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50

mathematical ideas
you really need to know

Tony Crilly

2

Contents
Introduction
01 Zero
02 Number systems
03 Fractions
04 Squares and square roots
05 π
06 e
07 Infinity
08 Imaginary numbers
09 Primes
10 Perfect numbers
11 Fibonacci numbers
12 Golden rectangles
13 Pascal’s triangle
14 Algebra
15 Euclid’s algorithm
16 Logic
17 Proof

3

18 Sets
19 Calculus
20 Constructions
21 Triangles
22 Curves
23 Topology
24 Dimension
25 Fractals
26 Chaos
27 The parallel postulate
28 Discrete geometry
29 Graphs
30 The four-colour problem
31 Probability
32 Bayes’s theory
33 The birthday problem
34 Distributions
35 The normal curve
36 Connecting data
37 Genetics
38 Groups
4

39 Matrices
40 Codes
41 Advanced counting
42 Magic squares
43 Latin squares
44 Money mathematics
45 The diet problem
46 The travelling salesperson
47 Game theory
48 Relativity
49 Fermat’s last theorem
50 The Riemann hypothesis
Glossary
Index

5

Introduction
Mathematics is a vast subject and no one can possibly know it all
...
The possibilities open to us here will lead to other times and different
cultures and to ideas that have intrigued mathematicians for centuries
...
From India and Arabia we derive our modern numbering system but it is one tempered
with historical barnacles
...

The technological triumphs of the modern age depend on mathematics and surely there is no
longer any pride left in announcing to have been no good at it when at school
...
The time pressure of
school does not help either, for mathematics is a subject where there is no merit in being fast
...
Some of the greatest mathematicians have been
painfully slow as they strove to understand the deep concepts of their subject
...
It can be dipped into at leisure
...
Beginning with Zero, or elsewhere if you wish,
you can move on a trip between islands of mathematical ideas
...
Alternatively you can move
from Golden rectangles to the famous Fermat’s last theorem, or any other path
...
Some of its major problems have been solved in recent
times
...

The Four-colour problem was solved with the aid of a computer, but the Riemann hypothesis, the
final chapter of the book, remains unsolved – by computer or any other means
...
The popularity of Sudoku is evidence that people can do mathematics
(without knowing it) and enjoy it too
...
You will see several leaders making entrances and exits in
some chapters only to reappear in others
...
But, real progress in mathematics is the work of ‘the many’
accumulated over centuries
...
There are everyday and advanced items, pure and applied mathematics, abstract and
concrete, the old and the new
...
There could have been 500 ideas but 50
are enough for a good beginning to your mathematical career
...
We learn that 1 is first
in the ‘number alphabet’, and that it introduces the counting numbers 1, 2, 3, 4, 5,
...
It is only later that we can count the number of apples in a box when there are
none
...
They failed
to give ‘nothing’ a name
...


How did zero become accepted?
The use of a symbol designating ‘nothingness’ is thought to have originated
thousands of years ago
...
A little later, the astronomer Claudius Ptolemy, influenced by the
Babylonians, used a symbol akin to our modern 0 as a placeholder in his number
system
...
This might be compared with the introduction of the
‘comma’ into language – both help with reading the right meaning
...

The seventh-century Indian mathematician Brahmagupta treated zero as a
‘number’, not merely as a placeholder, and set out rules for dealing with it
...
In thinking of zero as a number rather than a placeholder, he
was quite advanced
...
Brought up in North
Africa and schooled in the Hindu-Arabian arithmetic, he recognized the power of
7

using the extra sign 0 combined with the Hindu symbols 1, 2, 3, 4, 5, 6, 7, 8 and
9
...
How could zero be integrated
into the existing system of arithmetic in a more precise way? Some adjustments
were straightforward
...
Meanings were needed to ensure that 0 harmonized with the rest of
accepted arithmetic
...
Adding 0 to a number leaves that
number unchanged while multiplying 0 by any number always gives 0 as the
answer
...
Subtraction is a simple
operation but can lead to negatives, 7 0 = 7 and 0 7 = 7, while division
involving zero raises difficulties
...
Suppose the
measuring rod is actually 7 units in length
...
If the length to be measured
is actually 28 units the answer is 28 divided by 7 or in symbols 2 8 ÷ 7 = 4
...
What now can be made of 0 divided by 7? To help suggest an answer
in this case let us call the answer a so that

By cross-multiplication this is equivalent to 0 = 7 × a
...
Clearly it is not 7 so a must be a zero
...
The danger point is division by 0
...
By admitting the possibility of 7/0 being a number we have the potential for
numerical mayhem on a grand scale
...
It is not permissible to get any sense from the operation of dividing 7
(or any other nonzero number) by 0 and so we simply do not allow this
operation to take place
...

The 12th-century Indian mathematician Bhaskara, following in the footsteps of
Brahmagupta, considered division by 0 and suggested that a number divided by
0 was infinite
...
For example, 7 divided by a tenth is 70, and by
a hundredth is 700
...
In the ultimate smallness, 0 itself, the answer
should be infinity
...
Wrestling with
infinity does not help; infinity (with its standard notation ∞) does not conform to
the usual rules of arithmetic and is not a number in the usual sense
...
This is not particularly illuminating but it is not nonsense
either
...
We
reach the conclusion that 0/0 can be anything; in polite mathematical circles it is
called ‘indeterminate’
...
Arithmetic can
be conducted quite happily without it
...
The progress of science has depended on
it
...
It has entered the non-scientific
language with such ideas as the zero-hour and zero-tolerance
...
If you step off the 5th Ave sidewalk
in New York City and into the Empire State Building, you are in the magnificent
entrance lobby on Floor Number 1
...
’
In Europe they do have a Floor 0 but there is a reluctance to call it that
...
It is in the kernel of
mathematical concepts which make the number system, algebra, and geometry
go round
...
In the
decimal system, zero serves as a place holder which enables us to use both huge
numbers and microscopic figures
...
The 19th-century American
mathematician G
...
Halsted adapted Shakespeare’s Midsummer Night’s Dream to
write of it as the engine of progress that gives ‘to airy nothing, not merely a local
habitation and a name, a picture, a symbol, but helpful power, is the
characteristic of the Hindu race from whence it sprang’
...
The modern day equivalent occurs in set theory where the concept of a set
is a collection of elements
...
Now that is an odd idea, but like 0 it is
indispensible
...
Different cultures
at differing periods of time have adopted various methods, ranging from the basic ‘one,
two, three, many’ to the highly sophisticated decimal positional notation we use today
...
We call it a place-value system because you can tell the ‘number’
by the positioning of a symbol
...
Vestiges of base 60 are still with us: 60 seconds in
a minute, 60 minutes in an hour
...

While our ancient ancestors primarily wanted numbers for practical ends, there
is some evidence that these early cultures were intrigued by mathematics itself,
and they took time off from the practicalities of life to explore them
...

The Egyptian system from the 13th century BC used base ten with a system of
hieroglyphic signs
...
Where it has the advantage is the way it can be used
to express both very small and very large numbers
...
To see this let’s look at the Roman system
...


The Roman system
The basic symbols used by the Romans were the ‘tens’ (I, X, C and M), and
the ‘halves’ of these (V, L and D)
...
It
has been suggested that the use of I, II, III and IIII derives from the appearance
of our fingers, V from the shape of the hand, and by inverting it and joining the
12

two together to form the X we get two hands or ten fingers
...
The Romans also used S for ‘a half’ and a system of fractions based
on 12
...
The ancient Romans preferred to write IIII with IV only being
introduced later
...
For example, the meaning of
MMMCDXLIIII only becomes transparent when brackets are mentally introduced
so that (MMM)(CD)(XL)(IIII) is then read as 3000 + 400 + 40 + 4 = 3444
...
A Roman skilled in the art would have
short cuts and tricks, but for us it’s difficult to obtain the right answer without
first calculating it in the decimal system and translating the result back into
Roman notation:

13

The multiplication of two numbers is much more difficult and might be
impossible within the basic system, even to Romans! To multiply 3444 × 394 we
need the medieval appendages
...
If you asked a vegetarian citizen
of Rome to record how many bottles of wine he’d consumed that day, he might
write III but if you asked him how many chickens he’d eaten, he couldn’t write 0
...
Some constructions were
never used by the Romans, like MCM for 1900, but were introduced for stylistic
reasons in modern times
...
The
fourteenth King Louis of France, now universally known as Louis XIV, actually
preferred to be known as Louis XIIII and made it a rule that his clocks were to
show 4 o’clock as IIII o’clock
...
The decimal system is
based on ten using the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
...
When we
write down the number 394, we can explain its decimal meaning by saying it is
composed of 3 hundreds, 9 tens and 4 units, and we could write
394 = 3 × 100 + 9 × 10 + 4 × 1

This can be written using ‘powers’ of 10 (also known as ‘exponentials’ or
‘indices’),
394 = 3 × 102 + 9 × 101 + 4 × 100
2

where 10 = 10 × 10, 101 = 10 and we agree separately that 100 = 1
...


The point of decimal
So far we have looked at representing whole numbers
...
572 where the decimal point indicates the beginning
of the negative powers of 10
...
572
...
For example, 1,356,936,892 can be written as
1
...
356936892 × 10E9’ on calculators
or computers
...
Sometimes we might want to
use bigger numbers still, for instance if we were talking about the number of
hydrogen atoms in the known universe
...
7×1077
...
7×10−77, with a negative power, is a very small number and
this too is easily handled using scientific notation
...


Zeros and ones
While base 10 is common currency in everyday life, some applications require
other bases
...
The beauty of binary is that any number can be expressed
using only the symbols 0 and 1
...

Powers of 2

Decimal

20

1

21

2

22

4

23

8

24

16

25

32

26

64

27

128

28

256

29

512

210

1024

How can we express 394 in binary notation? This time we are dealing with
powers of 2 and after some working out we can give the full expression as,
394 =1×256+1×128+0×64+0×32+0×16+1×8+0×4+1×2+0×1

so that reading off the zeros and ones, 394 in binary is 110001010
...
These are the octal system (base 8) and the hexadecimal system
16

(base 16)
...
In this base 16 system, we customarily
use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
...
It’s as easy as ABC,
which bear in mind, is really 2748 in decimal!

the condensed idea
Writing numbers down

17

03

Fractions

A fraction is a ‘fractured number’ – literally
...
Let’s take the traditional example, the
celebrated cake, and break it into three parts
...
The unlucky person only gets ⅓
...

