Professor Stewart's Hoard of Mathematical Treasures (34 page)

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.

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?

 

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.

Emperor Frederick II’s question in the previous puzzle
³⁴
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
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
a
²
+
b
²
=
c
²
, then its area is ab/2. Let
y
=
c
/2. Then a calculation shows that
y
²
- ab/2 and
y
²
+ 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 6
²
= 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.