Multimedia Design Limited presents
Do You Like Maths?
Infinity can be infinitesimal...
Pi goes on and on forever and ever, getting smaller and smaller and smaller...
The Roman number system was better for counting than is the modern number system...
Prepare to have your world rocked!
Martin Cross is a computer programmer. He lives in Ipswich. He is single. He is aged forty-six.
Do you like Numbers?
We are going to see infinity. That is, we are going to see something that infinity is, and something that it is not.
Infinity is not a number.
Infinity is not large.
Martin Cross
Self-published by the author through Multimedia Design Limited.
Multimedia Design Limited, 50 Norfolk Road, Ipswich. IP4 2HB.
Typeset in Zurich Light, Serpentine, Perpetua, Humanist 521 and Verdana. Printed with Microsoft Word 2000 using a Samsung ML-4500. Cover designed with Excel 2000, CorelDRAW 8.0 and Bryce 4.0. Cover design by the author. Cover printed using a Canon iP4200.
Thanks to:
Richard for editing; Jeremy and Kelvin, Uncle Frank & the Lees, Emerald & Paul for everything; Cathie, Duncan, Lil & Noush and Nan P and the family; Clive, Mez, Jelly and the Gang; and Suzanne, Fiona, Matt W and Matt P.
This publication may be liberally lent, re-sold, hired out or otherwise circulated fully according to the wish of the purchaser.
" A treatise on algebra or geometry, which cost but a very few shillings, afforded me close study for a year." - George Smith, Carpenter (b. 1800).
If you currently have GCSEs coming up, you should not just rely on native ability. The exam is not there to tell you how good you are at a subject you may like. If you want to find out how clever you are, find out how to take an IQ test. The exam is there to tell others how good you can be at a subject, whether or not you like it.
To get the best grade in an exam which will mark you for life, you need to be aware of 'exam technique', the practicing of answers to questions that are on the curriculum, such as those from previous papers.
Combining this with natural ability will allow you to exceed yourself on the day and benefit indefinitely from then on, throughout your future life.
It is 25 years since I took my 'O' Levels and you may feel you've nothing to learn from me, through an essay. Bear in mind however that my curriculum is 'learning for the sake of learning', not for an exam result. It is learning for philosophy - the love of wisdom - and although that is a different curriculum than yours at school, you may well benefit from such a fresh approach to the same subject, however ambitious and naturally capable you are already.
If you have GCSEs upcoming then the central idea that is presented in this essay, that infinity is not large, will probably not faze you, and may just remind you of something you already know. If your 'O' Levels were as long ago as mine, then it may be something you will find much less easy to swallow. In the same way as a clever student should swallow their pride and embrace exam technique, the would-be authority, or teacher, needs to present knowledge and learning in such a way that it avoids the trap of argument. I hope to do that here by writing directly to the 16 year old exam student.
To him or her I can say that, at first, we think of infinity simply as going on and on forever getting larger and larger. But this is wrong and moreover, there is a better way to think of it. Not as going on endlessly but stubbornly unchanging; but as going - yes - on and on: but endlessly reinventing – or rediscovering – itself, in the process.
Here's an example. We think of infinity as larger than the largest thing we can think of because this captures the sense of it, we think. We do not need to imagine a heat hotter than the hottest star, or a blue more blue than the most intense blue, because these things are the same idea in kind. But are they? For example, infinity is the bluest blue, but the bluest blue is not infinity. Infinity is white. This is because white is all the colours; it is the colour that is not a colour.
So infinity, unlike the largest large, can be both 'more' and 'different', in one. And this is what makes it so exciting, and so new, and such a discovery.
Read on to find out more.
"Do you like maths?" I should think the usual answer to this particular question would be: nobody likes maths - except those people who are very good at it!
But I like maths and I'm not very good at it - I got the bottom grade at school, when I just scraped a pass at 'A' Level. So perhaps maths was a bit too difficult for me for a subject. But numbers, that was what I liked, and that's what the ancient Greeks and Romans liked so much. They were the first to explore the system of 'natural numbers' which we use for counting.
Whoah! Hang on a second. There I go with what I want to say before we have even got started! I am letting my enthusiasm get the better of me. What do you need to know before we can start?
Who am I? Well, I work as a computer programmer. Do you know what you want to do for your work? I am hoping that if you like numbers it will be something where what you like is a help to you, like programming.
In my work as a computer programmer I need a good grasp of logic. My best subject in school was English, and logic originally comes from 'logos', the Greek for 'word', so maybe that explains it. I am still writing. I have explained the difference between maths and logic to myself as: maths is the *similarity* between numbers and logic is the *difference* between numbers. Thus, maths is the gentle art of calculation, whilst the brutal art of computation is what I do.
I have worked for very clever people. People with Firsts (from Cambridge!) and others with four 'A' Levels, and the ones who were better than I at software were so because of the maths. But it has not held me back in my career as long as I could do the logic.
Now, what is it that we have in common? Well, let’s see if it can be demonstrated.
The Guardian Newspaper runs a series of intellectual conundra - often mathematical - called Pyrgic Puzzles, which you can see on the Web, through:
http://www.google.co.uk/search?q=guardian+pyrgic+puzzle.
One February in 2002 I read this question in the newspaper:
How many times can you take 3 from 39?
(Guardian Pyrgic Puzzles 9/2/02)
Of course you can see as I did that it is a trick question. It would be too easy to divide 3 into 39 to be an interesting question for those of us who do like numbers. You won't be too surprised then at the answer they gave: "Usually once, then you are taking 3 from 36. Actually as many times as you like, the answer is always 36." The trick being of course that they have looked at it as the logical question, not the mathematical one.
But, being as cantankerous in middle age as I was when I was young doing my exams, I didn't like this answer at all. I thought it broke the rules, but not in a good, way. It seems to me that the better question is: “Can one take 3 from 36 - and reach zero (i.e. as in the spirit of the maths question) in another way – and if so, in how many ways?”
In maths, we tend to go straight to the answer by the fastest route; it is fun for the pure novelty of it to take a slower route, isn’t it? I hope you agree. My answer is in the Appendix.
I hope you agree because we'll be using this kind of logic to give ourselves a basis for our interest in numbers. I'll be doing that with infinity which, as the answer in the appendix showed me, is a wonderfully rich example, and I'll be doing it with number bases, from the decimal system that we use to the binary system of computers. This will be enough from which to make a review that is shorter than a whole book, but longer than a newspaper column. That will share my enthusiasm, and understanding.
Maths is ruthless if you are too quick with your first answer or too keen on a clever answer, isn't it? You always want to look at a question to see how interesting it is, but maths exams are deliberately designed not to cater to this. You are to answer the question that has been asked, not the one you thought was asked. So that is one reason why it is so worth answering earlier papers questions, so that you don't fall into the first obvious hole.
Maths O Level questions used to trip me up. Maybe they still would. But if you stay interested in numbers, like I did, and like Marilyn Vos Savant clearly did, in the Appendix, then you may achieve something in addition to ‘jumping through the hoop’ of the exam.
Counting Ones
So I got the bottom grade at school, when I just scraped a pass at 'A' Level. And so perhaps maths was a bit too difficult for me for a subject. But numbers, that was what I liked, and that’s what the ancient Greeks and Romans liked so much. They were the first to explore the system of 'natural numbers' which we use for counting. You may know as well that there are other types of counting: real numbers like 1.2, imaginary numbers based on i, transfinite numbers, negative numbers like -1 and positive, rational and irrational numbers like e; even surreal numbers, and more. But all these numbers are based on the natural numbers 1,2,3 etc. and that is all I need to understand bases, and to be a computer programmer.
What is a base? Well, it is easiest to explain with a computer. You see, computers have a base of two. This means that they cannot count past 1. Doesn't that sound odd? It is true though, computers count like this: 0, 1, 10, 11, 100, 101, 110, 111 - do you know what the next number in the sequence would be? You can work it out, logically.
If we write out this new binary system, based on two, against our decimal system, based on ten, it may help for comparison:
Figure 1: Binary numbers against their decimal equivalents
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
0 |
1 |
10 |
11 |
100 |
101 |
110 |
111 |
1000 |
1001 |
1010 |
1011 |
1100 |
1101 |
1110 |
1111 |
10000 |
Binary is hard to read and long-winded, but it has one great virtue...