Here is another example
...
Here the fraction is written as ⅘
...
That would be written as
⅕ and we see that ⅕ + ⅘ = 1 where 1 represents the original price
...
The
bottom number is called the ‘denominator’ because it tells us how many parts
make the whole
...
So a fraction in established notation always looks
like

In the case of the cake, the fraction you might want to eat is ⅔ where the
denominator is 3 and the numerator is 2
...

We can also have fractions like 14/5 (called improper fractions) where the
18

numerator is bigger than the denominator
...
This comprises the whole
number 2 and the ‘proper’ fraction ⅘
...

Fractions are usually represented in a form where the numerator and
denominator (the ‘top’ and the ‘bottom’) have no common factors
...
If we write the fraction 8/10 = 2×4/2×5 we can ‘cancel’ the
2s out and so 8/10 = ⅘, a simpler form with the same value
...

The rational numbers were the numbers the Greeks could ‘measure’
...
Multiplication of whole numbers is so troublesome that ingenious ways
had to be invented to do it
...

Let’s start by multiplying fractions
...
The £30 is divided
into five parts of £6 each and four of these five parts is 4 × 6 = 24, the amount
you pay for the shirt
...
If you go into the shop you can now get the shirt for £12
...
To multiply two fractions together you just multiply
the denominators together and the numerators together:

If the manager had made the two reductions at a single stroke he would have
advertised the shirts at four-tenths of the original price of £30
...

Adding two fractions is a different proposition
...
We simply add the two numerators
together to get 3/3, or 1
...

Adding fractions requires a different approach
...
First
multiply the top and bottom of ⅔ by 5 to get 10/15
...
Now both fractions have 15 as a common
denominator and to add them we just add the new numerators together:

Converting to decimals
In the world of science and most applications of mathematics, decimals are
the preferred way of expressing fractions
...
8
...
But how
could we convert, say ⅞, into decimal form? All we need to know is that when
we divide a whole number by another, either it goes in exactly or it goes in a
certain number of times with something left over, which we call the ‘remainder’
...
It doesn’t go, or you could say it goes 0 times with
remainder 7
...
’
• Now divide 8 into 70 (the remainder of the previous step multiplied by 10)
...

So we write this alongside our first step, to make ‘0
...

Because 7 × 8 = 56, the answer is 7 with remainder 4
...
87’
• Divide 8 into 40 (the remainder of the previous step multiplied by 10)
...
When we get remainder 0 the recipe is
complete
...
The final answer is ‘0
...

20

When applying this conversion recipe to other fractions it is possible that we
might never finish! We could keep going forever; if we try to convert ⅔ into
decimal, for instance, we find that at each stage the result of dividing 20 by 3 is 6
with a remainder of 2
...
In this case we have the infinite decimal
0
...
6 to indicate the ‘recurring decimal’
...
The fraction 5/7 is
interesting
...
714285714285714285
...
If any fraction results in a repeating
sequence we cannot ever write it down in a terminating decimal and the ‘dotty’
notation comes into its own
...


Egyptian fractions

Egyptian fractions
The Egyptians of the second millennium
21

BC

based their system of fractions on

hieroglyphs designating unit fractions – those fractions whose numerators are 1
...
It was
such a complicated system of fractions that only those trained in its use could
know its inner secrets and make the correct calculations
...
These were
their ‘basic fractions’ from which all other fractions could be expressed
...
A feature of the system is that
there may be more than one way of writing a fraction, and some ways are
shorter than others
...
For instance, a full analysis of
the methods for finding the shortest Egyptian expansion awaits the intrepid
mathematical explorer
...
This activity was prized by the fraternity who followed their leader
Pythagoras, a man best remembered for ‘that theorem’
...
Pythagoreans
believed mathematics was the key to the nature of the universe
...
To the Pythagoreans 1 was the most important number, imbued with
spiritual existence
...
Continuing to count up the dots
of the subsequent squares gives us the ‘square’ numbers, 1, 4, 9, 16, 25, 36, 49,
64,… These are called ‘perfect’ squares
...
The Pythagoreans didn’t stop with squares
...


The triangular numbers resemble a pile of stones
...
What
is the triangular number which comes after 10, for instance? It will have 5 dots in
the last row so we just add 10 + 5 = 15
...
But there is a more striking link
...

Adding two successive triangular numbers

That’s right! When you add two successive triangular numbers together you
get a square number
...

Consider a square made up of 4 rows of 4 dots with a diagonal line drawn
through it
...
This observation holds for any sized
square
...
The area
of a square whose side is 4 is 4 × 4 = 42 = 16 square units
...


The square x2 is the basis for the parabolic shape
...
A parabola
has a focus point
...

In a car headlight a light bulb at the focus sends out a parallel beam of light
...


Square roots
If we turn the question around and want to find the length of a square which
has a given area 16, the answer is plainly 4
...
The symbol √ for square roots has been employed since the
1500s
...
For
example,
,
,
,
,
, and so on
...
These are 2,
3, 5, 6, 7, 8, 10, 11, …

There is a brilliant piece of alternative notation for square roots
...
This is the
basis for logarithms, invented after we learnt in around 1600 that a problem in
multiplication could be changed into one of addition
...

25

These numbers all have square roots, but they are not equal to whole numbers
...

Let’s look at

...
If you work out
on your calculator you will get 1
...
Is this the square root
of 2? To check we make the calculation 1
...
414213562
...
999999999
...
414213562 is only an
approximation for the square root of 2
...
The number
is important in mathematics, perhaps not quite
as illustrious as π or e (see pages 20–27) but important enough to gets its own
name – it is sometimes called the ‘Pythagorean number’
...
Suppose we have a line AB whose
length we wish to measure, and an indivisible ‘unit’ CD with which to measure it
...
If we
place the unit down m times and the end of the last unit fits flush with the end of
AB (at the point B) then the length of AB will simply be m
...

The Greeks believed that at some point using n copies of AB and m units, the
26

unit would fit flush with the end-point of the mth AB
...
For example if 3 copies of AB are laid side by side and 29 units fit
alongside, the length of AB would be 29/3
...
By
Pythagoras’s theorem the length of AB could be written symbolically as
so the
question is whether
?
From our calculator, we have already seen that the decimal expression for
is potentially infinite, and this fact (that there is no end to the decimal
expression) perhaps indicates that
is not a fraction
...
3333333… and that represents the fraction ⅓
...


Is

a fraction?

This brings us to one of the most famous proofs in mathematics
...
Firstly it is assumed that
cannot be a fraction and ‘not a fraction’ at
the same time
...
There is no
middle way in this logic
...
They assumed
that it was a fraction and, by strict logic at every step, derived a contradiction, an
‘absurdity’
...
Suppose We can assume a bit more too
...
This is OK because if
they did have common factors these could be cancelled before we began
...
)
2
We can square both sides of
to get 2 = m /n2 and so m 2 = 2n2
...
Next m itself cannot be odd (because the square of an odd
number is odd) and so m is also an even number
...
As m is even it must be twice something which
we can write as m = 2k
...

Combining this with the fact that m 2 = 2n2 means that 2n2 = 4k2 and on
cancellation of 2 we conclude that n2 = 2k2
...
We have thus deduced
by strict logic that both m and n are both even and so they have a factor of 2 in
common
...
The conclusion therefore is that
cannot be a fraction
...
Numbers which cannot be expressed as
fractions are called ‘irrational’ numbers, so we have observed there are an infinite
number of irrational numbers
...
Forget all the other constants of nature, π
will always come at the top of the list
...


π or pi, is the length of the outside of a circle (the circumference) divided by
the length across its centre (the diameter)
...
Whether the circle is big or
small, π is indeed a mathematical constant
...

For a circle of diameter d and radius r :
circumference = πd = 2πr area = πr2
For a sphere of diameter d and radius r :
surface area = πd2 = 4 πr2
volume = 4/3 πr3

Archimedes of Syracuse
The ratio of the circumference to the diameter of a circle was a subject of
ancient interest
...

It was Archimedes of Syracuse who made a real start on the mathematical
theory of π in around 225 BC
...

Mathematicians love to rate their co-workers and they place him on a level with
Carl Friedrich Gauss (The ‘Prince of Mathematicians’) and Sir Isaac Newton
...
He was hardly an ivory tower figure though – as well
as his contributions to astronomy, mathematics and physics, he also designed
weapons of war, such as catapults, levers and ‘burning mirrors’, all used to help
keep the Romans at bay
...


Given that π is defined as the ratio of its circumference to its diameter, what
does it have to do with the area of a circle? It is a deduction that the area of a
circle of radius r is πr2 , though this is probably better known than the
circumference/diameter definition of π
...


How can this be shown? The circle can be split up into a number of narrow
equal triangles with base length b whose height is approximately the radius r
...

Let’s take 1000 triangles for a start
...
We can join together each adjacent pair of these triangles to
form a rectangle (approximately) with area b × r so that the total area of the
polygon will be 500 × b × r
...
The more triangles we take
the closer will be the approximation and in the limit we conclude the area of the
30

circle is
...

Archimedes estimated the value of π as bounded between
and

...
The honour for designating the actual symbol π goes to the little known
William Jones, a Welsh mathematician who became Vice President of the Royal
Society of London in the 18th century
...


The exact value of π
We can never know the exact value of π because it is an irrational number, a
fact proved by Johann Lambert in 1768
...

The
first
20
decimal
places
are
3
...
16227766016837933199 and this was adopted around A D
500 by Brahmagupta
...

π can be computed from a series of numbers
...
Euler
found a remarkable series that converges to π:

The self-taught genius Srinivasa Ramanujan devised some spectacular
approximating formulae for π
...
While Lambert had proved it could not be
a fraction, in 1882 the German mathematician Ferdinand von Lindemann solved
the most outstanding problem associated with π
...
By solving this ‘riddle of the ages’
31

Lindemann concluded the problem of ‘squaring the circle’
...
Lindemann proved conclusively that it cannot be
done
...

