This is an annoying feature of the decimal representation: some numbers can be written in two apparently different ways. But the same goes for fractions: and are equal, for instance. No worries. You get used to it.
Ghost of a Departed Quantity
After decades of institutionalized denial, research mathematician reveals: .999... can be less than one, almost everywhere.
It took mathematicians centuries of effort to hammer out a logically rigorous theory of limits, infinite series and calculus, which they called ‘analysis’. All the seductive but logically incoherent ideas about infinitely large and infinitely small numbers - infinitesimals - were safely banished. The philosopher George Berkeley had scathingly referred to infinitesimals as ‘ghosts of departed quantities’, and everyone agreed he was right. However, calculus worked anyway, thanks to limits, which exorcised the ghosts.
Infinity, big or small, was a process, not a number. You never added all the terms of an infinite series: you added a finite number, and asked how that sum behaved as the number of terms grew ever larger. You approached infinity, but you never got there. Similarly, infinitesimals don’t exist. No positive number can be smaller than any positive number, because then it has to be smaller than itself.
But, as I say somewhere else in this book, you should never give up on a good idea just because it doesn’t work. Around 1960, Abraham Robinson made some surprising discoveries at the frontiers of mathematical logic, reported in his 1966 book Non-Standard Analysis. He proved that there are extensions of the real number system (called ‘non-standard reals’ or ‘hyperreals’) that share almost all the usual properties of real numbers, except that infinite numbers and infinitesimals genuinely do exist. If n is an infinite number, then 1/n is infinitesimal - but not zero. Robinson showed that the whole of analysis can be set up for hyperreals, so that, for example, an infinite series is the sum of infinitely many terms, and you do get to infinity.
An infinitesimal is now a new kind of number that is smaller than any positive real number, but it is not itself a real number. And it is not smaller than any positive hyperreal number. But you can convert all finite hyperreals to real numbers by taking the ‘standard part’, which is the unique real number that is infinitesimally close.
There is a price to pay for all this. The proof that hyperreals exist is non-constructive - it shows they can occur, but doesn’t tell you what they are. However, any theorem about ordinary analysis that can be proved using non-standard analysis has some standard-analysis proof. So this is a new method for proving the same theorems about ordinary analysis, and it is closer to the intuition of people like Newton and Leibniz than the more technical methods introduced later.
There have been some attempts to introduce non-standard analysis into undergraduate mathematics teaching, but the approach remains a minority sport. For more information, go to: en.wikipedia.org/wiki/Non-standard_analysis
As I was writing this book, and had just finished the previous item on 0.9, Mikhail Katz emailed me a paper, written with Karin Usadi Katz, that uses non-standard analysis to place that expression in a different light. They point out that in ordinary analysis there is an exact formula
for any finite decimal 0.999 . . . 9. Now let n be an infinite hyperreal. The same formula holds, but when n is infinite, ()n is not zero, but infinitesimal. The departed quantity does indeed leave behind a ghost.
Similar remarks hold for the infinite series that represents 0.. None of this contradicts what I said earlier about 0.9 and 0.3, because then I was talking about standard analysis, and the standard part of 1 - ()n is 1 when n is infinite. But it shows that the intuitive feeling some people have, that ‘there’s a little bit missing’, can be given a rigorous justification if it is interpreted in an entirely reasonable way. I don’t think we should teach that approach in school, but it should make us more sympathetic to anyone who suffers from that particular difficulty.
Katz and Katz’s paper contains a lot more about this issue, and poses the key question: ‘What does the teacher mean to happen exactly after nine, nine, nine when he writes dot, dot, dot?’ The standard analysis answer is to take ‘...’ as indicating passage to a limit. But in non-standard analysis there are many different interpretations. The traditional one assigns the largest possible sensible value to the expression - which is 1. But there are others.
Nice Little Earner
Smith and Jones were hired at the same time by Stainsbury’s Superdupermarket, with a starting salary of £10,000 per year. Every six months, Smith’s pay rose by £500 compared with that for the previous 6-month period. Every year, Jones’s pay rose by £1,600 compared with that for the previous 12-month period. Three years later, who had earned more?
Answer on page 317
A Puzzle for Leonardo
In 1225, Emperor Frederick II visited Pisa, where the great mathematician Leonardo (later nicknamed Fibonacci; see Cabinet, page 98) lived. Frederick had heard of Leonardo’s reputation, and - as emperors do - he thought it would be a great idea to set up a mathematical tournament. So the emperor’s team, which consisted of John of Palermo and Theodore, but not the emperor, battled it out head-to-head with Leonardo’s team, which consisted of Leonardo.
Among the questions that the emperor’s team set Leonardo was this: find a perfect square which remains a perfect square when 5 is added or subtracted. They wanted a solution in rational numbers - that is, fractions formed by whole numbers.
Help Leonardo solve the emperor’s puzzle.
Answer on page 318. Or see the next item.
Congruent Numbers
Emperor Frederick II’s question in the previous puzzle34 leads into deep mathematical waters, and only recently have mathematicians begun to plumb their murky depths. The question is: what happens if we replace 5 by an arbitrary whole number? For which whole numbers d can we solve
y2 - d = x2 , y2 + d = z2
in rational numbers x, y, z?
