Ian Stewart
Page 16
One day Nugent was filling in the form, as usual, when he suddenly noticed something remarkable: the two numbers (that is, sequences of digits) recorded in the two rows of boxes were identical.
What was the number concerned?
Answer on page 321
Multiplying with Sticks
We all know how to measure a length when our ruler or tape measure is too short. We measure as far as we can, mark the end point, then continue measuring from there, and add the distances together. This puts into practice a basic principle of Euclid’s geometry: if you place two lines end to end - pointing in the same direction - then their lengths add.
This means that you can make an adding machine from two sticks. Just make marks along the edge distances 1, 2, 3, 4, and so on; then position the sticks to perform the addition sum.
The number on the top stick is 3 more than the corresponding one on the bottom stick.
Big deal, I hear you thinking, and it’s true that this gadget isn’t terribly practical. But a close relative is - or, to be honest, was. To get it, we change the markings, replacing each number by the corresponding power of 2.
Now the numbers on the top stick are the corresponding numbers on the bottom stick, multiplied by 8. Our adding-sticks have become multiplying-sticks. This trick works because of the well-known formula
2a×2b = 2a+b
Well, that’s fantastic. Now we can multiply powers of 2.
Back in the days when computers and calculators were undreamt of, and would have been seen as magic, multiplying two numbers was really hard work. But astronomers need to do a lot of multiplications to keep track of the stars and planets. So, around 1594, James Craig, court doctor to King James VI of Scotland, told John Napier, Baron of Murchiston, about something called prosthapheiresis. It sounds painful, and in a way it was: the Danish mathematicians had discovered how to multiply numbers using a formula discovered by François Viète:
Using tables of sines and cosines, you could use this formula to turn a multiplication problem into a short series of addition problems. It was a bit complicated, but it was still quicker than conventional multiplication methods.
For years Napier had been thinking about efficient methods for doing sums, and it dawned on him that there was a better way. The formula for multiplying powers of 2 works for powers of any fixed number. That is,
na×nb = na+b
for any number n. If you set n to something close to 1, such as 1.001, then the successive powers will be very closely spaced, so any number that interests you will be close to some power of n. Now you can use the formula to convert multiplication to addition. For instance, suppose I want to multiply 3.52 by 7.85. Well, to a good approximation
(1.001)1259 = 3.52
(1.001)2062 = 7.85
Therefore,
3.52×7.85 = (1.001)1259×(1.001)2062 = (1.001)1259+2062 = (1.001)3321 = 27.64
The exact answer is 27.632. Not bad!
Pages from Napier’s logarithm tables.
For more accuracy, you should replace 1.001 by something more like 1.0000001. Then you just draw up a table of the first million or so powers of that number, and you’ve got a quick way to multiply numbers to about 9-digit accuracy, just by adding the corresponding powers. Perversely, Napier chose to use powers of 0.9999999, which is less than 1, so the numbers got smaller as the powers got larger.
Fortunately, Henry Briggs, an Oxford professor, took an interest and sorted out a better way. The upshot of all this was the concept of a logarithm, which turns the calculations back to front. For example, since (1.001)1259 = 3.52, the logarithm of 3.52 to base 1.001 is 1,259. In general, log x (to base n) is whichever number a satisfies
na = x
Now the formula for na+b can be reinterpreted as
log xy = log x + log y
whichever base you use. For practical purposes, base 10 is best, because we use decimals. Mathematicians prefer base e, which is roughly 2.71828, because it is better behaved with respect to the operations of calculus.
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 25 = 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 slipstick35 - 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:
en.wikipedia.org/wiki/Slide_rule
www.sliderule.ca/
www.sliderules.info/
As Long as I Gaze on Laplacian Sunrise
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?
Answer on page 321
Another Take on Mathematical Cats
• Did Erwin Schrödinger have a cat? Yes and no.
• Did Werner Heisenberg have a cat? I’m not sure.
• Did Kurt Gödel have a cat? If he did, we can’t prove it.
• Did Fibonacci have a cat? He certainly had a lot of rabbits.
• Did René Descartes have a cat? He thought he did.
• Did Augustin-Louis Cauchy have a cat? That’s a complex question.
• Did Georg Bernhard Riemann have a cat? That hypothesis has not yet been proved.
• Did Albert Einstein have a cat? One of his relatives did.
• Did Luitzen Brouwer have a cat? Well, he didn’t not have one.
• Did William Feller have a cat? Probably.
• Did Ronald Aylmer Fisher have a cat? The null hypothesis is rejected at the 95% level.
Bordered Prime Magic Square
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.
The Green-Tao Theorem
An arithmetic sequence36 is a list of numbers such that successive differences are all equal - for example,
17, 29, 41, 53, 65, 77, 89
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:
7, 37, 67, 97, 127, 157
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 a
ny 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:
Length K Prime arithmetic sequence (0≤ n ≤ K - 1)
3 3 + 2n
4 5 + 6n
5 5 + 6n
6 7 + 30n
7 7 + 150n
8 199 + 210n
9 199 + 210n
10 199 + 210n
11 110,437 + 13,860n
12 110,437 + 13,860n
13 4,943 + 60,060n
14 31,385,539 + 420,420n
15 115,453,391 + 41,44,140n
16 53,297,929 + 9,699,690n
17 3,430,751,869 + 8,729,721n
18 4,808,316,343 + 717,777,060n
19 8,297,644,387 + 4,180,566,390n
20 214,861,583,621 + 18,846,497,670n
21 5,749,146,449,311 + 26,004,868,890n
22 1,351,906,725,737,537,399 + 13,082,761,331,670,030n
23 117,075,039,027,693,563 + 1,460, 812,112,760n
24 468,395,662,504,823 + 45,872,132,836,530n
25 6,171,054,912,832,631 + 81,737,658,082,080n
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
2^2^2^2^2^2^2^2^100k
where a^b represents ab. 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
3, 11, 23, 71, 191, 443
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.
Peaucellier’s Linkage
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:
en.wikipedia.org/wiki/Peaucellier-Lipkin_linkage
A Better Approximation to π
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,
22/7 = 3.142857142857. . .
whereas
π = 3.141592653589. . .
A more accurate approximation is
355/113 = 3.141592920353. . .
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
3/8 = 0.375
and one that repeats over and over again is
5/12 = 0.4166666...
In a sense, the decimals for 3/8 also repeat for ever, because we can write
3/8 = 0.37500000000. . .
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.
355/113 =3.14159292035398230088495575221238938 053097345132743362831858407079646017 699115044247787610619469026548672566 37168. . .
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.
Strictly for Calculus Buffs
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 x4 (1 - x4 )/(1 + x2) 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.
The Statue of Pallas Athene
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 o
ne-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.]
Answer on page 322
How much gold?
Calculator Curiosity 3
Get your calculator, and work out:
6×6
66×66
666×666
6,666×6,666
66,666×66,666
666,666×666,666
6,666,666×6,666,666
66,666,666×66,666,666
At least, do that until your calculator runs out of digits. After which you should be able to guess what happens anyway.
Answer on page 322
Completing the Square
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:
Start here!
Answer on page 322
The Look and Say Sequence
One of the strangest sequences in mathematics was invented by John Horton Conway. It begins
1 11 21 1211 111221 312211 13112221 1113213211