The actual calculation of π continued apace
...
In modern times
the quest for calculating π to more and more decimal places gained momentum
through the modern computer
...
By 2002, π had been
computed to a staggering 1,241,100,000,000 places, but it is an ever growing
tail
...
Are the digits of π
random? Is it possible to find a predetermined sequence in the expansion? For
instance, is it possible to find the sequence 0123456789 in the expansion? In the
1950s this seemed unknowable
...
L
...
J
...
In
fact these digits were found in 1997 beginning at the position 17,387,594,880,
or, using the equator metaphor, about 3000 miles before one lap is completed
...

π in poetry
If you really want to remember the first values in the expansion of π perhaps a little poetry will
help
...


32

The letter count of each successive word in Keith’s version provides the first 740
digits of π
...
But the extensive calculations are not just for fun
...

Perhaps the strangest episode in the story of π was the attempt in the Indiana
State Legislature to pass a bill that would fix its value
...
J
...
A practical problem encountered in this piece of legislation
was the proposer’s inability to fix the value he wanted
...
While π is more august
and has a grand past dating back to the Babylonians, e is not so weighed down by the
barnacles of history
...
Whether it’s populations, money or other physical quantities,
growth invariably involves e
...
71828
...
It came to light in the early 17th century when several
mathematicians put their energies into clarifying the idea of a logarithm, the
brilliant invention that allowed the multiplication of large numbers to be
converted into addition
...
Jacob
Bernoulli was one of the illustrious Bernoullis of Switzerland, a family which
made it their business to supply a dynasty of mathematicians to the world
...


Money, money, money
Suppose we consider a 1-year time period, an interest rate of a whopping
100%, and an initial deposit (called a ‘principal’ sum) of £1
...
Likewise, if we have greater
principal sums like £10,000 we can multiply everything we do by 10,000
...
So we shall have the
princely sum of £2
...
For the first half-year we gain an interest
of 50 pence and our principal has grown to £1
...
So, by the end of the full year we would have this amount and the 75
pence interest on this sum
...
25 by the end of the year! By
compounding the interest each half-year we have made an extra 25 pence
...
By compounding every half-year we gain an extra
£250
...
Carrying
out a similar calculation, we find that our £1 has grown to £2
...
Our money
is growing and with our £10,000 it would seem to be advantageous if we could
split up the year and apply the smaller percentage interest rates to the smaller
time intervals
...
Of course, the only realistic compounding period is per day
(and this is what banks do)
...
Is this a good thing or a bad thing? You know the answer: if you
are saving, ‘yes’; if you owe money, ‘no’
...


The exact value of e
Like π, e is an irrational number so, as with π, we cannot know its exact value
...
71828182845904523536…
Using only fractions, the best approximation to the value of e is 87/32 if the
top and bottom of the fraction are limited to two-digit numbers
...

This second fraction is a sort of palindromic extension of the first one –
mathematics has a habit of offering these little surprises
...
In this, for
example, 5! = 5×4×3×2×1
...
In its mathematical
properties, e appears more ‘symmetric’ than π
...
If you know your
American history then you might remember that e is ‘2
...
There are many such devices for
remembering e but their interest lies in their quaintness rather than any
mathematical advantage
...
In
1840, French mathematician Joseph Liouville showed that e was not the solution
of any quadratic equation and in 1873, in a path-breaking work, his countryman
Charles Hermite, proved that e is transcendental (it cannot be the solution of any
algebraic equation)
...

Nine years later, Ferdinand von Lindemann adapted Hermites’s method to prove
that π was transcendental, a problem with a much higher profile
...
Is e raised to the power
of e transcendental? It is such a bizarre expression, how could this be otherwise?
Yet this has not been proved rigorously and, by the strict standards of
mathematics, it must still be classified as a conjecture
...
Close, but not close enough
...
The values of eπ and πe are
36

close but it is easily shown (without actually calculating their values) that eπ > πe
...
14069 and πe = 22
...

The number eπ is known as Gelfond’s constant (named after the Russian
mathematician Aleksandr Gelfond) and has been shown to be a transcendental
...


Is e important?
The chief place where e is found is in growth
...
Connected with this are the curves depending on
e used to model radioactive decay
...
Pierre
Montmort investigated a probability problem in the 18th century and it has since
been studied extensively
...
What is the probability that no one
gets their own hat?
It can be shown that this probability is 1/e (about 37%) so that the probability
of at least one person getting their own hat is 1 – 1/e (63%)
...
The Poisson distribution which deals with rare
events is another
...
The list is endless
...

When we think of the famous numbers of mathematics we think of 0, 1, π, e and
the imaginary number i = √–1
...

Perhaps e’s real importance lies in the mystery by which it has captivated
generations of mathematicians
...
Just why an author
like E
...
Wright should put himself through the effort of writing an e-less novel –
presumably he had a pen name too – but his Gadsby is just that
...


the condensed idea
The most natural of numbers

38

07

Infinity

How big is infinity? The short answer is that ∞ (the symbol for infinity) is very big
...
For every huge number produced, say 101000 , there is always a bigger one,
such as 101000 + 1
...

Mathematics uses infinity in any which way, but care has to be taken in treating
infinity like an ordinary number
...


Counting
The German mathematician Georg Cantor gave us an entirely different concept
of infinity
...
The idea on which Cantor’s theory depends has to
do with a primitive notion of counting, simpler than the one we use in everyday
affairs
...
How would
he know how many sheep he had? Simple – when he lets his sheep out in the
morning he can tell whether they are all back in the evening by pairing each
sheep with a stone from a pile at the gate of his field
...
Even without using numbers, the farmer is being
very mathematical
...
This primitive idea has some surprising consequences
...
For
example N = {1, 2, 3, 4, 5, 6, 7, 8,
...
Once we have a set, we can talk about subsets, which are smaller sets
within the larger set
...
} and E = {2, 4, 6, 8,
...
If we were to ask ‘is there the
same number of odd numbers as even numbers?’ what would be our answer?
Though we cannot do this by counting the elements in each set and comparing
answers, the answer would still surely be ‘yes’
...

Cantor would agree with the answer, but would give a different reason
...

The idea that both sets O and E have the same number of elements is based on
the pairing of each odd number with an even number:

If we were to ask the further question ‘is there the same number of whole
numbers as even numbers?’ the answer might be ‘no’, the argument being that
the set N has twice as many numbers as the set of even numbers on its own
...
We could do better with the one-to-one
correspondence idea
...


Cardinality
The number of elements in a set is called its ‘cardinality’
...
The cardinality
of the set {a, b, c, d, e} is 5 and this is written as card{a, b, c, d, e} = 5
...
For the cardinality of the whole
numbers N, and any set in a one-to-one correspondence with N, Cantor used the
40

symbol
(ℵ or ‘aleph’ is from the Hebrew alphabet; the symbol
is read as
‘aleph nought’)
...

Any set which can be put into a one-to-one correspondence with N is called a
‘countably infinite’ set
...
For example, the list of odd numbers is simply 1, 3, 5,
7, 9,
...


Are the fractions countably infinite?
The set of fractions Q is a larger set than N in the sense that N can be thought
of as a subset of Q
...
Nevertheless it can be done
...
To start, we write
down a row of all the whole numbers, positive and negative alternately
...
Below this row we write those fractions
which have 3 as denominator, again omitting those which have already been
recorded
...
For example, 209/67 is in
the 67th row, around 200 places to the right of 1/67
...
If we start on the top row and move to the right at each
step we will never get to the second row
...
Starting at 1, the promised linear list
begins: 1, −1, ½, ⅓, −½, 2, −2, and follows the arrows
...
So we can conclude that the set of
fractions Q is countably infinite and write card(Q) =

...
These are
the irrational numbers – they ‘fill in the gaps’ to give us the real number line R
...
So, how
could we make a list of the real numbers? In a move of sheer brilliance, Cantor
showed that even an attempt to put the real numbers between 0 and 1 into a list
is doomed to failure
...

42

Suppose you did not believe Cantor
...
500000000000000000
...
31830988618379067153
...
If you could not produce one then Cantor
would be correct
...
a1a2a3a4a5
...
b1b2b3b4b5
...
c1c2c3c4c5
...
d1d2d3d4d5
...

where x1 differs from a1 , x2 differs from b2 , x3 differs from c3 working our way
down the diagonal?’ His x differs from every number in your list in one decimal
place and so it cannot be there
...

In fact, no list is possible for the set of real numbers R, and so it is a ‘larger’
infinite set, one with a ‘higher order of infinity’, than the infinity of the set of
fractions Q
...


the condensed idea
A shower of infinities

43

08

Imaginary numbers

We can certainly imagine numbers
...
But the
mathematical use of imaginary is nothing to do with this daydreaming
...
Do imaginary numbers exist or not?
This was a question chewed over by philosophers as they focused on the word
imaginary
...
They are as much a part of everyday life as the number 5 or π
...
And
by adding a real number and an imaginary number together we obtain what’s
called a ‘complex number’, which immediately sounds less philosophically
troublesome
...
So what number, when squared, gives −1?
If you take any non-zero number and multiply it by itself (square it) you
always get a positive number
...

Even if we have forgotten the school rule that ‘two negatives make a positive’ we
may remember that the answer is either −1 or +1
...
So we must conclude −1 × −1 = 1, which is
positive
...

This caused a sticking point in the early years of complex numbers in the 16th
century
...
The development of complex numbers is the ‘completion of the real
numbers’ to a naturally more perfect system
...
When Michael
Faraday discovered alternating current in the 1830s, imaginary numbers gained a physical reality
...


The square root of –1
We have already seen that, restricted to the real number line,

there is no square root of −1 as the square of any number cannot be negative
...
Or we could take the
bold step of accepting √−1 as a new entity, which we denote by i
...
What they are we do
not know, but we believe in their existence
...
So in our
new system of numbers we have all our old friends like the real numbers 1, 2, 3,
4, π, e,
and
, with some new ones involving i such as 1 + 2i, −3 + i, 2 +
3i,
,
, e + πi and so on
...