So you can see that binary 10 is decimal 2; binary 100 is our 4; 1000 is 8, and so on. When you also know that 4 is 2 squared, 8 is 2 cubed, and so on through 16, then you can sere that 1011 is 8 + 4 + 2 instead of one thousand and eleven, so it is easy enough to convert any number. Try it yourself: what is 1001?
Phew! Imagine what decimal 1,000,000 would look like in binary! (If you have a computer, you have the calculator in Windows and you can use its scientific mode to see for yourself).
But here is a question that might interest you - which is easier to count in: decimal or binary?
At first glance, you would think that this is a stupid question. Binary has nothing to recommend it, you would think - just compare 101101, with 45 in decimal! Before long you are writing so many digits they won't fit on the line! What would be the use of that?
But binary has at least one great virtue: it is simple. In binary 2 x n = 2^{n}. Its base is also its power. Although it is painful to count up in binary, as I made fun of, counting down in binary is a different matter altogether. To see what I mean, consider counting 100 dominos.
If you had a pile of 100 dominos how would you count them quickly and accurately? Nowadays, to count something important like money, they have machines to do this boring job over and over again, so this question is just for fun. But try to imagine a pile of 1000 dominos.
To count them accurately, you could make 10 piles of 10, and then 10 piles of 100. That would probably take about two or three minutes, and be perfectly fine. But what if there was a better way? Because dominos, like banknotes, are a uniform size and shape, you can physically group them. So another way to count the pile of one thousand would be to take one domino, and double it; and double it again to make four, and then again, and again, and again. Very quickly you are at 512 which is more than half of the 1000 dominos. You have counted this number as easily as you would count to 9, since 512 = 2 doubled nine times, which is 2 to the power of 9.
To count the pile that is left, divide your pile of 512 into two equal piles. Is the pile that is left greater or lesser? Well 512 from 1000 leaves 488, but half of 512 is 256. 488 is greater than 216 so you can instantly take a further 216 from the uncounted pile and add them to the counted pile. In one further step you have now counted 728 dominos out of the 1000.
Finishing off is not quite so impressive as it gets slower not quicker, but the principle is just as easy. Halve your 256 and see if there are that many in the uncounted pile, if there are, they are counted, if not then do nothing (ie add zero). 256 from 488 is 232, which is 11100111, which means you will do this 8 more times (since there are 8 digits in the binary number). Six times you will add something to the counted pile and twice you will add nothing. Adding these 6 + 2 steps onto the nine steps from earlier makes seventeen steps, so you have counted out 1000 dominos as easily as you can count from 1 to 17!
It would be interesting to do this for real, wouldn't it? I wonder where you would get 1000 dominos from? It would be even better for us if it was 10000 - and even better still would be 1024 dominos! That is exactly 2 to the power of ten, so you could count that in just ten steps.
And if that is so (and if you were brave enough!), you could bet say, another computer programmer a sum of money that counting in binary was quicker than counting in decimal. Take a pile of dominos that is between, say, seventy and 130, by asking a third impartial individual to add between -30 and +30 to a pre-counted pile of 100. If you practised, you would win every time.
I thought this out whilst writing this article. (It just goes to show!) With Appendix Two, I searched out the writing from the Internet, but not so anything else written here. It is all unique to this publication and not generally available on the Internet. (Maybe it will become so, as word of this writing, and the other ideas here, spreads online...)
You would need to practise though. You remember I said that after counting out 512 in our pile of dominos, we need to quickly and easily halve it? But how do you quickly and easily halve 512 dominos? The clue is, you would need to organise how you count 1, then 2, 4, 8, 16 and so on. If you try it, I think you will quickly see what I mean.
Number Bases
I'll keep using computers, but only to give you examples about which I know. So let's get back to counting.
So why, if base two is so ideal for counting things, including money, do we use base ten? What made the Greeks and Romans decide it was the best? A common answer is because we have ten fingers and ten toes, and indeed this is a good answer. That simple capacity allows us to count up to one hundred (what mathematicians call ‘an order of magnitude higher’) without even having to have a pencil and paper. But do you think this is the whole answer? In that case, why not Base Five, or Base Twenty?
Base Twenty would require us to have twenty names for the numbers zero to nineteen instead of combining the digits 0-9. However, if you think about it – we already do have! 17 is not Onety-seven, as it should be; it is seventeen. Ten is not Onety; it is ten. (Onety – I’ve only just thought of that myself.) You may think that a base of twenty is utterly ridiculous but not so. The Babylonian counting system, which is not from so very long ago compared with the Greeks, used a number base of sixty.
How could you possibly handle a base of sixty? Well, in the absence of computers and with just a basic educational system, perhaps most people did not want to have the hassle of knowing maths. We would think of them today as impoverished but, a long time ago, maybe life was different. Maybe it really was easier for some people if they couldn’t do arithmetic. Then, for those people that had to, the important thing was to be able to do arithmetic the easy way, using numbers that worked together. Now, this is the great strength of the base of sixty.
Although our natural numbers multiply together very well – and I hope you know your times table for that will be useful to you your whole life. But of course you do! Where was I? ... Oh yes. Natural numbers, such as 1, 2, 3, etc can be multiplied very easily by most people in their heads. 3 times 3 is 9; 3 times 9 is 27; 3 times 27 is 3-times-2-shifted-over-and-added-to-3-times-seven. (By the way, mathematicians wouldn’t use hyphens to each other to join things together. They use brackets to join things together, whereas we use brackets like this, to add something on the end or underneath, don’t we?)
Hold on, I am getting to the point! The point is, natural numbers multiply very well but they don’t divide so very easily. What is 2/7ths, or 3/5ths, or 22/7ths?
The point about 60 is that it divides as easily as it multiplies: 60 splits obviously into 6 and 10, and 6 splits neatly by 1, 2 and 3. In addition, six multiplies to 12, 24, 36, 72, 144, 288 and 60 multiplies to 120, 240, 360 etc. So mathematicians constantly find these numbers combine and divide nicely with each other: 288 + 72 = 360, for instance. Does 360 divide by 24? It does so very neatly, exactly 25 times!
There are 360 degrees in a circle, 60 degrees is one end of a right-angled triangle, 30 degrees the other. There are 24 hours in a day and twelve signs in the zodiac. There used to be twelve pennies to the shilling and twenty shillings to the pound and shopkeepers would talk in terms of a dozen (12) or a gross (12*12=144). That isn’t so common today, since decimalisation in 1971. Just about everyone prefers not having twenty shillings to the pound otherwise there would have been many, many more complaints. We tried base twenty but we didn't like it enough to keep it!
Okay, so if not Base Twenty what about Base Five? That would have the distinct advantage that you could count an order of magnitude higher (five lots of five) on the visible two hands, without having to take off your footwear! One could count easily to twenty-five only, but if you’ve ever had to do a mundane job of keeping count you may have had the same experience as me: that the easiest way is to make a tally of five. It is easier than doubling, just make a mark for each count – one, two three, four, and when you get to five, put a line diagonally crossing through the four. Repeat this up to, say, twenty-eight, and you may see how easy it is to tally using this system.
Oddly enough the reason I think we don't use Base Five is because numbers were not designed originally for the ordinary person to use. Again, it is because there are so many of us in just a few years. It is only in the last few hundred years that numbers have become so commonplace and central to ordinary life. Before the Fifteenth Century even the abacus wasn't in widespread use, and numbers were only used by experts. Imagine trying to introduce a week with ten days, or a clock with ten hours, or a pack of cards with ten cards in five suits; or an alphabet with twenty letters.
We can say that other number bases have been tried but have not proved popular. In the olden days they weren't able to spend so long in maths classes at school and so it was a big advantage to have just a few simple rules to remember to do both addition and division. Nowadays, even a base of twenty is too cumbersome for the kind of work we all do in our head every day. In some ways then, we think faster than our ancestors did. We have more intelligence but not necessarily more wisdom - and there are such a lot more of us now compared with then, so that we might easily be led astray if we were not careful - for instance by using up millions of years worth of oil in a few decades. (I'm afraid that is very much a problem for future generations.)
