Professor Stewart's Hoard of Mathematical Treasures (38 page)

Because π is irrational - not equal to an exact fraction - its decimal expansion never repeats the same block of digits over and over again. This was proved in 1770 by Johann Lambert.

The next two approximations to π are 103,993/33,102 and 104,348/33,215.

In 1944, D. P. Dalzell published a short note containing the curious formula
which relates π and its commonest approximation, 22/7, to an integral. You can verify the formula using no more than school calculus, because
where the integral of each term is a standard result. The last term gives π and the rest give 22/7. This particular formula is significant, though, because the function being integrated is positive in the range from 0 to 1. The integral from 0 to 1 is just the average value, so this must also be positive. Since the function concerned is not always zero, we deduce that π is less
than 22/7. This is a fairly simple way to prove that the usual approximation is not exact.

The formula also leads to an estimate of the error, because the maximum value of
x
⁴
(1 -
x
⁴
)/(1 +
x
²
) between 0 and 1 is 1/256, so the average is at most 1/256. Therefore
With more effort you can prove that the error is at most 1/630.

This formula turns out to be part of a more extensive story (see page 322 for references). In 2005, Stephen Lucas started thinking about the improved approximation to π, 355/113, which we’ve just encountered. Lucas found the formula
which in the circumstances is quite elegant. Again the function being integrated is positive, so the formula proves that π is (slightly) smaller than 355/113.

According to a puzzle book published in the Middle Ages, the statue of the goddess Pallas Athene was inscribed with the following information:

‘I, Pallas, am made from the purest gold, donated by five generous poets. Kariseus gave half; Thespian an eighth. Solon gave one-tenth; Themison gave one-twentieth. And the remaining nine talents’ worth of gold was provided by the good Aristodokos.’

How much did the statue cost in total? [A talent is a unit of weight, roughly 1 kilogram.]

 

How much gold?

Get your calculator, and work out:

At least, do that until your calculator runs out of digits. After which you should be able to guess what happens anyway.

 

The traditional 3×3 magic square looks like this.

The traditional magic square.

Each cell contains a different number, and each row, column and diagonal sums to 15.

Your task is to find a square satisfying the same conditions, but with an 8 at top centre, like this:

Answer on page 322

One of the strangest sequences in mathematics was invented by John Horton Conway. It begins

Answers on page 323

The things of this world cannot be made known without a knowledge of mathematics. Roger Bacon

 

I had a feeling once about Mathematics - that I saw it all ... I saw - as one might see the transit of Venus or even the Lord Mayor’s Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable, but it was after dinner and I let it go. Sir Winston Spencer Churchill

 

Mathematics seems to endow one with something like a new sense. Charles Darwin

 

For a physicist, mathematics is not just a tool by means of which phenomena can be calculated; it is the main source of concepts and principles by means of which new theories can be created. Freeman Dyson

 

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. Albert Einstein

 

Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. Stephen Hawking

 

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. Robert A. Heinlein

 

Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff to any degree of fineness; but, nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulae will not get a definite result out of loose data. Thomas Henry Huxley

Medicine makes people ill, mathematics make them sad, and theology makes them sinful. Martin Luther

 

I tell them that, if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh. Thomas Mann

 

The greatest unsolved theorem in mathematics is why some people are better at it than others. Adrian Mathesis
³⁷

She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and-so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had come easy to Scarlett in her schooldays. Margaret Mitchell

 

The advancement and perfection of mathematics are intimately connected with the prosperity of the State. Napoleon I

 

Mathematical propositions express no thoughts ... we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. Ludwig Wittgenstein

 

[Mathematics] is an independent world. Created out of pure intelligence. William Wordsworth

 

I’m sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it. Malcolm
X

 

Like the crest of a peacock, so is mathematics at the head of all knowledge. An old Indian saying

Fermat’s Last Theorem states that two non-zero integer cubes can’t add up to a cube, and ditto for fourth, fifth or higher powers. It was famously proved by Andrew Wiles in 1994-5 (Cabinet, page 50). One of the first people to make inroads into the problem was Euler, who proved the Last Theorem for cubes: two non-zero cubes cannot add up to a cube. But he also noticed that three cubes can add up to a cube. In fact,
Euler guessed (the fancy word is ‘conjectured’) that you need to add at least four fourth powers to get a fourth power, at least five fifth powers to get a fifth power, and so on.