Adding and multiplying
Now that we have complex numbers in our mind, numbers with the form a +
bi, what can we do with them? Just like real numbers, they can be added and
multiplied together
...
So 2 + 3 i
added to 8 + 4i gives (2 + 8) + (3 + 4)i with the result 10 + 7i
...
If we want to multiply 2 + 3i by 8
+ 4i we first multiply each pair of symbols together and add the resulting terms,
16, 8i, 24i and 12i2 (in this last term, we replace i2 by −1), together
...

(2 + 3i) × (8 + 4i) = (2 × 8) + (2 × 4i) + (3i × 8) + (3i × 4i)

With complex numbers, all the ordinary rules of arithmetic are satisfied
...
In fact the complex
numbers enjoy all the properties of the real numbers save one
...


The Argand diagram
The two-dimensionality of complex numbers is clearly seen by representing
them on a diagram
...

Every complex number has a ‘mate’ officially called its ‘conjugate’
...

The mate of 1 − 2i, by the same token, is 1 + 2i, so that is true mateship
...
In the
case of adding 1 + 2i and 1 −2i we get 2, and multiplying them we get 5
...
The answer 5 is the square of the ‘length’ of
the complex number 1 + 2i and this equals the length of its mate
...

The separation of the complex numbers from mysticism owes much to Sir
William Rowan Hamilton, Ireland’s premier mathematician in the 19th century
...
It only acted as a
placeholder and could be thrown away
...
Shorn of i, addition becomes
(2, 3) + (8, 4) = (10, 7)

and, a little less obviously, multiplication is
(2, 3) × (8, 4) = (4, 32)

47

The completeness of the complex number system becomes clearer when we
think of what are called ‘the nth roots of unity’ (for mathematicians ‘unity’ means
‘one’)
...
Let’s take z6 = 1 as an
example
...

More is true
...
In general the n roots
of unity will each lie on the circle and be at the corners or ‘vertices’ of a regular
n-sided shape or polygon
...
Complex numbers are 2-dimensional, but what is special about
2? For years, Hamilton sought to construct 3-dimensional numbers and work out
a way to add and multiply them but he was only successful when he switched to
four dimensions
...
Many wondered about 1648

dimensional numbers as a possible continuation of the story – but 50 years after
Hamilton’s momentous feat, they were proved impossible
...
Occasionally we have to go back to basics
...
Can we get more basic than this?

Well, 4 = 2 × 2 and so we can break it down into primary components
...
These are composite numbers
for they are built up from the very basic ones 2, 3, 5, 7,
...
These are the prime numbers,
or simply primes
...
You
might wonder then if 1 itself is a prime number
...
This enables
theorems to be elegantly stated
...

For the first few counting numbers, we can underline the primes: 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
...
Prime
numbers are important because they are the ‘atoms’ of mathematics
...

The mathematical result which consolidates all this has the grand name of the
‘prime-number decomposition theorem’
...

We saw that 12 = 2 × 2 × 3 and there is no other way of doing it with prime
components
...
As
another example, 6,545,448 can be written, 23 × 35 × 7 × 13 × 37
...
One of the first
methods for finding them was developed by a younger contemporary of
Archimedes who spent much of his life in Athens, Erastosthenes of Cyrene
...
Today he’s noted for his sieve for finding prime numbers
...
He underlined 2 and
struck out all multiples of 2
...
Continuing in this way, he sieved out all the composites
...

So we can predict primes, but how do we decide whether a given number is a
prime or not? How about 19,071 or 19,073? Except for the primes 2 and 5, a
prime number must end in a 1, 3, 7 or 9 but this requirement is not enough to
make that number a prime
...
By the way,
19,071 = 32 × 13 × 163 is not a prime, but 19,073 is
...
Let’s see how many primes there are in each segment of 100 between 1
and 1000
...
For n = 1000 the formula gives
the approximate value of 172
...
It had always been assumed this was the case for any value of n,
but the primes often have surprises in store and it has been shown that for n =
10371 (a huge number written long hand as a 1 with 371 trailing 0s) the actual
number of primes exceeds the estimate
...


How many?
There are infinitely many prime numbers
...
Euclid’s beautiful proof goes like this:
Suppose that P is the largest prime, and consider the number N
= (2 × 3 × 5 ×
...
Either N is prime or it is not
...
If N is not a prime it must be
divisible by some prime, say p, which is one of 2, 3, 5,
...

This means that p divides N – (2 × 3 × 5 ×
...
But this
number is equal to 1 and so p divides 1
...
Thus, whatever the nature of N, we
arrive at a contradiction
...
Conclusion: the number of primes
is limitless
...
One which has held the record recently is the
enormous Mersenne prime 224036583 − 1, which is approximately 107235732 or a
number starting with 1 followed by 7,235,732 trailing zeroes
...

Twin primes are pairs of consecutive primes separated only by an even
number
...
On the numerical front, it is known there are
27,412,679 twins less than 10 10
...
274% of the numbers in this range
...

Christian Goldbach conjectured that:
Every even number greater than 2 is the sum of two prime numbers
...
In 1770
Edward Waring, a professor at Cambridge, posed problems involving writing whole numbers as the
addition of powers
...
In numerology, take the
unrivalled cult number 666, the ‘number of the beast’ in the biblical book of Revelation, and which
has some unexpected properties
...


For instance, 42 is an even number and we can write it as 5 + 37
...
The conjecture is true for a huge range of numbers – but it
has never been proved in general
...
The Chinese mathematician Chen
Jingrun made a great step
...

The great number theorist Pierre de Fermat proved that primes of the form 4k
53

+ 1 are expressible as the sum of two squares in exactly one way (e
...
17 = 12 +
42 ), while those of the form 4k + 3 (like 19) cannot be written as the sum of two
squares at all
...

So, for example, 19 = 12 + 12 + 12 + 42
...

We described the prime numbers as the ‘atoms of mathematics’
...
Has mathematics stood still?’ If we limit ourselves to the
counting numbers, 5 is a prime number and will always be so
...
As the product of two Gaussian
integers, 5 and numbers like it are not as unbreakable as was once supposed
...
There
are perfect squares, but here the term is not used in an aesthetic sense
...
In another direction, some
numbers have few divisors and some have many
...
When the addition of the divisors of a number equals the
number itself it is said to be perfect
...
Why? Because the number of prime
numbers between 1 and 10 (namely 2, 3, 5, 7) equalled the non-primes (4, 6, 8,
9) and this was the smallest number with this property
...

It seems the Pythagoreans actually had a richer concept of a perfect number
...
These categories were defined in
terms of the relationships between them and their divisors
...

Whether a number is superabundant is determined by its divisors and makes a
play on the connection between multiplication and addition
...
For such a small number as 30 we can see the divisors
are 1, 2, 3, 5, 6, 10 and 15
...
The number
30 is superabundant because the addition of its divisors (42) is bigger than the
number 30 itself
...
So the number 26 is deficient because its divisors 1, 2 and 13 add up
to only 16, which is less than 26
...

A number that is neither superabundant nor deficient is perfect
...
The first perfect
number is 6
...
The
Pythagoreans were so enchanted with the number 6 and the way its parts fitted
together that they called it ‘marriage, health and beauty’
...
He believed that the perfection
of 6 existed before the world came into existence and that the world was created
in 6 days because the number was perfect
...
Its divisors are 1, 2, 4, 7 and 14 and, when we
add them up, we get 28
...
After 28, you have wait until 496 for the next
56

perfect number
...
For the next perfect numbers we have
to start going into the numerical stratosphere
...
The balance of opinion suggests that they, like the
primes, go on for ever
...
If we have a perfect
number of beads, they can be arranged around a hexagonal necklace
...


Mersenne numbers
The key to constructing perfect numbers is a collection of numbers named
after Father Marin Mersenne, a French monk who studied at a Jesuit college with
René Descartes
...
Mersenne
numbers are constructed from powers of 2, the doubling numbers 2, 4, 8, 16,
32, 64, 128, 256,
...
A Mersenne number is a
number of the form 2n − 1
...
But it is those Mersenne numbers that are also prime that can be used to
construct perfect numbers
...
The Mersenne numbers could
only be prime if the power was a prime number, but was that enough? For the
first few cases, we do get 3, 7, 31 and 127, all of which are prime
...
But primes are not constrained by simplicity, and it was found
that for the power 11 (a prime number), 211 – 1 = 2047 = 23 × 89 and
consequently it is not a prime number
...
The Mersenne
numbers 217 – 1 and 219 – 1 are both primes, but 223 – 1 is not a prime, because
Just good friends
The hard-headed mathematician is not usually given to the mystique of
numbers but numerology is not yet dead
...
Later
they became useful in compiling romantic horoscopes where their mathematical
properties translated themselves into the nature of the ethereal bond
...
Why so? Well, the divisors of 220
are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and if you add them up you get
284
...
If you figure out the divisors of 284 and add them up,
you get 220
...

Mersenne Primes
Finding Mersenne primes is not easy
...
The great
Leonhard Euler contributed the eighth Mersenne prime, 231 – 1 = 2,147,483,647, in 1732
...
But with
powerful computers the Mersenne prime industry had moved on and in the late 1970s high school
students Laura Nickel and Landon Noll jointly discovered the 25th Mersenne prime, and Noll the 26th
Mersenne prime
...

58

223 – 1 = 8,388,607 = 47 × 178,481

Construction work
A combination of Euclid and Euler’s work provides a formula which enables
even perfect numbers to be generated: n is an even perfect number if and only if
n = 2p – 1(2p – 1) where 2p – 1 is a Mersenne prime
...
This
formula for calculating even perfect numbers means we can generate them if we
can find Mersenne primes
...
Writing at the beginning of the 19th century, the table maker Peter
Barlow thought that no one would go beyond the calculation of Euler’s perfect
number
230 (231 – 1) = 2,305,843,008,139,952,128

as there was little point
...