It is right and proper that no power of two is a 'round' number. So, we computer programmers have had to invent bases that are 'friendly' to binary. You may have heard of Octal, which is base 8 (or 2^{3}) and Hexadecimal, which is base 16 (or 2^{4}). If there was a power of two which coincided with a power of ten, then computers would be not nearly so unhappy with decimal.
Computers have to handle a lot of different numbers quickly, and they need to be as good with big numbers as with small. When you think that the computer industry first became really successful in the nineteen-sixties as the big banks started using them, you can see why. A bank account may contain a few cents or many thousands of dollars, and at any time a small amount can be paid out or a large amount paid in. These days, economics is one of those areas which relates to very large numbers indeed. A millionaire seems wealthy beyond imagining but Multimillionaires had been around since well into the Nineteenth Century. The richest man in the world in the sixties, at the start of my lifetime, was oil billionaire J Paul Getty. Ten times richer than he - so a different order of magnitude - is the richest man in the world today - Bill Gates. (These billions are the American definition of the billion by the way, as 1,000,000,000).
If there are six billion people in the world today, then Bill Gate's wealth amounts to $10 for every man, woman and child on the face of the Earth, including you and me. Of course I owe him a lot more than that since he has indirectly given me a career. It is impossible not to feel grateful when I think of the pleasure I have had, and the great vision Microsoft has shown in leading from the front for all of the past ten years (to 2007), I think.
It is a great strength of modern life that the huge populations which have sprung up in the last 100 years manage to work together. It is not often talked about. The oil and other natural physical resources that we have put to use have indirectly also created a framework of roles. As you go out to work and try to forge ahead in a career of your choice, you will find this out for yourself. You will find this is more often assumed than explained. It is a virtual (i.e. metaphysical) world, which one is expected to understand; the world of work. In some ways, I think Bill Gates deserves much more than he has received.
Most people would say that 60 billion dollars is enough for any individual and sadly, I think it is too much even for the visionary of the PC. Unfortunately, rather than thanks, admiration and most importantly recognition, there has only been vast and unimaginable wealth. Originally, he used to say he planned to give it away. Sorry to say, he has recently given himself a new job in spending the money. It is a subtle, but absolute, difference.
I'd have liked to see the word 'genius' applied to Bill Gates because Microsoft Office and Windows were creations of that, to me. I'd have liked to see him rewarded with glory and I'd have liked to see him preferring that to the money.
'Genius', like 'Saint', is a word that people seem to have become uncomfortable with. As an alternative, we routinely talk of 'thinking outside the box'. The mind is like a 'black box' which you cannot see into, as opposed to a 'white box', which you can see the workings of. Thus, the 'black box' of your - and Bill Gates' - mind has within it the white box of conscience. (This box is neither smaller nor bigger, but I’m saying one ‘contains’ the other, for convenience).
The important thing is, it is a white box because we know how it works. Remember? The conscience will always tell you when you have done wrong, but it cannot tell you what is the right thing, for you, to do. That you have to find out for yourself, mistakes and all. When you are thinking 'inside the box' as in normal working life, people don't know what you are thinking or why you are thinking it. They only know what you are thinking if you tell them. And they only know why that is what you think if you tell them that too. (This can get complicated...)
So when people talk of thinking 'outside the box', it is a convenience. You would never really want to be 'out of your mind' (or indeed, 'off your box'!)
In the end, all Bill Gates did receive was the money and, in a world where fifty percent of the population owns less than one percent of the world's wealth (UN figures), the last thing we need is yet another bottomless lake of private wealth.
But that is another story, and even the entire wealth of an individual like Bill Gates does not compare to the earnings of a complete country, which for a big economy like America is measured in trillions - billions of billions. In one year, the earnings of the United States are nearly one thousand times Bill Gates' entire fortune, at $13 trillion. This is just the earnings for the year of course, according to the Financial Times (March 2006). There are much, much bigger figures elsewhere, and the biggest figures of all are out in space. For instance, do you know what the Sun weighs?
This may sound an odd question. The Sun is made up of gas, isn't it? Surely gas doesn't weigh anything! Well, the short answer is, it does when there is enough of it! The Sun is a million times the volume of the Earth and the concentration of all that gas creates gravity, which compresses the gas at the centre. To cut a long story short, scientists estimate the sun weighs 2.2 octillion tons.
You can probably guess what an octillion is, if you recognised the tri- prefix when we talked of a trillion. The next one is quad-, so quadrillion, then quin-tillion, and so on. In the American system, each multiple is 1000 times larger, so they go up by 10^{3}. A million is 10^{6}, hence billion = 10^{9},
Trillion = 10^{12}
Quadrillion = 10^{15}
Quintillion = 10^{18}
Hexillion = 10^{21}
Sextillion = 10^{24}
Octillion = 10^{27}
Etc
In the English system, each number goes up by a million, so a billion is a million million (10^{12}) and a trillion is a million billion (10^{18}). The English system is much better!
Trillion = 10^{18}
Quadrillion = 10^{24}
Quintillion = 10^{30}
Hexillion = 10^{36}
Etc
A 1 with 27 zeros after it looks like this:
1,000,000,000,000,000,000,000,000,000
which is awfully cumbersome and quite difficult to grasp. We much prefer a unit that we can relate to, and that is one reason to use light year' instead of the -illion suffix when discussing really large numbers like those out in space.
The speed of light is about 186,000 miles per second. It is so fast it is measured in seconds not years because you can't see any delay in it here on Earth. Unlike sound, which travels slowly by comparison with light, if you are a long way from something, you can sometimes notice the delay in the speed of sound. So, in a storm you can see lightning seconds before you hear the thunder.
By divine coincidence, the speed of light is just fast enough to give a noticeable delay when you leave Earth. We did notice the delay when we went to the moon. The moon is far enough away that it takes light over a second to get to here from there. And the 2 Octillion tons of the Sun are about eight light minutes from Earth. There is no star nearer than four light years from Earth.
It turns out that the unit of the light year brings the Universe down to a manageable scale. Even though we have to deal with billions of light years, we do not have to manage distance magnitudes of one octillion light years. And even though the Universe is expanding, this is not a problem anyone is like to face in the future.
Light is the fastest thing in the Uiverse - it is a limit. So, if we want to imagine bigger numbers then we literally do have to imagine them. The largest round number is one you will have heard of elsewhere. A Googol is the name that has been given to 10 to the power of 100, and a googolplex, which is the largest number so far, is also appealingly round: a googol to the power of a googol. What may surprise you is that this is only marginally larger than Archimedes thought of, two thousand years ago. In imagining numbers larger than 10,000, he finally stopped at 10^{80,000,000,000,000,000}, which he calls the "myriakis-myriostas periodu myriakis-myriston arithmon myriai myriades."
"-plex"ing the number of course instantly makes it much larger and the temptation is to do this again, and perhaps again. The reason for not doing so however is the same now as it was in Archimedes day - the number has to have a meaning other than sheer largeness.
A common-sense definition of infinity appears to be "the largest number you can think of... + 1'. (We're not concerned with an exact definition - we know you can't define infinity exactly.) My largest real number so far was the weight of the Sun. I don't want to be saying it in octillions - that may be very convenient for a lot of zeros but you can imagine how difficult it is for one to even start speaking 1,111,111,111,111,111,111,111,111. It takes longer and longer to say the next number. We begin to wonder if "adding one" is having quite the effect we have assumed.
It does not, unless we are very careful. For example, a googol, which is ten to the power 100, is roughly the same order of magnitude as 70!. The '!' stands for 'factorial', which simply means 70x69x68x...1, as you probably know. Factorials are interesting because they are used in permutations, which are the number of ways things can combine in a given order. The number of ways 70 people can queue for the cinema is 70! Although this is not so 'round' a number as the googol, it is both quicker to say and shorter to write; and it is also more meaningful than a googol. Until it was taken for the name of the search engine, a googol had no other meaning. It goes to show that factorials as a way of counting large numbers can be better in several ways than using powers.
We could add one to 70!, and one, and one, and so on; but still eventually, we would have the difficulty of what comes next. How do we notate these bigger and bigger numbers? Maybe eventually, it takes longer to speak the next number than it did to speak all the previous numbers that went before it. Or to put it the other way, using powers of ten as a way of counting large numbers only seems to work initially. As time goes on, the same system would become less and less efficient. Ultimately, within a few million years, say, it is shown to be no better than tallying. You can see what made Archimedes stop.
