Professor Stewart's Hoard of Mathematical Treasures (37 page)

All very well, but what does this have to do with sticks? Well, what we’re doing, in effect, with those powers of 2, is marking each number at a distance along the stick given by its logarithm. For example, since 2
⁵
= 32, the logarithm of 32 to base 2 is 5, so we write 32 five units along the stick.

We have now invented the slide rule, which is basically a table of logarithms written in wood. We were anticipated around 1600 by William Oughtred and others, who over the centuries added many more scales for trigonometric functions, powers, and other mathematical operations. The slide rule - colloquially called a slipstick
³⁵
- was widely used by scientists and (especially) engineers until about 40 years ago, when it was rendered obsolete by electronic calculators.

A slide rule from the sixties.

Today the slide rule is mostly a quaint reminder of the predigital age. I own two: one I used at school, mainly in physics lessons, and a bamboo one I bought in a flea market. To find out more, visit:

Pierre Simon de Laplace is best known for his work in celestial mechanics, but he was also one of the pioneers of probability theory. Now, pioneering work is often sloppy, because the basic issues haven’t been properly explored; that’s what pioneers are for, in fact.

Laplace argued that, if we observe the Sun rising every morning for
n
- 1 days, then we can infer that the probability that it will not rise the next morning is 1/n. After all, out of n mornings, it has risen on n - 1, so only 1 is left for it not to rise.

Ignoring the dodgy assumptions here, there is a reassuring deduction: since the Sun has now risen for hundreds of billions of consecutive mornings, the probability that it won’t rise tomorrow is staggeringly small.

Unfortunately, Laplace’s argument has a sting in the tail. Accepting his value for the successive probabilities, what is the probability that the Sun will always rise?

 

Recall that a magic square is a square array of numbers, such that all rows, columns and diagonals have the same sum.

Bordered prime magic square.

Allan Johnson, Jr, discovered a 7×7 magic square composed entirely of primes. Moreover, it is bordered: that is, the smaller 5×5 and 3×3 squares indicated by the bold lines in the picture are also magic.

An arithmetic sequence
³⁶
is a list of numbers such that successive differences are all equal - for example,
where each number is 12 greater than the one before. This is called the common difference.

In this particular list, which has seven terms, many numbers are prime, but some (65 and 77) aren’t. However, it is possible to find seven primes in arithmetic sequence:
with common difference 30.

Until recently, very little was known about the possible lengths of prime arithmetic sequences. There are infinitely many of length 2, because any two primes form an arithmetic sequence (there is only one difference, which equals itself) and there are infinitely many primes. In 1933 Johannes van der Corput proved that there are infinitely many prime arithmetic sequences of length 3, and there the matter rested.

Experiments, using computers when the numbers get big, found examples of prime arithmetic sequences with any length up to (as I write) 25. Here’s a table:

There are others, but these have the smallest final term for given k.

In 2004, to general astonishment, the whole topic was blown out of the water by Ben Green and Terence Tao, who proved that there exist arbitrarily long prime arithmetic sequences. Their proof combined half a dozen different areas of mathematics, and it even gave an estimate of how small the primes could be, for a given k. Namely, they need be no larger than
where a^b represents
a
^b
. These numbers are mind-bogglingly large, and it is conjectured that they are much larger than necessary, and can be replaced by k! + 1. Here
k
! =
k
× (
k
- 1)× (
k
- 2) × ‧‧‧ × 3 × 2 × 1 is the factorial of k.

This theorem has many consequences. It implies that there exist arbitrarily large magic squares in which every row and every column consist of primes in arithmetic sequence. Indeed, the same goes for magic d-dimensional hypercubes, for any d.

In 1990, before Green and Tao proved their theorem, Antal Balog proved that, if that result were correct, then there would exist arbitrarily large sets of primes with the curious feature that the average of any two of them is also prime - and all these averages are different. For example, the six primes
form such a set, with all 15 averages (such as (3 + 11)/2 = 7 and (23 + 443)/2 = 233) being distinct primes. So now Balog’s result is proved as well.

In the opposite direction, it has been known for a long time that every prime arithmetic sequence has finite length. That is, if you continue any arithmetic sequence for long enough you will hit a number that is not prime. This doesn’t contradict the Green-Tao Theorem, because some other arithmetic sequence could contain more primes. So all lengths here are finite, but there is no upper limit to their sizes.

In the early days of steam engines, there was a lot of interest in mechanical linkages that could turn rotary motion into straight-line motion, such as a wheel driving a pump. One of the neatest arrangements, which is mathematically exact, is Peaucellier’s linkage, invented in 1864 by the French army officer Charles-Nicolas Peaucellier. It was invented independently by a Lithuanian named Lippman Lipkin.

Peaucellier’s linkage.

The two black blobs are fixed pins that let the links rotate; the grey ones are pins that link the rods together, also allowing them to rotate. The two rods marked a have the same length, and the four rods marked b have the same length. As pin
X
moves round the circle - which it must do because one rod is fixed to the centre of the circle - pin
Y
moves up and down along the straight line drawn in grey. The linkage limits the position of
X
to an arc of the circle, so
Y
is limited to a segment of the line.

The (fairly complicated) proof that it works, an animation of the linkage, and an explanation of the deeper mathematical ideas behind it can be found at:

The famous approximation to π is 22/7, which is convenient for school calculations because it’s nice and simple. It is
not
exact - in decimals,
whereas

A more accurate approximation is
which agrees with π to six decimal places - not bad for such a simple fraction. In fact, there is a rigorous sense in which 355/113 is the best approximation to π using numbers of that size.

The decimal for 22/7 keeps repeating the same sequence of digits, 142857, indefinitely. As mentioned on page 172, this is a general feature of fractions: if you write a fraction as a decimal, then either it stops, or it ‘recurs’: it goes on for ever, repeating the same string of digits over and over again. Conversely, all decimals that stop or recur are equal to exact fractions.

An example of a fraction whose decimal representation stops is
and one that repeats over and over again is

In a sense, the decimals for 3/8 also repeat for ever, because we can write
with a repeating string 0. But terminating zeros are usually omitted.

It may not look as though the decimal for 355/113 repeats,
but actually it does - after the 112th decimal place! It is no coincidence that 112 = 113 - 1, but it would take too long to explain why. If you take the calculation that far, you’ll get after which the digits repeat again, starting from immediately after the decimal point.