Odd perfect numbers
No one knows if an odd perfect number will ever be found
...
The English mathematician James Joseph
Sylvester declared the existence of an odd perfect number ‘would be little short of
a miracle’ because it would have to satisfy so many conditions
...
It is one of the oldest problems in mathematics, but if an
odd perfect number does exist quite a lot is already known about it
...


the condensed idea
The mystique of numbers

59

60

11

Fibonacci numbers

I n The Da Vinci Code, the author Dan Brown made his murdered curator Jacques
Saunière leave behind the first eight terms of a sequence of numbers as a clue to his
fate
...
Welcome to the most famous
sequence of numbers in all of mathematics
...

The sequence is widely known for its many intriguing properties
...
For example 8 = 5 + 3, 13 = 8 + 5,
...
All you have to remember is to begin with the
two numbers 1 and 1 and you can generate the rest of the sequence on the spot
...
Classical musical composers have used it as an
inspiration, with Bartók’s Dance Suite believed to be connected to the sequence
...
618 as a salute to the ultimate ratio of the Fibonacci
numbers, a number we shall discuss a little later
...
Fibonacci posed the following problem of rabbit generation:
Mature rabbit pairs generate young rabbit pairs each month
...
By the end of the first month they will
have matured, by the end of the second month the mature pair is still there and
they will have generated a young rabbit pair
...
Miraculously none of the rabbit pairs die
...
The generations can be shown in a ‘family tree’
...
We see the number of pairs
is 8
...
This shows that the birth of rabbit pairs
follows the basic Fibonacci equation:
number after n months = number after (n – 1) month
+ number after (n – 2) months

Properties
Let’s see what happens if we add the terms of the sequence:
62

The result of each of these sums will form a sequence as well, which we can
place under the original sequence, but shifted along:

The addition of n terms of the Fibonacci sequence turns out to be 1 less than
the next but one Fibonacci number
...
+ 987, you just subtract 1 from 2584 to get 2583
...
If the other alternation
is taken, such as 1 + 3 + 8 + 21 + 55, the answer is 88 which is a Fibonacci
number less 1
...
We get a
new sequence by multiplying each Fibonacci number by itself and adding them
...
For example,
1 + 1 + 4 + 9 + 25 + 64 + 169 = 273 = 13 × 21

Fibonacci numbers also occur when you don’t expect them
...
What if we want to count the
number of ways the coins can be taken from the purse to make up a particular
amount expressed in pounds
...

63

The value of £4, as we draw the coins out of the purse, can be any of the
following ways, 1 + 1 + 1 + 1; 2 + 1 +1; 1 + 2 + 1; 1 + 1 + 2; and 2 + 2
...
If
you take out £20 there are 6,765 ways of taking the £1 and £2 coins out,
corresponding to the 21st Fibonacci number! This shows the power of simple
mathematical ideas
...
Let’s do it for a few terms 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
...
It takes its place
amongst the top mathematical constants like π and e, and has the exact value

and this can be approximated to the decimal 1
...
With a little
more work we can show that each Fibonacci number can be written in terms of
Φ
...
The first few prime numbers in the
Fibonacci sequence are 2, 3, 5, 13, 89, 233, 1597 – but we don’t know if there
are infinitely many primes in the Fibonacci sequence
...
A spectacular member of the family is one we may associate
with a cattle population problem
...
It is only
the mature pairs which can reproduce
...


Thus the generation skips a value so for example, 41 = 28 + 13 and 60 = 41
+ 19
...
For the
cattle sequence the ratios obtained by dividing a term by its preceding term
approach the limit denoted by the Greek letter psi, written ψ, where
ψ = 1
...


This is known as the ‘supergolden ratio’
...

Rectangles are present within the artists’ community – Piet Mondrian, Ben Nicholson and
others, who progressed to abstraction, all used one sort or another
...
Of these, perhaps the golden rectangle
has found greatest favour
...

Let’s look at some other rectangles first
...
4142
...
4142 × b
...
The A-formulae system used for paper sizes has a highly
67

desirable property, one that does not occur for arbitrary paper sizes
...
They are two smaller versions of
the same rectangle
...

Similarly a piece of A5-size paper generates two pieces of A6
...
The smaller the
number on the A-size the larger the piece of paper
...
4142 would do the trick? Let’s fold a rectangle, but this time
let’s make it one where we don’t know the length of its longer side
...
If we now fold the rectangle, the length-towidth ratio of the smaller rectangle is 1/½x, which is the same as 2/x
...
The true value of x is therefore √2 which is
approximately by 1
...


Mathematical gold
The golden rectangle is different, but only slightly different
...

The key property of the golden rectangle is that the rectangle left over, RNPS,
is proportional to the large rectangle – what is left over should be a mini-replica
of the large rectangle
...
The length-towidth ratio is again x/1
...

By equating them, we get the equation

68

which can be multiplied out to give x2 = x + 1
...
618
...
If you type 1
...
618 which is the same as x + 1 = 2
...
This number is the
famous golden ratio and is designated by the Greek letter phi, Φ
...


Going for gold
69

Now let’s see if we can build a golden rectangle
...
The length
OS = ½, and so by Pythagoras’s theorem (see page 84) in the triangle ORS, OR
=
Using a pair of compasses centred on O, we can draw the arc RP and we’ll find
that OP = OR = √5/2
...


History
Much is claimed of the golden ratio Φ
...
More than this is the danger of claiming the golden ratio was
there before the artefact – that musicians, architects and artists had it in mind at
the point of creation
...
The progress
from numbers to general statements without other evidence is a dangerous
argument to make
...
At its time of construction the golden ratio was
certainly known but this does not mean that the Parthenon was based on it
...
74 which is close to 1
...

In his 1509 book De divina proportione, Luca Pacioli ‘discovered’ connections
between characteristics of God and properties of the proportion determined by Φ
...
Pacioli was a Franciscan monk who wrote
influential books on mathematics
...
His other claim to fame is that he taught mathematics to
70

Leonardo da Vinci
...
Later, Gustav Fechner, a German experimental psychologist, made
thousands of measurements of rectangular shapes (playing cards, books,
windows) and found the most commonly occurring ratio of their sides was close
to Φ
...
He placed great
emphasis on harmony and order and found this in mathematics
...
One of his planks was the
‘modulator’ system, a theory of proportions
...
Le
Corbusier was inspired by Leonardo da Vinci who, in turn, had taken careful
notes on the Roman architect Vitruvius, who set store by the proportions found
in the human figure
...

This is how we build the supergolden rectangle MQPN
...
Join the diagonal MP and mark the intersection
o n RS as the point J
...

We’ll say the length RJ is y and the length MN is x
...
We get the supergolden
rectangle by making the rectangle RJKN proportional to the original rectangle
MQPN, that is y/(x− 1) = x/1
...
The cubic equation has one positive real
solution ψ (replacing x with the more standard symbol Φ) whose value is
ψ = 1
...


the number associated with the cattle sequence (see page 47)
...


the condensed idea
Divine proportions

72

13

Pascal’s triangle

The number 1 is important but what about 11? It is interesting too and so is 11 × 11 =
121, 11 × 11 × 11 = 1331 and 11 × 11 × 11 × 11 = 14,641
...
But where do we find it?

Throwing in 11° = 1 for good measure, the first thing to do is forget the
commas, and then introduce spaces between the numbers
...


Pascal’s triangle is famous in mathematics for its symmetry and hidden
relationships
...
The many connections of Pascal’s triangle
with other branches of mathematics have made it into a venerable mathematical
object, but its origins can be traced back much further than this
...

73

The Pascal pattern is generated from the top
...
To construct further rows we continue
to place 1s on the ends of each row while the internal numbers are obtained by
the sum of the two numbers immediately above
...
The English mathematician G
...

Hardy said ‘a mathematician, like a painter or a poet, is a maker of patterns’ and
Pascal’s triangle has patterns in spades
...
If we work out (1 + x) × (1
+ x) × (1 + x) = (1 + x)3 , for example, we get 1 + 3x + 3x2 + x3
...
The scheme followed is:

If we add up the numbers in any row of Pascal’s triangle we always obtain a
power of 2
...

This can be obtained from the left-hand column above if we use x = 1
...
If we
draw a vertical line down through the middle, the triangle has ‘mirror symmetry’
– it is the same to the left of the vertical line as to the right of it
...
Under the diagonal made up of 1s we have the diagonal
made up of the counting numbers 1, 2, 3, 4, 5, 6,
...
(the numbers which can be made up
of dots in the form of triangles)
...
These numbers correspond to
tetrahedra (‘three-dimensional triangles’, or, if you like, the number of cannon
balls which can be placed on triangular bases of increasing sizes)
...
Each number is three
times the previous one with the one before that subtracted
...
Based on this, the next number in the sequence will be 3 × 34 – 13 =
89
...
and these are
generated by the same ‘3 times minus 1’ rule
...
But there’s more
...


Even and odd numbers in Pascal’s triangle

Pascal combinations
The Pascal numbers answer some counting problems
...
Let’s call them Alison, Catherine, Em m a, Gary, Jo hn, Matthew and
Thomas
...
Mathematicians
find it useful to write C(n,r) to stand for the number in the nth row, in the rth
position (counting from r = 0) of Pascal’s triangle
...
The number in the 7th row of the triangle, in the 3rd position, is 35
...
This accounts for the fact that C(7,4) = 35 too
...


The Serpiński gasket
76

0s and 1s
In Pascal’s triangle, we see that the inner numbers form a pattern depending
on whether they are even or odd
...


Adding signs
We can write down the Pascal triangle that corresponds to the powers of (−1
+ x), namely (−1 + x)n
...
However it is
the diagonals which are interesting here
...
are the coefficients of the expansion while the terms in the
next diagonal along are the coefficients of the expansion
(1 + x)−1 = 1 − x + x2 − x3 + x4 − x5 + x6 − x7 +
...