Now, where was I? Something about the Greeks... ah yes - Archimedes!
Archimedes died about 200 BC, and afterwards the Ancient Greeks were followed by the famous Roman Empire. Most people know of the Roman system of counting because it is so memorable. I know I said earlier that we don't like Base 5 but what I didn't say is that we used to! It is a complicated story but let me come to it; for the great strength of the Roman system comes with a great weakness.
The Roman system elegantly combined a decimal base 10 system with base 5 markers. The symbols for base 10 were X for 10, C for 100 and M for 1000. The markers for multiples of five were V for 5, L for 50 and D for 500.
But before all that, you start off by tallying, so I is followed by II, which comes before III. Now we are nearer to five than we are to one, so we stop tallying and switch to maths instead. The next number is one-from-five - IV, or four - and then five, V, is followed by one-after-five, VI; and VII and VIII, from tallying. We're closer to ten now so again we use one-from-ten, IX, and then X, XI, XII, XIII. As numbers get larger, the same idea applies, so XXX is 30 and XL is forty. LX is sixty. CCC is 300 and CD is 400. The number 48 is XLVIII. The number one hundred and one is CI.
It is the last number, CI, that shows the great strength of the Roman system purely for counting. Because they are always counting away or toward the nearest round number, they avoid the problem that scuppers us. Whereas we effectively count by including the previous number, making our numbers longer and longer and longer (whatever system of notation we devise), the Roman system starts with the largest number they can think of, and then simply makes it larger or smaller until it fits.
The Roman system appears perfect for counting. It acknowledges the simplicity of tallying, the logic of addition and subtraction, and the naturalness of fives-and-tens, yet it failed. Part of the reason why is given by the example of 8, or VIII. It is just too tempting to write it as IIX. (I kept doing this by accident when I was drafting my work). But the weakness in the system was evident at the time. One Ancient Roman, who was buried in York, has a headstone that today says that he died aged XXIIX and that he commanded the VIIII Legion.
But did the Roman system fail because it was overtaken by history, and not through fault, or because it was logically flawed, in this slight way?
It is a question that has a striking parallel to computers today. Wonderful as computers are, undoubtedly also there are many people who strongly resent them. Even though I am on the other side of the fence, I can acknowledge the reasons for this. Just like with VIII and IIX, there are often two or even three or four ways to do just the same thing on a computer. It is a great weakness - especially as the ways to do things become more and more for historical reasons, rather than because they are intuitively logical as we were promised.
If the computer goes the way of the Roman number system then it will be replaced by the mobile phone - which will certainly be easier to use. But the mobile phone in this guise is a tool of business. For all the many frustrations and impositions of the computer, I love it for its ability to give me 'the means of production', in an arbitrary world.
The Roman system is a curio now which we use for watch-faces and encyclopaedia volumes. It does not survive as a living and evolving system for counting anywhere that I can think of. If it were introduced today from scratch would it be a different story as a result of that second chance? To be so, there would have to be a natural and elegant solution to the problem that we only partially solve with our '-illions'. And if that is a natural reflection of infinity, maybe it is not really a problem at all.
Oh dear, listen to me running on. I'd better get on with it or we'll be here all day!
The reason I am making such a big point out of something that you probably did not disagree with at the start is because mathematics uses the term 'countable infinity'.
Admiitedly, infinity is not countable, simply by its own definition. We know this by common sense, but in maths terms we would say that it disobeys an axiom of maths addition. In maths, it is axiomatic that a+b = b+a for any number a or b. Infinity does not obey this when infinity = infinity + 1. Therefore you cannot do arithmetic on infinity. Maths still does have this phrase 'countable infinity' but it is a modern idea and we have not yet reached it in our review of history.
For the moment there is a mismatch here: a 'hole'; which any of us who wants to is free to explore for themself in their own mind.
I will try to explain.
The second rule, or axiom, of maths says the same thing about multiplication:
a x b = b x a
We happily apply this with zero: anything mutiplied by zero is zero, and of course we can divide by zero. Anything divided by zero is infinity.
So what about infinity? What is something divided by infinity?
You might think that people would say "zero" but mathematicians are cannier than that. Infinity is not a number so they would say that this is not computable. Meanwhile, common sense would say that it is infinitesimal: any number n - except zero - when divided by zero gives a number that is so small, it approaches zero, we would say.
It isn't mathematicians who are wrong - they say there is no answer. What they do not say is why we are mathematically wrong in our answer. (And we are physically wrong, by the way).
Investigating this ‘hole’ is what we have started to do. We have started back in history, to get a perspective on the idea of 'the largest number you can think of' + 1. And we have already made some progress. We can see the problem that adding one to a number makes it longer and longer in our decimal base system of counting, and that means adding more and more zeros to it. We saw that for large numbers the Roman system was better, however the Roman system did not last long enough to address the problem of a systematic way of counting large numbers. In the coming pages we will go on to look at what happened afterward. It will be a lot of fun, as long as you like numbers.
It is a problem we will return to, but now does infinity even exist? Well, yes, undoubtedly infinity does exist in maths. For example, Pi has an approximate value as we will see later, but it is the 'real' value (i.e. the Greek symbol) which is the widely-used value in maths.
Infinity exists in the real world too, so you and I are infinite. The best analogy for me - because it is the complete opposite - is the computer: you and I live on in our children's and descendants minds, and in their consciences, so unlike a computer even death does not switch us off...
Most well known of all is that we live in an expanding, infinite Universe. This is modern knowledge but physicists have believed it for over a Century. Not just big; not very big; not very, very-big-and we-don't-quite-know-how-big; but infinite. Without qualification or caveat.
And then there is God. Although I don't understand eternity or what scientists fully mean by saying that the Universe is infinite, I do know that God is nothing less than infinite. In my experience, people pretty much act the same whether they say they believe in God or not. I don't know what you think but although I say I believe now (and used to say I didn't) I have always been very keen that there should be a God. I don't mind if the Universe should turn out to be finite after all - and very big! But I would be a bit disappointed to discover there was no God!
You may feel the opposite. If I were discussing this with someone who did not want there to be a God, I would come back to suggesting that people are infinitely varied, and it would be very disappointing to discover you or I were limited; finite. But I am prepared to consider it. Maybe a better idea than God will come along(!) It sounds quizzical but numbers have had better ideas, subsequently.
An idea which has proven its strength, and which would have been new to Archimedes, is the idea of zero. Surprisingly, zero is a recent invention, even though now it seems as necessary and important as the number one. Yet the truth is that zero was invented about 200 AD by the Hindus of India.
We have seen that Roman numbers don't require zero because of their way of counting. Numbers which grow from right to left, as 1000...0, need to have zero's but Roman numbers grow from left to right, which is no less logical. You don't need a zero then because when the number is zero it simply isn't there! And when you want to go below 1, you use fractions and again these are ratios of one number against another. No need for zero then either.
No need for zero then as long as what you are doing is counting. It seems as if zero only becomes necessary when you start working out what things are other than by counting. When you start using formulas, and equals signs, then it becomes pretty difficult to work out the result of 2 - 2 without a zero.
That said, the Greek's managed pretty spectacularly without this help! When I think back to my 'O' Levels and all those formula for triangles, circles and tangents, I remember being both impressed and stretched. It was the famous Greek names like Pythagoras which had managed to both map and master this science of Algebra, without even having the help of a 'zero'.
The Babylonians had a special symbol for the absence of a number around 300 BC. They didn't think of this 'absence of a number' as a kind of number any more than we think that the "absence of an ear" is a kind of ear.
The absence of an ear can indeed be a kind of ear. If you imagine someone having an accident, with a rough edge of skin left behind around the hole itself, then as the purpose of the outer ear is to scoop sound into the hole, then the rough edge of ear will be acting as an ear, however poorly, because it is doing the same thing.
Another 'grey area', of conflicting rules, is created by thinking that the outer ear as separate to the inner ear, since it is the inner ear that contains the delicate "drum" by which we actually do hear. In some cases then the absence of the ear is indeed the ear, in others, the presence of the ear is itself not the ear!
