on a sunny day.

I did not end up going to Leicester University.

But in their open day they did introduce me to the pigeonhole principle, by which one may prove all sorts of odd things. One of these, for example, is that there are at least two people in Newcastle with the exact same number of hairs on their head.

You can look up more details of the pigeonhole principle here, as I would rather assume it is known and then use it rather than recapitulate the whole thing.

A while ago I came across a number called Graham’s number, which was a peculiar for the fact that it was immensely large, no one has calculated it, but we do not that it ends in a 7 (when written in base 10, at least), which is the kind of quirky thing that really piques my interest. [I ought here to note that the episode of QI on which I first saw this was repeated on Monday night, after I wrote most of this, but before I put it online]

So I got thinking what is the potentially the smallest number that no person has ever written down, spoken aloud or actually even thought about. I wanted to ensure that I would be right so where I have had to make estimations, I have erred on the side of caution, leading me to suspect that though I am convinced I am right, I have over-shot the mark in at least one respect.

The first trouble was to estimate how many people have ever lived. Here, we are instantly presented with a problem of trying to define the demarcation of the first homo sapiens as opposed to an earlier ancestor and to then consider at what point in human evolution numeracy developed. As I had no idea I resorted to Wikipedia, who gave a statistic cited from an American study that estimated there had been between 100,000,000,000 and 115,000,000,000 people who have ever lived. So naturally, I added on a bit (just to be on the safe side) and assumed for the purpose of my calculation 120,000,000,000.

Next, I had to estimate how long they live for. Again, without any detailed research to hand, I made a guess by using the current average age of around 80 years. I suspect that over the course of human history, it has not been less than this, so my estimate is suitably conservative (if that phrase is not an oxymoron).

Of this, there are likely to be times (such as childhood and old age) when the ability to count to large numbers will not be present. So I took off 10 years, which I think is not unreasonable.

Next, how much of that time is spent asleep. I have heard that people spend a third of their lives asleep, and that the average person gets 8 hours sleep a night. Personally, I don’t know where these people get the time from. I get 6 hours a night, so I estimated that each person was only awake for 52.5 years.

Of course, most people do not spend every waking moment thinking about numbers. As a mathematician by training and an accountant by profession, I probably do it more than most, although even then I would estimate that I don’t spend more than 5% of my waking time thinking about numbers. There are far more everyday concerns that take up much of my thinking time. Again, erring on the side of caution, I plumped for 10%.

This means that on my grossly optimistic assumptions, the average human can spend 165,672,864 seconds in their lifetime thinking about numbers. Given our earlier estimate of the number of people, this gives the total thinking time to date as somewhere in the region of 19,880,743,680,000,000,000 seconds.

Now, even though it can be very quick to count to 10, the numbers we are interested are not likely to be small. So how long does it take to say them? Of course, this will depend on language, so I admit my figure is a plucked out of thin air. I would opt for 2 seconds. I think when you get the scale of the hundreds of thousands, that’s not unreasonable. Order of magnitude higher than that will probably take considerably longer, so 2 is a fair estimate to use for a conservative guess.

So what’s the answer then? I believe that there is a number which is less than 9,940,371,840,000,000,000 which no person in human history has ever spoken, written or thought about.

I am sure that this is far too high an estimate, as we have considered numbers like a googol and googolplex which are many orders of magnitude larger and I haven’t taken into account repetition. Goodness knows how many times the number 100 has been considered by humans over the years!

I know for certain that the number in question cannot be 4,724,557,109,087,242 because I just thought about it. In fact, any number I think about is, by definition, the wrong answer, because as soon as I think of it, it can no longer remain “un-thought-of.” I’d love to think that I “discovered” a number by being the first one to think about it. Of course, by continuity, we know that it must have existed, but I have no way of verifying if I was the first one to think of it.

It strikes me a little bit of quantum mechanics where a system will collapse into its eigenstates as soon as it is observed. Truly fascinating and enjoyable.

That’s why I love science!

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