The Leibniz harmonic triangle
The German polymath Gottfried Leibniz discovered a remarkable set of
numbers in the form of a triangle
...
But unlike Pascal’s triangle, the number in one
row is obtained by adding the two numbers below it
...
To construct this triangle we can progress from the top and move from left
to right by subtraction: we know 1/12 and 1/30 and so 1/12 − 1/30 = 1/20, the
number next to 1/30
...
Just as we
did before, we can write these Leibnizian numbers as B(n,r) to stand for the nth
number in the rth row
...
So it is with Pascal’s triangle and
its intimate connections with so many parts of mathematics – modern geometry,
combinatorics and algebra to name but three
...


the condensed idea
The number fountain

79

14

Algebra

Algebra gives us a distinctive way of solving problems, a deductive method with a twist
...
For a moment consider the problem of taking the
number 25, adding 17 to it, and getting 42
...
We are given the
numbers and we just add them together
...
This is where backwards thinking comes in
...


Word problems which are meant to be solved by algebra have been given to
schoolchildren for centuries:
My niece Michelle is 6 years of age, and I am 40
...

In x years from now Michelle will be 6 + x years and I will be 40 + x
...
When I am 51 Michelle
will be 17 years old
...
She will be 34 when I am 68
...
They have no terms like x2 or
x3 , which make equations more difficult to solve
...
In
past times, x2 was represented as a square and because a square has four sides
the term quadratic was used; x3 was represented by a cube
...
To progress from numbers to
letters is a mental jump but the effort is worthwhile
...
Unfortunately it
resulted in an episode when mathematics was not always on its best behaviour
...
Girolamo Cardano from Milan persuaded Tartaglia to tell
him of his methods but was sworn to secrecy
...


Origins
Algebra was a significant element in the work of Islamic scholars in the ninth
century
...
Dealing with practical problems in terms of linear and quadratic
equations, al-Khwarizmi’s ‘science of equations’ gave us the word ‘algebra’
...

Girolamo Cardano’s great work on mathematics, published in 1545, was a
watershed in the theory of equations for it contained a wealth of results on the
cubic equation and the quartic equation – those involving a term of the kind x4
...
For example, the quadratic equation ax2 + bx
+ c = 0 can be solved using the formula:

If you want to solve the equation x2 – 3x + 2 = 0 all you do is feed the values
a = 1, b = −3 and c = 2 into the formula
...
What puzzled mathematicians was that they
81

could not produce a formula which was generally applicable to equations
involving x5 , the ‘quintic’ equations
...
He actually proved a negative concept, nearly
always a more difficult task than proving that something can be done
...

Abel convinced the top rung of mathematicians, but news took a long time to
filter through to the wider mathematical world
...


The modern world
For 500 years algebra meant ‘the theory of equations’ but developments took
a new turn in the 19th century
...
They could even represent higherdimensional objects such as those found in matrix algebra (see page 156)
...

A significant event in modern algebra occurred in 1843 when the Irishman
William Rowan Hamilton discovered the quaternions
...
For many years he tried three-dimensional symbols, but no
satisfactory system resulted
...

Success came rather unexpectedly
...
This flash of
inspiration came to him as he walked with his wife along the Royal Canal to
Dublin
...
Without hesitation, the
38-year-old vandal, Astronomer Royal of Ireland and Knight of the Realm, carved
the defining relations into the stone on Brougham Bridge – a spot that is
82

acknowledged today by a plaque
...
He lectured on it year after year and published two
heavyweight books on his ‘westward floating, mystic dream of four’
...
In 1844 the German linguist and mathematician Hermann Grassmann
published another algebraic system with rather less drama
...
Today both quaternions and Grassmann’s
algebra have applications in geometry, physics and computer graphics
...
This had been used as a basis for geometry by Euclid but it wasn’t
applied to algebra until comparatively recently
...
In this modern
algebra, the pervading idea is the study of structure where individual examples
are subservient to the general abstract notion
...

The most fundamental algebraic structure is a group and this is defined by a
list of axioms (see page 155)
...
All these new words were
imported into mathematics in the early 20th century as algebra transformed itself
into an abstract science known as ‘modern algebra’
...
Pronounced ‘Al Gore rhythm’ it is a concept useful to
mathematicians and computer scientists alike
...


Firstly, an algorithm is a routine
...
We can see why computers like algorithms because they
are very good at following instructions and never wander off track
...
There is a considerable risk
of it all going horribly wrong
...
There
are often several methods available to do the same task and the best one must
be chosen
...
Some may be quick but produce the
wrong answer
...
There must be hundreds of recipes
(algorithms) for cooking roast turkey with stuffing
...
So we
have the ingredients and we have the instructions
...
The only thing missing in this recipe, usually present in a mathematical
algorithm, is a loop, a tool to deal with recursion
...

In mathematics we have ingredients too – these are the numbers
...
The gcd of two whole numbers is the greatest number that divides into
both of them
...


The greatest common divisor
The gcd in our example is the largest number that exactly divides both 18 and
85

84
...
So 6 will
also divide both numbers
...
On checking, these candidates do not divide 84 so 6 is the
largest number that divides both
...

T he gcd can be interpreted in terms of kitchen tiling
...
In this case, we can see that a 6 × 6 tile will do
the trick
...
There is also a related concept, the least common
multiple (lcm)
...
The link between the lcm and gcd is highlighted by the fact that the lcm
of two numbers multiplied by their gcd is equal to the multiplication of the two
numbers themselves
...

Geometrically, the lcm is the length of the side of the smallest square that can
be tiled by 18 × 84 rectangular tiles
...
We have already calculated gcd(18, 84)
= 6 but to do it we needed to know the divisors of both 18 and 84
...
Then, comparing them, the number 2 is common to both and is the
highest power of 2 which will divide both
...
We concluded: 2 × 3 = 6 is the largest
number that divides both
...
We’d first have to factorize
both these numbers, and that would be only the start
...


The algorithm
There is a better way
...
’
The algorithm Euclid gives is beautifully efficient and effectively replaces the
effort of finding factors by simple subtraction
...

The object is to calculate d = gcd(18, 84)
...
It
does not divide exactly but goes 4 times with 12 (the remainder) left over:
84 = 4 × 18 + 12

Since d must divide 84 and 18, it must divide the remainder 12
...
So we can now repeat the process and divide 18 by 12:
18 = 1 × 12 + 6

to get remainder 6, so d = gcd(6, 12)
...
6 is the largest number which will divide both 0 and 6 so
this is our answer
...


Uses for the gcd
The gcd can be used in the solution of equations when the solutions must be
whole numbers
...

Let’s imagine Great Aunt Christine is going for her annual holiday to Barbados
...
When he arrives back in Belgravia, John’s
nine-year-old son James pipes up ‘that can’t be right, because the gcd 6 doesn’t
divide into 652’
...

James knows that there is a solution in whole numbers to the equation 18x +
84y = c if and only if the gcd 6 divides the number c
...
James does not even need to know how many suitcases x, y of
either weight Aunt Christine intends to take to Barbados
...
They
don’t have to be prime themselves but just have to be prime to each other, for
example gcd(6, 35) = 1, even though neither 6 nor 35 is prime
...

Let’s look at another problem: Angus does not know how many bottles of
wine he has but, when he pairs them up, there is 1 left over
...
How many bottles does he
have? We know that on division by 2 we get remainder 1 and on division by 5
we get remainder 3
...
Running along the odd numbers we quickly find that 13 fits the bill (we
can safely assume Angus has more than 3 bottles, a number which also satisfies
the conditions)
...

Let’s now add another condition, that the number must give remainder 3 on
division by 7 (the bottles arrived in packs of 7 bottles with 3 spares)
...
to take account of this, we find
that 73 fits the bill, but notice that 143 does too, as does 213 and any number
found by adding multiples of 70 to these numbers
...
If Angus has between 150 and 250 bottles then the theorem
nails the solution down to 213 bottles
...


The condensed idea
A route to the greatest

89

16

Logic

‘If there are fewer cars on the roads the pollution will be
acceptable
...
If there is road pricing the summer will be
unbearably hot
...

The conclusion is inescapable: pollution is acceptable
...
We are only
interested in its validity as a rational argument
...


Two premises and a conclusion
As it stands the newspaper passage is quite complicated
...
His approach
was based on the different forms of the syllogism, a style of argument based on
three statements: two premises and a conclusion
...
In this
example, the conclusion has a certain inevitability about it whatever meaning we
attach to the words ‘spaniels’, ‘dogs’ and ‘animals’
...
Yet both instances of the syllogism have the
same structure and it is the structure which makes this syllogism valid
...
This is what makes a valid
argument useful
...
For example, another might be

Is this a valid argument? Does it apply to all cases of As, Bs and Cs, or is there
a counterexample lurking, an instance where the premises are true but the
conclusion false? What about making A spaniels, B brown objects, and C tables?
Is the following instance convincing?

Our counterexample shows that this syllogism is not valid
...
Our first example was known as BARBARA because it
contains three uses of ‘All’
...
Aristotle’s logic – his theory of the syllogism – was thought
to be a perfect science well into the 19th century
...
It deals with propositions
or simple statements and the combination of them
...
It used to be called
the ‘algebra of logic’, which gives us a clue about its structure, since George
Boole realized that it could be treated as a new sort of algebra
...


And truth table

Let’s try it out and consider a proposition a, where a stands for ‘Freddy is a
spaniel’
...
If I am thinking of my dog
named Freddy who is indeed a spaniel then the statement is true (T) but if I am
thinking that this statement is being applied to my cousin whose name is also
Freddy then the statement is false (F)
...


Not truth table
92

If we have another proposition b such as ‘Ethel is a cat’ then we can combine
these two propositions in several ways
...
The
connective V corresponds to ‘or’ but its use in logic is slightly different from ‘or’ in
everyday language
...
This conjunction of propositions can be summarized in a truth table
...
The algebra of logic becomes clear when we combine these
propositions using a mixture of the connectives with a, b and c like a ⋀ (b Vc)
...
There is a parallel between the
algebra of logic and ordinary algebra because the symbols Λ and V act similarly
to × and + in ordinary algebra, where we have x × (y + z) = (x × y) + (x × z)
...

Other logical connectives may be defined in terms of these basic ones
...