In fact, in computing, we have invented a different value, NULL, which is used to denote a value which is unkown and absent, as opposed to the value of zero, which is known and absent. This is a very recent innovation, being something that is particular used in databases. (It was part of the SQL language Ted Codd invented in the sixties.)
Try thinking of it by analogy to the world of law, for there is something to learn about both worlds in doing so. In the area of law too a great value is put upon clarity and logic, for these are ways to show that something is fair and just. The laws have to be made up and improved as we go along, though. There is no law-generating ‘formula’ that can be trusted to ‘generate law’.
This is a very great deal of work. We are very careful then about the new laws we create. Our experts, the lawyers, have the job of knowing the laws so that we can make the best use of those we have. And when lawyers want to know whether or not they can win a case they often are asking whether the outcome is known (it has happened before and so the result should be the same as previously) or it is the first time (and so a new judgement will require the judge to set a precedent).
Where there was a previous outcome, in law it is called a 'precedent', and in computing databases it is called a 'default' value. So, a value, (often zero), would mean there is a precedent while NULL would mean that there is no precedent, as yet.
It is not quite true then to think that zero is a number just like the other integers, 1, 2, 3, 4, etc. And there is something to be gained from thinking about why not, just as there is a great deal to be gained from thinking about why infinity is not a number. As shall be seen.
This brings me to the idea of a number so close to zero that it does not have an exact value: the idea of the infinitesimally small. I wanted to explain just now that zero is well understood, and that it is best understood alongside NULL. Unlike zero, 'infinitesimal' is not well understood, and I want to show you this.
Let's take a rule of thumb definition for infinitesimal as we did above for infinity. Could we start with 'the smallest number you can think of -1'? No, that is no use. Take one away from a number smaller than one and you get a negative number, which is not what we mean by 'small' at all. Clearly, the 'largest negative number' has the same problem when you take one away from it as the largest positive number does when you add one to it!
So how about: infinitesimal = 1/infinity? No, that won't do either. Logically, you can't divide something by infinity any more than you can divide by zero. There is a spurious plausibility to the idea that 1/0 = infinity (because then conversely 1/infinity would be 0). This is clearly nonsense though, for what is 0/infinity? It is precisely the danger of thinking of infinity as if it were a number. A rule for 1/infinity is worse than useless for, without a consistent axiom (like a + b = b + a for any number) it stops you thinking.
So let's start thinking now by looking beyond simple 'common sense'. One may say that the 'common sense' definition of infinitesimal is that it is 0.0000001 (or however many zeros you care). I would say this is a good child's definition of the notion but it is not good enough for us. I don't mind the zero's but where I suggest you quarrel is with the '1' - what is this 'unit' out of which infinitesimality is made? Can one now add infinitesimals?
In maths the idea of zero is a recent, and very successful, addition. But I am beginning to wonder if it can be just added, or has it pushed aside the word 'infinitesimal' which no longer seems to me to quite fit in.
Do you know Zeno's Paradox? It goes like this: a frog is trying to jump to the edge of a lake. It progresses by jumping half the distance of its previous leap each time. It never reaches the edge of the lake but it comes infinitesimally close. Zeno was a precursor of Aristotle so there would have been no problem with infinitesimal in his day.
We have moved over to physics with this 'model' of the infinitesimally small, however there is a significant problem in modern physics, even at my 'O' Level knowledge of it. I am sure you will have heard of Quantum Mechanics which is the branch of physics that deals with the subatomic scale. Quantum theory says that when you get to the smallest possible sizes then a principle of 'uncertainty' comes into action. Light for example, behaves as both a particle and a wave, but on the smallest possible scale you cannot know both the position (as for a particle) and the speed or direction (as for a wave) both at the same time. This is not simply because the values cant be measured, it is more subtly that, in some way, the values are not yet decided.
We still commonly think of 'infinitesimal' in the way that Zeno thought of it, but that is no longer right because of Uncertainty. And on the macroscopic level, physics has the same problem with infinitesimal that maths does. For nowhere is there anything that is truly zero. Even out in space, in the absolute coldness of literally ‘nothing’, absolute zero is not the stasis which we think of, for that would violate Uncertainty. (For physics, I recommend you read James Gleick’s biography of Richard Feynman. It deserves the Nobel Prize, and this is a paraphrasing from that book, titled ‘Genius’.)
Can I tell you a short story? It won’t take long and I think it illustrates the area we've been looking into.
A man and his doctor agree to a friendly game of golf together. As they are making their leisurely way around the golf course, the conversation naturally turns to matters medical. The doctor remarks, "Not many people realise it, but most men have one testicle that is larger than the other". "Not me, Doctor," returns the golfing patient "I'm the opposite!”
“You see, I have one smaller than the other."
Zero is available as a synonym for infinitesimally small in physics as it was in maths. Or we could have a new word, 'finitesimal', for something that is vanishingly small - but still finite.
All we can do for the current moment is to take the English definition: something that is infinitesimal goes on and on, forever and ever, getting smaller and smaller. But here again something is not right. There is a thing that goes on forever and ever getting smaller and smaller, but it is never described as infinitesimal. In fact it is always described as infinite. That something is Pi.
It may help to remind myself that Pi is the value which relates the edge of either a circle or a sphere, to its centre, or to its area. The known Universe is thought of as a sphere, and the area of that sphere can be calculated using Pi, accurate to 39 places, to the nearest atom.
Computers have been used to calculate Pi to between ten and one hundred million decimal places - that is the same order of magnitude as a googol! When calculating the result of a single mathematical formula, a computer is just as likely to give you the wrong answer as the correct one (sadly, even when I am the one programming it), so only a human can tell whether the result is correct or not. But calculating the result of a single mathematical formula repeatedly, which is called iteration, is what computers are so good at. I'd hate to think what a boring job that would have been otherwise, wouldn't you?
People will say the value of Pi - 3.142... - is not "small", and I would agree: it is neither small nor big. But - and here is the logic one cannot argue with - the next digit down is always smaller than the current digit - a maximum of 9/10ths of it. As long as Pi continues, it is going on getting smaller and smaller and smaller; way beyond any need for it!
You can see this by extending Pi to be more accurate than that first approximation, giving us 3.1416. Obviously, this is less than 3.142 - by four tenths. And the next order of accuracy will be a maximum of 3.14160 (approx.), which appears to be the same as 3.1416, but only because it is rounded. (The real value - 3.141595 approx - is obviously less than 3.1416 - by one-tenth, approx. - and our next approximated value of 3.141593 is less than 3.141595 - by three tenths).
(I suspect that older people might have more trouble with this than younger people. I know it took a while for it to fit comfortably with me. One point of explanation to offer to those like us is that we will be probably be much more used to rounding up approximations than rounding down. The above works either way - each digit is smaller than the last - but be aware that, because of rounding, any one digit can be exactly the same as the last.)
Now when you start to think about it, Pi doesn't really feel infinite, either. Not in the way that I believe it should be used; literally as in-finite: containing everything that is finite. Pi goes on and on forever but it is always a number. However far you get it remains an endless, unrepeating stream of digits. So Pi does not feel infinite because it is infinitesimal; it goes on forever and ever, getting smaller and smaller.
(Do you remember hearing about a book called "The God of Small Things"? It won the Booker Prize in 1997. For many years I assumed this was what it was about. Finally, after a few years, I found someone who had a copy and was prepared to lend it to me. What a shock after all that time to discover it was a novel!)
Well done! We are over half way, and getting somewhere at last. Although you are probably beginning to think more about your tea, and I know I am, let’s continue the good work for a little longer.
Consideration of Pi brings us to the work of Ramanujan, as described in an excellent book called 'The Man Who Knew Infinity'. This brings us nearly up to date as Ramanujan, a great Indian Mathematician, lived only about 100 years ago. He was very interested in infinite series, and it was this that led him to an investigation of Pi. It turns out that Pi is often part of the solution to an infinite series.
Here are some infinite series and their totals:
Pi/4 = 1 -1/3 -+1/5 -1/7 + 1/9 -...
Pi/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 *...
(Pi-3)/4 = 1/(2*3*4) + 1/(4*5*6) + 1/(6*7*8) +...