Leonardo called such d ‘congruent numbers’, a term still used today despite it being a bit confusing - number theorists habitually use the word ‘congruent’ in a completely different way. Congruent numbers can be characterised as the areas of rational Pythagorean triangles - right-angled triangles with rational sides. This isn’t obvious, but it’s true: Leonardo’s method of solution, explained in the answer to the previous problem, hints at this result. If the triangle has sides a, b, c with a2 + b2 = c2, then its area is ab/2. Let y = c/2. Then a calculation shows that y2 - ab/2 and y2 + ab/2 are both perfect squares. Conversely, we can construct a Pythagorean triangle from any solution x, y, z, d, with d equal to the area.
The familiar 3-4-5 triangle has area 3×4/2 = 6, so 6 is a congruent number. Here the recipe tells us to take y = 5/2. Then
To get d = 5, we have to start with the 40-9-41 triangle, with area 180 = 5×36. Then divide by 62 = 36 to get the triangle with sides 20/3, 3/2, 41/6, whose area is 5. Now
and we have recovered Leonardo’s answer to the emperor’s question.
The question now remains: which whole numbers d can be the area of a Pythagorean triangle with rational sides? The answer is not obvious. It turns out to be linked to a different equation,
p2 = q3 - d2q
which has solutions p, q in whole numbers if and only if d is congruent.
Some numbers are congruent, some aren’t. For example, 5, 6, 7 are congruent, but 1, 2, 3, 4 aren’t. It need not be straightforward to decide: for example, 157 is a congruent number, but the simplest right triangle with area 157 has hypotenuse
The best test currently known depends on an unproved conjecture, the Birch-Swinnerton-Dyer conjecture, which is one of the Clay millennium mathematics prizes (Cabinet, page 127) with a million dollars on offer for a proof or disproof. Frederick II didn’t realise what he was starting.
Present-Minded Somewhere Else
Norbert Wiener pioneered the mathematics of random processes, as well as the new subject of cybernetics, in the first half of the 20th century. He
was a brilliant mathematician, and notorious for forgetting things. So when the family moved to a new house, his wife wrote the address on a slip of paper and gave it to him. ‘Don’t be silly, I’m not going to forget anything as important as that,’ he said, but he put the paper in his pocket anyway.
Norbert Wiener.
Later that day, Wiener became immersed in a mathematical problem, needed some paper to write on, took out the slip bearing his new address, and covered it in equations. When he had finished these rough calculations, he crumpled the paper into a ball and threw it away.
As evening approached, he recalled something about a new house but couldn’t find the slip of paper with its address. Unable to think of anything else to do, he walked to his old house, and noticed a little girl sitting outside it.
‘Pardon me, my dear, but do you happen to know where the Wieners have mov—’
‘That’s OK, Daddy. Mommy sent me to fetch you.’
It’s About Time
Crossnumber grid.
A crossnumber is like a crossword, but using numbers instead of words. All the clues for this one are about time, and are prefaced by the phrase ‘the number of ...’.
Across Down
1 Days in a normal year 1 Days in October
3 Minutes in a quarter of an hour 2 Seconds in an hour and a half
4 Seconds in one hour, 24 minutes and 3 seconds 3 Hours in a week
4 Hours in 20 days 20 hours
6 Seconds in five minutes 5 Hours in a fortnight
7 Hours in a normal year 6 Seconds in one hour 3 seconds
8 Hours in 4 days 9 Hours in a day and a half
10 Days in a leap year
Answer on page 319
Do I Avoid Kangaroos?
• The only animals in this house are cats.
• Every animal that loves to gaze at the moon is suitable for a pet.
• When I detest an animal, I avoid it.
• No animals are meat-eaters, unless they prowl by night.
• No cat fails to kill mice.
• No animals ever take to me, except those in this house.
• Kangaroos are not suitable for pets.
• Only meat-eaters kill mice.
• I detest animals that do not take to me.
• Animals that prowl at night love to gaze at the moon.
If all these statements are correct, do I avoid kangaroos, or not?
Answer on page 319
The Klein Bottle
In the late 1800s, there was a vogue for naming special surfaces after mathematicians: Kummer’s surface, for instance, was named after Ernst Eduard Kummer. The mathematicians tended to be German, and the German word for surface is Fläche, so this was the ‘Kummersche Fläche’. I’m delving into the linguistics here, because it led to a pun being used to name a mathematical concept. That still happens, but this may well have been the first occasion. The pun derives from a very similar word, Flasche, which means ‘bottle’. At any rate, the scene was set: when Felix Klein invented a bottle-shaped surface in 1882, it was naturally called the ‘Kleinsche Fläche’. And inevitably this rapidly mutated into ‘Kleinsche Flasche’ - the Klein bottle.
I don’t know whether the pun was deliberate, or a mistranslation. At any rate, the new name was so successful that even the Germans adopted it.
Klein’s surface . . .
... interpreted as a bottle.