Now if we look again at the newspaper leader, we can write it in symbolic
form to give the argument in the margin:

93

Is the argument valid or not? Let’s assume the conclusion P is false, but that
all the premises are true
...
It will then be impossible to have the premises true but
the conclusion false
...
As C VS is true, the fact that C is false means that S is true
...
That is, ¬H is false
...
The content of the
statements in the newspaper leader may still be disputed, but the structure of the
argument is valid
...
S
...
This uses the universal quantifier, ∀, to mean ‘for all’,
and the existential quantifier, ∃, to mean ‘there exists’
...
This suggests
confused thinking, but it is really about a widening of the traditional boundaries
of logic
...
So we had the set of
spaniels, the set of dogs, and the set of brown objects
...
If we meet a pure bred ‘Rhodesian
ridgeback’ in the park we are pretty sure it is not a member of the set of spaniels
...
What if
we had the set of heavy spaniels
...
Mathematics allows us to
be precise about fuzziness
...
It has moved on
from Aristotle and is now an active area of modern research and application
...
The quest for cast iron rational
arguments is the driving force of pure mathematics
...


Proofs are not arrived at easily – they often come at the end of a great deal of
exploration and false trails
...
A successful proof carries the mathematician’s
stamp of authenticity, separating the established theorem from the conjecture,
bright idea or first guess
...
To this add insight
...
Progression on the basis of
unproven facts carries the danger that theories may be built on the mathematical
equivalent of sand
...


What is a proof?
When you read or hear about a mathematical result do you believe it? What
would make you believe it? One answer would be a logically sound argument
that progresses from ideas you accept to the statement you are wondering about
...
Depending on the quality of the proof you are
either convinced or remain sceptical
...


The counterexample
96

Let’s start by being sceptical – this is a method of proving a statement is
incorrect
...
Suppose you hear a
claim that any number multiplied by itself results in an even number
...
If
we have a number, say 6, and multiply it by itself to get 6 × 6 = 36 we find that
indeed 36 is an even number
...
The
claim was for any number, and there are an infinity of these
...
Trying 9, say, we find that 9 × 9 =
81
...
This means that the statement that all numbers
when multiplied by themselves give an even number is false
...
A
counterexample to the claim that ‘all swans are white’, would be to see one black
swan
...

If we fail to find a counterexample we might feel that the statement is correct
...
A proof has to be
constructed and the most straightforward kind is the direct method of proof
...
If we can do this we
have a theorem
...
But we may be able to
salvage something
...
Changing the hypothesis is something we can do
...

First we try some other numerical examples and we find this statement
verified every time and we just cannot find a counterexample
...
As you know, an even number is one which
is a multiple of 2, that is 6 = 2 ×3
...
But in
this argument there is nothing which is particular to 6, and we could have started
with n = 2 × k to obtain
n × n = 2 × (k + k +
...
Our proof is now complete
...
Nowadays they use a filled-in square
...


The indirect method
In this method we pretend the conclusion is false and by a logical argument
demonstrate that this contradicts the hypothesis
...

Our hypothesis is that n is even and we’ll pretend n × n is odd
...
+ n and there are n of these
...
Thus n is odd, which contradicts the
hypothesis
...
The full-strength indirect
method is known as the method of reductio ad absurdum (reduction to the
absurd), and was much loved by the Greeks
...

The classical proof that the square root of 2 is an irrational number is one of this
form where we start off by assuming the square root of 2 is a rational number
and deriving a contradiction to this assumption
...
are all true
...
This specific technique (not to be confused with scientific induction) is
widely used to prove statements involving whole numbers
...
As a practical
example, think of the problem of adding up the odd numbers
...
Now 9 is 3 × 3 = 32 and 16 is 4 × 4 = 42, so could it be
that the addition of the first n odd numbers is equal to n2? If we try a randomly
chosen value of n, say n = 7, we indeed find that the sum of the first seven is 1
+ 3 + 5 + 7 + 9 + 11 +13 = 49 which is 72
...

This is where mathematical induction steps in
...
This metaphor applies to a row of dominos standing on their
ends
...
This is clear
...
We can apply this thinking to the
odd numbers problem
...
Mathematical induction sets up a chain reaction whereby
P1, P2, P3,
...
The statement P1 is trivially true because 1 = 12
...
We use the result at
one stage to hop to the next one
...


Difficulties with proof
Proofs come in all sorts of styles and sizes
...
Some others detailing the latest
research have taken up the whole issue of journals and amount to thousands of
pages
...

There are also foundational issues
...
If the assumption that a solution of an
equation does not exist leads to a contradiction, is this enough to prove that a
solution does exist? Opponents of this proof method would claim the logic is
merely sleight of hand and doesn’t tell us how to actually construct a concrete
99

solution
...
They pour scorn on the classical
mathematician who regards the reductio method as an essential weapon in the
mathematical armoury
...


the condensed idea
Signed and sealed

100

18

Sets

Nicholas Bourbaki was a pseudonym for a self-selected group of French academics who
wanted to rewrite mathematics from the bottom up in ‘the right way’
...
The axiomatic method was
central and the books they put out were written in the rigorous style of ‘definition,
theorem and proof’
...


Georg Cantor created set theory out of his desire to put the theory of real
numbers on a sound basis
...


The union of A and B

What are sets?
A set may be regarded as a collection of objects
...
The objects themselves are called ‘elements’ or ‘members’ of the
set
...
An example is A = {1, 2, 3, 4, 5} and we can write 1 ∈ A for
membership, and 6 ∈ A for non-membership
...
If A and B are two sets then the
set consisting of elements which are members of A or B (or both) is called the
‘union’ of the two sets
...
It can also be
described by a Venn diagram, named after the Victorian logician the Rev
...
Euler used diagrams like these even earlier
...


The intersection of A and B

If A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 10, 21}, the union is A ∪ B = {1, 2,
3, 4, 5, 7, 10, 21} and the intersection is A ∩ B = {1, 3, 5}
...


The complement of A

The operations ⋂ and ⋃ on sets are analogous to × and + in algebra
...
The Indian-born
British mathematician Augustus De Morgan, formulated laws to show how all
three operations work together
...

Cantor defined sets as the collection of elements with a specific property
...
}, all the whole numbers bigger than
10
...

Following Cantor’s lead, we can write the set as A = {x: x is a whole number >
10}, where the colon stands for ‘such that’
...
In this case A is itself an abstract thing, so it is possible to
have A ∈ A
...
The British
philosopher Bertrand Russell hit upon the idea of a set S which contained all
things which did not contain themselves
...

He then asked the question, ‘is S ∈ S?’ If the answer is ‘Yes’ then S must
satisfy the defining sentence for S, and so S∉S
...
Russell’s question ended with this statement, the basis of Russell’s
paradox,
S ∈ S if and only if S ∉ S

It is similar to the ‘barber paradox’ where a village barber announces to the
locals that he will only shave those who do not shave themselves
...

If does shave himself he should not
...
For
mathematicians it is simply not permissible to have systems that generate
contradictions
...

Another way to avoid these antinomies was to formalize the theory of sets
...
The Greeks tried something similar
103

with a problem of their own – they didn’t have to explain what straight lines
were, but only how they should be dealt with
...
This effectively debarred such dangerous creatures as the set of all sets
from appearing
...
In 1931,
Gödel proved that even for the simplest of formal systems there were statements
whose truth or falsity could not be deduced from within these systems
...
They were undecidable statements
...
This result applied to the Zermelo–
Fraenkel system as well as to other systems
...
Loosely speaking, the cardinality measures the ‘size’ of a set
...
The set Q can be put in a list but the set R cannot
(see page 31)
...
Mathematicians denote card(Q) by)
, the Hebrew ‘aleph
nought’ and card(R) = c
...


The continuum hypothesis
Brought to light by Cantor in 1878, the continuum hypothesis says that the
next level of infinity after the infinity of Q is the infinity of the real numbers c
...
Cantor struggled with it and though he
believed it to be true he could not prove it
...

104

The problem was so important that German mathematician David Hilbert
placed it at the head of his famous list of 23 outstanding problems for the next
century, presented to the International Mathematical Congress in Paris in 1900
...

He did prove (in 1938) that the hypothesis was compatible with the Zermelo–
Fraenkel axioms for set theory
...
This is equivalent to showing the
axioms and the negation of the hypothesis is consistent
...

This state of affairs is similar in nature to the way the parallel postulate in
geometry (see page 108) is independent of Euclid’s other axioms
...
In a
similar way, the continuum hypothesis can be accepted or rejected without
disturbing the other axioms for set theory
...


the condensed idea
Many treated as one

105

19

Calculus

A calculus is a way of calculating, so mathematicians sometimes talk about the ‘calculus
of logic’, the ‘calculus of probability’, and so on
...


Calculus is a central plank of mathematics
...
Historically it is associated with Isaac Newton and
Gottfried Leibniz who pioneered it in the 17th century
...
In fact,
both men came to their conclusions independently and their methods were quite
different
...
Each generation bolts on
techniques they think should be learned by the younger generation, and these
days textbooks run beyond a thousand pages and involve many extras
...
The words are derived
from Leibniz’s differentialis (taking differences or ‘taking apart’) and integralis
(the sum of parts, or ‘bringing together’)
...
Calculus is really one subject, and you
need to know about both sides
...

I’m very good at integral and differential calculus
...
Imagine we are standing on a bridge high above a gorge and are
106

about to let a stone drop
...
We can also do
impossible things like stopping the stone in mid-air or watching it in slow motion
over a short time interval
...
Nothing surprising
in that; the stone is attracted to the earth and will fall faster and faster as the
hand on our stopwatch ticks on
...


What is the stone’s speed at a given instant of time, say when the stopwatch
reads exactly 3 seconds after it has been released? How can we work this out?
We can certainly measure average speed but our problem is to measure
instantaneous speed
...
By taking smaller and smaller time
intervals the average speed will be closer and closer to the instantaneous speed
at the place where we stopped the stone
...