Pi/4 the first series above, discovered by Liebnitz, is "useless" for calculating Pi since the first five hundred terms are required to give three-decimal place accuracy. Ramanujan discovered a series in which the very first term gave Pi to eight decimal places and, years later, provides the basis of the fastest known algorithm for computer estimation.
Now, what we have been considering is the infinite series of natural numbers and Ramanujan would have approved of our curiosity. It would be only natural to wonder if there could be an infinite series that provided a faster way of counting than integers. Even though it takes us into a much more complicated area of maths, there is just as interesting a reason to do so, if we look at prime numbers.
Prime numbers are numbers that are only divisible by one and themselves. The ancient Greeks first discovered them. These numbers are, in a way, the basis of integers, since every integer can be made from primes and there are far less primes than there are integers when you start counting. For any number that you can say in words, like 1768211, you can say the same number as 1117 times 1583, using two primes. In this case it is slower to say the primes, but since every integer can be made from primes, counting primes represents a complete, increasingly fast way of counting integers by implication. And it was proven (in 300 BC!) that there are an infinite number of primes.
(Can I still say that, by the way: "an infinite number of primes"? It's not right, is it? I certainly can't say it is finite, so I suppose I should say an infinitesimal number of primes. Hmmm, I think I'll prefer "an infinitely-long number of primes", pedantic though it may sound.)
As we start off counting just primes - 1, 2, 3, 5, 7, 9, 13, etc - there are very many of them but as we keep counting there are less and less. There are a non-finite number of possible primes just as there are a non-finite number of integers, but there are fewer primes than integers in any given range. So, counting ‘in primes’ is quicker than counting in integers, and at an accelerating rate. If we have to give up the idea of counting to infinity we can still be curious about counting, itself. This would be far quicker, like a shortcut.
But now here is a quite different problem: we do not know how to count just primes! 2000 years after the Greeks, there is no way to calculate the list of primes; they can only be computed, by trial and error. In fact today, this principle is used in the practice of cryptography. As recently as the Second World War, cracking cyphers such as the Enigma code helped us win. But now, we have cyphers based on extremely long primes that cannot practically be factorised by trial-and-error, and so are entirely unbreakable.
There is no way to count primes. There may be in the future due to the amazing concept of quantum computers, although that will be a development in logic not in maths. Meanwhile, this is a problem not of size but of knowledge.
If we want it to, it may well teach us something very profound. It may well be that Infinity is not about bigness, or size after all. Maybe it has more to do with what you know, and don't know - and what you can learn, than it does size. There is a non-finite amount to know, but knowing a non-finite amount still leaves a non-finite amount to learn. As a naturally curious person, I find that heart-warming, but sobering.
So far we've raised some exciting questions regarding ordinary common sense and done so quite fairly, using no more than our own common sense and general knowledge, to do it. There is another reason to feel intrigued, as well, because I am not the only one swimming against the tide of mathematical history. There is a famous and recent story about cleverness versus wisdom.
Imagine a game show. You are shown three doors behind one of which is a car. You get to pick a door and there is a 1/3 chance of success. If you do not win the car, you are then offered a choice of sticking with the same door or switching to a different door. Now, before you choose, the game show host (who knows) opens the door that does not have a car behind it, out of the two. Do you stick with your choice or switch?
In a column in the magazine, the clever and wise writer showed that contrary to your wishes you should always switch. In doing so, she justified her notoriety by standing up to the 9/10 of mathematicians the length and breadth of the United States who were utterly convinced she was wrong. (See Appendix Two for the full story).
I like this story because the woman was clever and had the courage to question "established wisdom". I like the idea of questioning something as obvious and simple as a game show and I like the idea of the underdog winning. There should be a use for extreme cleverness. In my pyrgic puzzle example above and in Appendix One, it was my feeling that the puzzle-setter was not being at all fair that made me want to question him, and it. In the end it is fairness that makes the trail that we follow which leads us from the beginning to the end.
And at long last we are approaching the end of this journey, that you have allowed me to take you on.
Some years ago, one of the many books I was always interested in reading was a book about the mathematician Paul Erdös. This book was called the man who loved only numbers and was a bestseller for the publishers Fourth Estate who, at this time, were having great success publishing clear but sophisticated non-fiction to an intelligent, educated audience.
This book was unusual in that it contained what appeared to be an error. It described the work of Georg Cantor who was a mathematician at around the same time as Ramanujan, and who was also very interested in infinity. Cantor believed he had found a proof of levels, or classes, of infinity. His proof, as described, seemed to me to have two problems, one of which was that it was a logical argument, not a mathematical proof. There was a second problem too. There is a wise axiom in philosophy called Occam's Razor which says that the simplest explanation is often the best. It seemed to me that all infinities are the same infinity, by Occam's Razor.
I wrote to the author by way of the publisher Fourth Estate and also wrote to the cleverest person I knew (my Uncle Frank Lee) the same letter, to see if I had made a mistake, but no-one wrote back to say I was wrong. (When I showed my Uncle this writing, I am sure he felt a bit guilty for not writing about that. Not rightly so, Uncle).
There wasn't much more to be done about it at the time, but I still have the book, written by Paul Hoffman, and the letter, dated 10th April 1999. This is his description of Cantor's theory, which is still the clearest I have found:
Cantor wants to prove that there are more 'real' numbers, like 0.135, than there are integers.
Hoffman writes: "Just start listing all the real numbers between 0 and 1 in no particular order, Cantor urged, and match them up with the counting numbers:
.12146789 . . . 1
.32769234 . . . 2
.71234568 . . . 3
.43567233 . . . 4
.64594678 . . . 5
. . . . . .
. . . . . .
It might seem that there should be enough counting numbers to match up with every real one. But the reals simply can't be put into one-to-one correspondence with the counting numbers. Assume the list does contain all the reals. According to Cantor's diagonal argument, construct a new number from the digits in boldface, .12264 . . . , by replacing every single digit by a different digit of your choosing. For example, you could add ONE to each digit to produce the number .23375 . . . This new number is clearly a real number since it should appear somewhere on the list. But it can't be anywhere on the list because, by its very construction, its first digit differs from the first number on the list, its second digit differs from the second number on the list, its third digit differs from the third number on the list, and so on. Thus the list of real numbers isn't complete; there are more reals than counting numbers to pair them with."
Isn't that a beautifully simple and clear explanation? It has a compelling simplicity. I have been especially struck by this when looking for alternative explanations to compare it with. I am greatly indebted to the Wikipedia website for the many times it has helped me, but its explanation of Cantor is impenetrable.
However what leapt out at me along with the good writing was what I thought was a simple error. In reading it twice I felt I could see what about it was wrong. I've mentioned the things that occurred to me later above, but this is what I wrote about in my letter.
Cantor is saying that you can transform each digit into a different digit to generate a new number, simply by adding ONE, according to Hoffman's example. But, I wrote, what if the digit is a 9? In that case, adding one to it may generate a carry digit (if you need to carry), or zero (if you don't). Now, what if the number you generate is all 9's? That is perfectly allowed. As the order of the digits is up to you this is just as valid as the example - i.e. the number you get happens to be .9999 . . . , then you either get 1 if you do carry, or zero, if you don't. But a real number is between one and zero according to the description.
Thus, the argument given is not complete: it works for some cases but not all. (You can't simply change the method of adding one either. The only method that wouldn't have this flaw would be one that included human ingenuity - the same problem as with an infinitely increasing count.)
This is the problem, but I was not mathematician enough to know whether it was Hoffman's problem, or Cantor's.
I still am not mathematician enough, but if you have managed to read this far then you are as well able to see the problem as I am. The method of 'carrying over' digits is the fundamental principle on which our decimal system, and in fact any number system from binary computers to that of ancient Rome, is founded. This is how we started out this investigation into numbers. It is such 'common sense' that I have been forced to conclude Hoffman could never have forgotten it. But Cantor could have.
The truth is that extreme cleverness was no advantage to Cantor. He was mentally unstable and suffered from a sad life. Even great mathematicians like Gauss have been agnostic in the face of Cantor's ideas, but cleverness is a gift of the mind. It is a gift that must be managed through the body by using the same principles of hard work and self-sacrifice that we all apply voluntarily. Cantor was emotionally attached to his work, but he was mistakenly attached to the conclusion, as it turns out, at the expense of the means. Loving your job doesn't necessarily make you good at it, just as hating your work does not necessarily make you bad at it.
