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!

Proof of why a transposition error is always divisible by 9

Here's a little proof I knocked up over the weekend. Sorry it's in a picture format, the platform on which this blog is based can't deal with simple mathematical symbols, so I had to convert a document into a couple of pictures. If you click on the pictures, you should be able to see a larger version of each of the two pages of the proof.

Book Review: Cycles of Time by Roger Penrose

This is a very interesting read on Penrose's new hypothesis: conformal cyclic cosmology. Before he gets to this in the third part of the book, he first needs to give the reader the background to his thinking. To that end, the first part of the book looks at the Second Law of Thermodymanics, which plays a pivotal role in this work. So if you don't yet have any idea what this is, I would recommend a little preliminary reading before tackling Cycles of Time.

If you are not familiar with Penrose's writings, then this perhaps is not the best starting point. He jumps straight into the Second Law and doesn't shy away from the necessary maths. For a science graduate, this is relatively easy reading, though those without a formal background in maths or physics may struggle, although Penrose's styles of diagrams are immensely helpful. One thing that is helpful is that even if you haven't grasped all the detail in a given section (and I certainly didn't) then that doesn't mean you cannot grasp any of the later concepts.

No one could ever accuse Penrose of patronising his audience, and though many topics will be familiar to scientists, Penrose's particular style always stretches you and makes you think in a slightly different way; so that which you thought you knew quite well suddenly has a few extra question marks posed against it. One thing that is very praiseworthy in this book is Penrose's modesty and his clearly laying out of what is well evidenced scientific consensus and what is his own minority view, as well as pointing out the drawbacks in his own theory. This style contrasts greatly with the brash optimism that Hawking & Mlodinow put forward in their book, The Grand Design, published within a few weeks of Cycles of Time. The fact that Penrose does this raises some interesting questions. For example, he does state that in order for his hypothesis to be correct, we would have abandon many well-established theories, such as the invariability of rest-masses of fundamental particles.

I could not claim to have fully understand all the nuances and detail of this book at the first, but that does not diminish my enjoyment of it or my ability to get the overall gist of it. I will be re-reading this book, going over each line in more detail in order to get the complete picture.

Book Review: The Num8er My5teries by Marcus du Sautoy

It's a bit hard to review this book without having in mind Alex's Adventures in Numberland, published in the same year. Both books cover similar ground, although the approaches differ greatly. Whereas Alex Bellos travelled and spoke to various people who had a particular passion for certain aspects of mathematics or numbers, du Sautoy's book has the distinct feeling to it that he just sat down and wrote most of it straight out of what was in his own head. The ending of the book somewhat confirms this, as he states the book came out of his giving the Royal Institution Christmas lectures in 2006, and a few other projects he had previously worked on.

The book is broken down very simply into just 5 chapters, each with a basic premise to be looked at. But here, du Sautoy's passion for mathematics breaks through and he veers wildly off course and looks down a few sidestreets along the way. So if you pick a point about three-quarters of the way through each chapter, whatever is being discussed may not seem to have an immediate connection to what the chapter started out talking about. But this is not a criticism; merely a point of observation. It may not be to some people's liking, though I think it adds to the charm of the book.

Consistent with the philosophy of most mathematicians, du Sautoy believes that the joy in maths is to be found in doing it for oneself, not merely in the exposition of another. To this end, there are consistent puzzles inserted throughout the book for the reader to follow up on. So the fact that it doesn't take long to read cover to cover (I did it in 4 days) belies the depth of material that the pages didn't have room for and are followed up online. The book does get gradually more and more technical, which may put off some readers. Towards the end, I had to pull out a pen and some paper to follow a few of the steps.

Overall, it's written in a really down-to-earth manner with du Sautoy's enthusiasm evident on almost every single page, especially those page numbers which are prime numbers which he conveniently instructed the printers to make bold! I would recommend this for anyone interested in mathematics, though I disagree with the age ranged suggested (1-101, even if he did mean it in binary!). I think it should fairly accessible to an average 10 year old or a smart 8 year old, but with plenty to interest adult readers as well.