The Klein bottle is important in topology, as an example of a surface with no edges and only one side. A conventional surface, such as the sphere - by which topologists just mean the thin skin of the sphere’s surface, and not a solid ball (which they call a ball) - has two distinct sides, an inside and an outside. You can imagine painting the inside red and the outside blue, and the two colours never meet. But you can’t do that with a Klein bottle. If you start painting what looks like the outside blue, you get to the bent tube where it becomes narrower, and if you follow that tube as it penetrates through the bulging body you end up painting what looks like the inside blue as well.
Klein invented his bottle for a reason: it came up naturally in the theory of Riemann surfaces in complex analysis, which classifies nasty kinds of behaviour - in a beautiful way - when you try to develop calculus over the complex numbers. The Klein bottle is reminiscent of an even more famous surface, the Möbius band (or strip), formed by twisting a strip of paper and gluing the ends together. The Möbius band has one side, but it also has an edge (Cabinet, page 111). The Klein bottle gets rid of the edge, which topologists find more convenient because edges can cause trouble. Especially in complex analysis.
There is a price to pay, however: the Klein bottle can’t be represented in ordinary 3D space without penetrating through itself. However, topologists don’t mind that, because they don’t represent their surfaces in 3D space anyway. They prefer to think of them as abstract forms in their own right, not relying on the existence of a surrounding space. In fact, you can fit a Klein bottle into 4D space without any interpenetration, but that brings its own difficulties.
One way to represent a Klein bottle, which doesn’t require any self-intersection, is to borrow a trick that is familiar to almost everyone nowadays from computer games. (The topologists thought of it long before, I hasten to add.) In many games, the flat rectangular computer screen is ‘wrapped round’ so that the left and right edges are in effect joined together. If an alien spaceship shoots off the right-hand edge, it immediately reappears at the left-hand edge. The top and bottom may also be wrapped round in this manner. Now, a computer screen doesn’t actually bend. Much. So the ‘wrapping round’ is purely conceptual, a figment of the programmer’s mind. But we can easily imagine that the opposite edges abut, work out what would happen if they did, and respond accordingly. And that’s what topologists do.
Specifically, they also start with a rectangle, and wrap its edges round so that in the imagination they join. But there’s a twist - literally. The top and bottom edges are wrapped round as usual, but the right edge is given a half-twist, interchanging top and bottom, before wrapping it round to meet the left edge. So when a spaceship shoots off the top, it reappears from the corresponding position at the bottom; but when it shoots off the right-hand edge, it reappears upside down and at the opposite end of the left-hand edge.
Conventional computer screen Klein bottle wrap-round. wrap-round.
Topologically, the conventional wrapped-round screen is a torus - like a car inner tube or (I have to say this because many people have never seen an inner tube, since most car tyres are tubeless) a doughnut. But only the sugary surface, not the actual dough. You can see why if you imagine what happens when you do actually join the edges - using a flexible screen. Joining top to bottom creates a cylindrical tube; then joining the ends of the tube bends it round into a closed loop.
Wrapping round without a twist creates a torus.
However, if you imagine a similar procedure for a Klein bottle, then the two ends of the cylinder don’t join up that way: one of them has to be given the opposite orientation. In 3D, one way to do this is to make it thinner, poke it through the side of the cylinder, poke it out of the open end, and then roll it back on itself like the neck of a sweater and finally join it to the other end of the cylinder. This leads to the standard ‘bottle’ shape, with a self-intersection where you poked it through. As Klein wrote: the shape ‘can be visualized by inverting a piece of a rubber tube and letting it pass through itself so that outside and inside meet’.
Joining the edges of a cylinder to make a Klein bottle.
With an extra dimension to play with, you can push the end of the cylinder off into the fourth dimension before poking it through where the cylinder would have been; then pull it back into normal 3D space once it’s inside, and carry on as normal. That way, there’s no self-intersection.
The Klein bottle has a remarkable property, which has been celebrated in a limerick, whose author - perhaps mercifully - remains unknown:
A mathemati
cian named Klein
Thought the Möbius band was divine.
Said he: ‘If you glue
The edges of two,
You’ll get a weird bottle like mine.’
Can you see how to achieve this?
Answer on page 320
For some brilliant visualisations, go to:
plus.maths.org/issue26/features/mathart/index-gifd.html
Another cute factoid: any map on the Klein bottle can be coloured with at most 6 colours, so that adjacent regions have different colours. This compares with 4 colours for the sphere or plane (Cabinet, page 10) and 7 for the torus. See:
mathworld.wolfram.com/KleinBottle.html
Accounting the Digits
In the Great Celestial Number Factory, where all numbers are made, the accountants keep tabs on how many times each digit 0-9 is used, to make sure that there are adequate stocks in the warehouse. They record these counts on a standard form, like this:
Typical inventory form.
So, for example, as the digit 4 occurs 3 times, Nugent writes ‘3’ in the lower row of boxes, underneath the printed 4. Numbers are written so that they end in the right-hand box, like the example, and leading zeros may or may not occur. (None of that matters for this puzzle, but people do worry... )
Ian Stewart Page 15