I both love and hate computer programming, which by writing it out may have taught me something I didn't know. What else have we learned that we didn't know before? I asked earlier if you had any feeling about infinity because I want to suggest that you do. But I can't describe it any better than, it is the feeling you get when you are absolutely certain you are right and then you find out something you'd overlooked: there's more to it, or there's something you'd forgotten, or there's something you'd never have dreamt could matter...
We have seen with Pi that infinity is not what people want us to think it is. Pi, is more, and more, and more, yet ultimately the same thing over and over: number. It is both infinitesimal, in the sense that it goes on forever and ever and ever getting smaller and smaller and smaller, and it is infinite, in the sense that it is never the same numbers, and there is no repeating sequence.
Because, by Occam's Razor, all infinities are the same infinity, then anything that is infinite will have this dual characteristic of also appearing infinitesimal.
We can see this best by comparing two infinities that are both different. When you do, one will stay infinite and the other will appear infinitesimal by comparison. Compared to God, the Universe is infinitesimal. Compared to the Universe, we are infinitesimal. Compared to us, Pi is infinitesimal.
And it is useful to continue. Compared to Pi, 3.333 recurring is infinitesimal. Compared to 3.333... well, 33 billion is finite.
What I set out to say to you is that "the largest number you can think of + 1" is not a better approximation to infinity than that number on its own because adding one to a very large number has decreasing as well asincreasing magnitude. Infinity is the twist in the tale, that puts the story back on track. You must have enjoyed that experience for yourself. It is when the plot takes an entirely unexpected, yet perfectly logical, turn. There is a phrase in the film industry for it: the Maguffin. And there are other phrases you might have come across: the ghost in the machine; the music of the spheres. Indeed, music, itself.
In your future life you will commonly come across two simplistic views of truth. You will rarely come across the third, that I am presenting here. Take a moment to explore it.
There is the Platonic view of truth; that it is like a fire-lit cave. This is the mystical view; that the light of the fire throws shadows, but it is only ever the shadows that we can see. we cannot look directly at the fire.
Another view is the "computer game" view, that truth is simply a matter of levels. When you are good enough at it, then you get to play at the next level, there being enough levels to make a good game.
The third view is that truth is a dimension, being neither one of the two views above, entirely. This is a composite of the two views. There is truth we can know but there is truth that we will know, and do not yet. Personally, music is one-such for me. I was tone-deaf and could not play a note, but glad to be, for I loved hearing music from the Beatles to Pink Floyd. I could not have loved it more and very probably would have loved it a lot less had I been in competition with it.
Truth is a dimension with an 'ahead', and therefore also, a 'behind'.
There is truth that we use today, every day, which was not always known. It would be just as hard to look back. "How could they have known that, but not this?"
we would say, if we knew.
The "next level" is actually a room next door to this one. You can simply walk into it, if you want. The room next door to that one is also next door to this one. Do you want to walk into the second, instead? And do you want to now, because you can, or do you want to wait, until it is the right time?
You do not need any advice from me and that is not what I offer. What I want to say to you is how fabulously lucky you are to have the choice to make.
Returning from an infinite choice to the subject of infinity, it is just as valuable to come back from low numbers to high. We can start with the idea that compared to 33 billion, 3.333 recurring is infinite. Is it true to say that compared to the integers the reals are infinite? For the moment, I am happy that by comparison, Pi is infinite, yet each of us as a unique individual even unto death, is infinite in a way that Pi is not. Still, compared to us the Universe is infinite, for within our uniqueness we individuals practically fulfill predetermined, predestined roles through our work, whilst the Universe is expanding as I write.
Compared to the Universe of course, God is infinite whilst even though the Universe is expanding and infinite, it also has a definite shape. (If you are reading this before the date given you will have to find out more on the Web, at
www.whatistheshapeoftheuniverse.co.uk
www.thisistheshapeoftheuniverse.co.uk )
And astonishingly, God is also infinitesimal through the practice of free will. I think one good way this truth is expressed is in the idea of a Personal God - your own individual equal in Him. (That sort of really is a God of Small Things). It is also there in the idea of free will; that following the principle established after the 2nd World War at Nuremberg: you have a duty to not follow orders that are unconscionable; a responsibility to exercise your freedom of will.
Infinity and infinitesimal are both unknown but infinity is also unexpected; intriguing; surprising. And so are numbers. The set of 'natural numbers' can be + or -, so there is a set of Negative numbers. There are Fractions. Real numbers. Imaginary numbers. Perfect numbers, prime numbers, irrational and rational reals; odds and evens. Transfinities. Surreals... and more.
It is said however that out of all these, God made the Integers. I think this is shown in our finding. We started with:
Zero, infinitesimal, 1,2,3,4...infinity
Which I think is ok for young people. For us, and for you as a young adult, the integers are, so elegantly:
Zero/NULL, 1, 2, 3, 4 ... infinitesimal/infinite
where the first and last are not numbers.
As it has sometimes also been said, “there are three types of people in the world: those who can count; and those who cannot count”.
Postscript: Was Cantor Wrong?
Is it possible to like numbers too much?
It is school-age maths that we have used for this article. Even though Georg Cantor's ideas are not part of the syllabus, they are accessible to all; a part of modern culture. So, the professor of maths at Oxford University finds a popular voice in the Guardian with this subject.
I set out to be the kind of teacher that is entertaining (I can't very well be the disciplinarian type, from here). This article may help you study for your exam, if it is coming up, and of course it was also written for those of us who can remember taking our exam.
One advantage of writing for pleasure is of having the chance to come away from the thing. But it is also very difficult to like something, or someone, greatly when you are a long way away. If you and a future partner are forced to spend time apart, but you can keep your feelings strong, then they are likely to be true feelings. And when you are too close to the thing, then liking it too much can destroy or spoil the very thing you like so much.
With a Postscript I can come back to the subject after a deliberate break. Having now lived with it, what do I conclude from re-reading my argument in the cold light of today? Was it really that Cantor liked numbers too much, or perhaps that the professor of maths at Oxford University, liked them too little?
I do not think there is much to be gained from criticisng Oxford Professors. (I think that is best left to Assistant Professors, who are in the best position to judge, and to compete). Nor do I think there is any more to be said about Cantor. He himself hardly benefitted from his work, however judged.
So was the author of the Fourth Estate book wrong? He wrote that Cantor's proof began by listing all the numbers between 0 and 1. My refutation responded by proposing that the number might be all 9s. In that case, this would make the number 1.0. It made no sense to me that the set of all real numbers would contain the number 1.0 If it did, then it would also contain the number 2.0 and all the numbers between, and so on. In other words, it would not be a properly defined set.
But it turns out the set of real numbers does contain the integers.
That is, it contains them, technically. We can use this term 'technically' for a solution to a big problem that creates a lot of smaller problems. Technically, I was wrong.
But Microsoft's Excel, the mathematical calculation program is also, technically, wrong though I would hate to hear it said. Other times we use the phrase ‘technically’ to acknowledge a fault we cannot defend in a solution we know cannot be improved. This leads me on to the last part of my introduction of the world of numbers in this article, from a computer programmer.
A real number is a number that has an integer part and a decimal part. The decimal part is a fraction. All decimas can be expressed as fractions but not all fractions can be expressed as decimals. Literally. It would be truer to call all real numbers approximate numbers.
An approximate number has the potential to extend into infinity both in the integer value and in the fractional part. An example is the speed of rotation of a vinyl record: 33 1/3 rpm. You cannot enter that number into a calculator, except as an approximation.
The same is true with Excel, but it is worse. Not only is 1/3 infinite in the computer but so is 1/10^{th}! This means a simple decimal sum can lead to a rounding error. Enter the following sum into Excel, Open Office or any equivalent:
= 1.00 * (0.50 – 0.40 – 0.10)
You will know the true result is zero but you will see a non-zero value. If you were doing something similar with rounding, for example:
=ROUND(1-(0.5-0.4-0.1),1)
then you would get 1 instead of zero. Very very occasionally, this really matters. The Internet again tells the story of the Patriot Missile system which used a clock interval of 0.1s. Over only 100 hours this gave a significant enough rounding error for the guidance system received the wrong flight data, costing the lives of 28 people in one case.
Oddly enough, at the heart of every computer is something called a Floating Point Unit. This gives the computer its own mini calculator separate from the CPU. Originally this was done for speed of operation. Programs like ‘Mathematica’ , show how to implement a 'pure', mathematical representation of numbers, but even the computer programming languages themselves, such as Visual Basic or C, can only hold numbers over a certain size as 'strings'; that is, as words.
If we allow that real numbers can extend to infinity in two directions we can see that a proof about infinity using them is flawed, but what can we do about it? How can the ordinary student - you and I - show to the expert or professional mathematician that his maths has more to it than he sees?
Well, we can apply the new knowledge perhaps.
Suppose we were to say to him that any point in space can be represented by two numbers, x and y? ("Nonsense," he might confidently reply. "I think you will find, young man, that space is three dimensional or perhaps four dimensional, and only a two-dimensional plane can be encoded with two numbers")
"What if they were radians?" we might begin our answer.
You may also remember them from school. Radians prove useful in trigonometry often as a measure of angles when calculating sin, cos and tan. The interesting thing about them from our point of view is that they are a base, like our base ten, two, five, etc., but they are a base in units of Pi. A single radian describes any point in 180 degrees of arc. (Two radians would describe the whole circle, so our number system would be better based on a new measure - say the Irradian - being a unit of 2 radians.)
So, any point on a plane can be encoded in a single real number, by using the integer value as a measure in polar coordinates, r; and the fractional measure as a measure in radians/Irradians.
"And two numbers would be all that was need to encode any point on a three (or four) D plane." we would continue.
"And it is only logic to see that a single real number encodes two classes of infinity." We would finish. There would be no need of mention of Georg Cantor.
So next time anyone tells you that Cantor proved that there are degrees of infinity, and that the set of real numbers is greater than the set of integers, ask them if they know that any point on a 2D plane can be represented by a single, real number?
It is only school-age maths, after all.
Email the author about this work: mc@getcross.co.uk
www.whatistheshapeoftheuniverse.co.uk
www.thisistheshapeoftheuniverse.co.uk
www.mmdl.co.uk
Pyrgic Puzzle
Clearly the trick here is to be that they have looked at it logically, rather than mathematically - but have they? I could just as well say once, because there is only one three in three-nine (rather than thirty-nine), if one is making a virtue of disregarding the rules of maths. Or by combining this with the logic of the answerer I could say twice, because three from thirty-nine leaves 36, and three from three-six leaves just the six.
But, if the virtue is really combining maths and logic, then you can go on and say three times, since 3 from 39 is 9 (which gives the first one) and 9 is the product of two 3's (giving the second and third). Or, if the virtue is just how clever-clever one is, one could say five times, since 3**3 + 3x3 + 3 also equals 39, and if we allow 3/3 as part of the formula, we can escalate this to six times (3**3 + 3x(3 + 3/3)), seven times (3**3 + 3x3 + 3*3/3)) and presumably, on and on, indefinitely.
The correct answer then is actually as many times as you like - and beyond three, you reach zero every time.
The "Monty Hall" Dilemma
Marilyn Vos Savant is someone academics hate. Flashy and confident, she bills herself as the person with the highest IQ ever recorded, a whopping 228, from the Guinness Book of World Records, and she has questioned not only Andrew Wiles proof of Fermat's Theorem in 1993, but Einstein's Theory of Relativity. She is married to Robert Jarvik, who invented an artificial heart of Pyrolytic Carbon. This story takes us back to 1990, when Marilyn was writing a regular column, "Ask Marilyn" in America's Parade Magazine.
In her column for September 9, 1990, vos Savant answered a well-known brainteaser submitted by one of her readers, the so-called Monty Hall Dilemma. It comes from the famous American TV gameshow: "Let's Make A Deal" where the contestant is given the choice of three doors. Behind one of the doors is the star-prize, a car, and behind the other two doors are consolation, or booby, prizes. You choose first - say Door One - whereupon, the host, the eponymous Monty Hall - who knows where the car is - picks one of the other two doors and opens it to reveal the first consolation prize.
You are now into the second and final round. One door is open, revealing a consolation prize. Two doors are closed, the first one you chose, which may or may not contain the car, and a third one. Only Monty Hall knows which of the two is the right one to choose, and he now asks you whether you want to stick with your original choice, or switch. Which should you do?
What do you think? Marilyn vos Savant gave as her answer in the column that you should switch and she gave her explanation. Suppose there are 10,000 doors. You make your first choice and the host then opens all the other doors except one. You'd switch to that one pretty fast, wouldn't you?
Well, evidently not. On publication her column was besieged by mail from disgruntled readers including many mathematicians, maintaining that the odds were fifty-fifty, not two-thirds, in favour of switching.
Common sense says that the conditions are unchanged from when you very first made your choice. The car has not moved, has it? Therefore, since it is clearly not behind the open door, it is fifty-fifty whether it is behind one door or the other. There is no advantage to switching and human nature being what it is, you are more likely not to switch out of loyalty to your original choice.
In this case however, common sense is wrong! It is not quite fifty-fifty between the two doors. Don't forget when they started out each door had a third chance of being right and 'your' door still only has a third chance of being right, even though there are only two doors! It may be utterly counter-intuitive, but because of the unbreakable laws of probability, when Monty Hall opened one of his two doors, he shifted the probability of it being right onto the other of the two doors. The probability of your door being right is therefore now one-third. If you change your selection, you will win two times out of three.
Because this is counter-intuitive, and even perhaps for an inherent feeling that one is defending a principle of loyalty, feelings ran high about vos Savant's answer to the question, but let's see what happened:
In her column of December 2, 1990 vos Savant printed some of the letters:
As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error....
Robert Sachs, Ph.D., George Mason University
You blew it, and you blew it big! I'll explain: After the host reveals a booby prize, you now have a one-in-two chance of being correct. Whether you change your answer or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!
Scott Smith, Ph.D., University of Florida
This time, to drive her analysis home, vos Savant made a table that exhaustively listed the six possible outcomes:
Choose No1 & stick with it
Door 1 |
Door 2 |
Door 3 |
Outcome |
Car |
Consolation |
Consolation |
Win |
Consolation |
Car |
Consolation |
Lose |
Consolation |
Consolation |
Car |
Lose |
1/3 chance of win
Choose No1 & change
Door 1 |
Door 2 |
Door 3 |
Outcome |
Car |
Consolation |
Consolation |
Lose |
Consolation |
Car |
Consolation |
Win |
Consolation |
Consolation |
Car |
Win |
2/3 chance of win
The table only considers the case where the car is behind the first door you chose because if it is behind one of the other doors you automatically win by changing. It demonstrates that whatever door the car is behind, she wrote; "when you switch, you win two out of three times and lose one time in three; but when you don't switch, you only win one in three times."
But the table did not silence her critics. In a third column on the subject (February 17, 1991), she said the thousands of letters she received were running nine to one against her and included rebukes from a statistician at the National Institutes of Health and the deputy director of the Center for Defense Information. The letters had gotten shrill, with suggestions that she was "a goat" and that women look at mathematical problems differently from men. "You are utterly incorrect about the game-show question," wrote E. Bay Bobo, a Ph.D. at Georgetown, "and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively toward the solution to a deplorable situation. How many irate mathematicians are needed to get you to change your mind?"
"When reality clashes so violently with intuition," vos Savant responded in her column, "people are shaken." This time she tried another tack. Imagine, she said, that just after the host opened the door, revealing a booby prize, a UFO lands on the game-show stage, and a little green woman emerges. Without knowing what door you originally chose, she is asked to choose one of the two unopened doors. The odds that she'll randomly choose the car are fifty-fifty. "But that's because she lacks the advantage the original contestant had - the help of the host.... If the prize is behind No.2, the host shows you No.3; and if the prize is behind No.3, the host shows you No.2. So when you switch, you win if the prize is behind No.3 or No.2. YOU WIN EITHER WAY! But if you don't switch, you win only if the prize is behind door No.1."
Vos Savant was completely correct, as mathematicians with egg on their faces ultimately had to admit.