107

We might be tempted to make the small extra time equal to zero
...
It has moved no distance
and taken no time to do it! This would give us the average speed 0/0 which the
Irish philosopher Bishop Berkeley famously described as the ‘ghosts of departed
quantities’
...
By
taking this route we are led into a numerical quagmire
...
The exact formula connecting the
distance fallen y and the time x taken to reach there was derived by Galileo:
y = 16 × x2

The factor ‘16’ appears because feet and seconds are the chosen measurement
units
...
But how can we calculate the speed of the stone at time x = 3?
Let’s take a further 0
...
5 seconds
...
5 seconds the stone has travelled y = 16 × 3
...
5 seconds it has fallen 196 − 144 = 52 feet
...
5 = 104 feet per second
...
5 seconds is not a small enough measure
...
05 seconds, and we see that
the distance fallen is 148
...
84 feet giving an average speed of
4
...
05 = 96
...
This indeed will be closer to the instantaneous
speed of the stone at 3 seconds (when x = 3)
...
After a little symbol shuffling we find this is
16 × (2x) + 16 × h

As we make h smaller and smaller, like we did in going from 0
...
05, we
see that the first term is unaffected (because it does not involve h) and the
second term itself becomes smaller and smaller
...
For example, the
instantaneous velocity of the stone after 1 second (when x = 1) is 16 × (2 × 1)
= 32 feet per second; after 3 seconds it is 16 × (2 × 3) which gives 96 feet per
second
...
This is the effect of
differentiation, passing from u= x2 to the derivative = 2x
...
Nowadays we frequently write u = x2 and its derivative as du/dx = 2x
...


The falling stone was one example, but if we had other expressions that u
stood for we could still calculate the derivative, which can be useful in other
contexts
...

109

Integration
The first application of integration was to measure area
...
By measuring the area of each and adding them up we get
the ‘sum’ and so the total area
...
The area of each of the rectangular strips is
udx, so the area A under the curve from 0 to x is

If the curve we’re looking at is u = x2, the area is found by drawing narrow
rectangular strips under the curve, adding them up to calculate the approximate
area, and applying a limiting process to their widths to gain the exact area
...
Like the derivative, there is a regular pattern for the integral of
powers of x
...


The star result
If we differentiate the integral A = x3/3 we actually get the original u = x2
...

Differentiation is the inverse of integration, an observation known as the
Fundamental Theorem of the Calculus and one of the most important theorems
in all mathematics
...
Wherever change is involved, there
we find Calculus
...
This means proving something cannot be done
...


To perform these tasks they only used the bare essentials:
• a straight edge for drawing straight lines (and definitely not to measure lengths),
• a pair of compasses for drawing circles
...
Without modern
measuring equipment the mathematical techniques needed to prove these results
were sophisticated and the classical construction problems of antiquity were only
solved in the 19th century using the techniques of modern analysis and abstract
algebra
...
First place the compass point at O and, with any radius mark off
OA and OB
...
Do the
same at B
...
The triangles AOP and BOP are identical in shape and therefore
the angles AÔP and BÔP will be equal
...

Can we use a sequence of actions like this to split an arbitrary angle into three
equal angles? This is the angle trisection problem
...
But, if we take the angle of 60 degrees,
for instance this angle cannot be trisected
...
So summarizing:
• you can bisect all angles all the time,
• you can trisect some angles all the time, but
• you cannot trisect some angles at any time
...

The story goes that the natives of Delos in Greece consulted the oracle in the face
of a plague they were suffering
...

Imagine the Delian altar began as a three-dimensional cube with all sides
equal in length, say a
...
The volume of each is a3 and b3 and they are related by b3 =
2a3 or b = 3√2 × a where 3√2 is the number multiplied by itself three times that
makes 2 (the cube root)
...
Unfortunately for them, this is
impossible with a straight edge and compasses no matter how much ingenuity is
brought to bear on the would-be construction
...

The phrase ‘squaring the circle’ is commonly used to express the impossible
...

These are irrational numbers (they cannot be written as fractions), but showing
the circle cannot be squared amounts to showing that π cannot be a solution of
any algebraic equation
...

114

Mathematicians generally believed that π was a transcendental but this ‘riddle
of the ages’was difficult to prove until Ferdinand von Lindemann used a
modification of a technique pioneered by Charles Hermite
...

Following Lindemann’s result, we might think that the flow of papers from the
indomitable band of ‘circle-squarers’ would cease
...
Still dancing on
the sidelines of mathematics were those reluctant to accept the logic of the proof
and some who had never heard of it
...
This is a
symmetrical many-sided figure like a square or pentagon, in which sides are all
of equal length and where adjacent sides make equal angles with each other
...


Constructing an equilateral triangle

The polygon with 3 sides is what we normally call an equilateral triangle and
is particularly straightforward to construct
...

Place the compass point at A and draw a portion of the circle of radius AB
...
The intersection
point of these two arcs is at P
...
The actual triangle is completed by joining AB, AP and BP
using the straight edge
...
The equilateral triangle is constructed by
finding the point P and for this only the compasses are required – the straight
edge was only used to physically join the points together
...
The Italian Lorenzo Mascheroni proved the same results 125
years later
...

A prince is born
Carl Friedrich Gauss was so impressed by his result showing a 17-sided polygon could
be constructed that he decided to put away his planned study of languages and become
a mathematician
...
The 17-sided polygon is the shape of the base of his memorial at
Göttingen, Germany, and is a fitting tribute to his genius
...
We have already constructed the 3-sided
polygon, and Euclid constructed the 5-sided polygon but he could not construct
the 7-sided polygon (the heptagon)
...
He deduced that it is not
possible to construct a p-sided polygon for p = 7, 11 or 13
...
Gauss actually went further and proved that a psided polygon is constructable if and only if the prime number p is of the form
Numbers of this form are called Fermat numbers
...

When we try n = 5, the Fermat number is p = 232 + 1 = 4,294,967,297
...
If
we put n = 6 or 7 into the formula the results are huge Fermat numbers but, as
with 5, neither is prime
...


the condensed idea
Take a straight edge and a pair of
compasses
...
Trigonometry is the theory which we use to ‘measure the triangle’,
whether it is the size of the angles, the length of the sides, or the enclosed area
...


The triangle’s tale
There is a neat argument to show that the angles in any triangle add up to
two right angles or 180 degrees
...

The angle A C which we’ll call x is equal to the angle BÂM because they are
alternate angles and MN and BC are parallel
...
The angle about the point A is equal to 180 degrees (half of 360
degrees) and this is x + y + z which is the sum of the angles in the triangle
...
Of course we are assuming the
triangle is drawn on a flat surface like this flat piece of paper
...

Euclid proved many theorems about triangles, always making sure this was
done deductively
...
Nowadays this is
118

called the ‘triangle inequality’ and is important in abstract mathematics
...
If a bale of hay were placed at one
vertex and the ass at the other, they argued, the animal would hardly traverse
the two sides to satisfy its hunger
...
The best known statement of it is in terms of
algebra, a2 + b2 = c2 but Euclid refers to actual square shapes: ‘In right-angled
triangles the square on the side subtending the right angle is equal to the squares
on the sides containing the right angle’
...
There are several hundred
proofs in existence
...

This is a proof ‘without words’
...

Since the four equal triangles (shaded dark) are common to both squares we
can remove them and still have equality of area
...

a2 + b2 = c2

The Euler line
Hundreds of propositions about triangles are possible
...
In any triangle ABC we mark the midpoints D, E, F of
its sides
...
Now
join A to E
...
It fact it does and the point G is called the
‘centroid’ of the triangle
...

There are literally hundreds of different ‘centres’ connected with a triangle
...

This is called the ‘orthocentre’
...
This is the centre of the circle which can be drawn through A,
B and C
...
In any triangle ABC the centres G, H and O, respectively the
centroid, orthocentre, and circumcentre, themselves lie along one line, called the
‘Euler line’
...


The Euler line

Napoleon’s theorem
For any triangle ABC, equilateral triangles can be constructed on each side and
from their centres a new triangle DEF is constructed
...

Napoleon’s theorem appeared in print in an English journal in 1825 a few
years after his death on St Helena in 1821
...

Unfortunately there is no evidence to take us further and ‘Napoleon’s theorem’ is,
like many other mathematical results, ascribed to a person who had little to do
with its discovery or its proof
...


121

Napoleon’s theorem

The essential data that determines a triangle consists of knowing the length of
one side and two angles
...

In surveying areas of land in order to draw maps it is quite useful to be a ‘flatearther’ and assume triangles to be flat
...
By
trigonometry everything is known about the triangle ABC and the surveyor
moves on, fixes the next triangulation point from the new base line AB or AC and
repeats the operation to establish a web of triangles
...

It was used as the basis for the Great Trigonometrical Survey of India which
began in the 1800s and lasted 40 years
...
To ensure utmost accuracy in measuring
angles, Sir George Everest arranged the manufacture of two giant theodolites in
London, together weighing one ton and needing teams of a dozen men to
transport them
...
Accuracy in measurement was
paramount and much talked about but it was the humble triangle which was at
the centre of operations
...
Once all the lengths in a triangle have
been computed, the calculation of area is straightforward
...
There are several formulae for the area A of a triangle, but
the most remarkable is Heron of Alexandria’s formula:
122

Building with triangles
The triangle is indispensable in building
...
You can push a square or
rectangle out of shape but not a triangle
...
One breakthrough
occurred in the building of bridges
...
It was patented in
1848 by James Warren and the first bridge designed in this way was constructed at
London Bridge Station two years later
...


It can be applied to any triangle and we don’t even have to know any angles
...
For example, if a triangle has sides 13, 14 and 15, the
perimeter is 13 + 14 + 15 = 42, so that s = 21
...
The triangle is a familiar object, whether to
children playing with simple shapes or researchers dealing on a day-to-day basis
with the triangle inequality in abstract mathematics
...
The triangle has received much attention but it is surprising
that so much is waiting to be discovered about three lines forming such a basic
figure
...
Artists do it all the time; architects lay out a sweep of new
buildings in the curve of a crescent, or a modern close
...
Sportspeople make their way up the pitch in a curve, and when they shoot for
goal, the ball follows a curve
...


Mathematicians have studied curves for centuries and from many vantage
points
...


125

The conic sections

Classical curves
The first family in the realm of the classical curves are what we call ‘conic
sections’
...
The conic is formed from the double cone, two ice-cream cones
joined together where one is upside down
...

We can think of a conic as the projection of a circle onto a screen
...
The image
on the ceiling will be a circle but if we tip the lamp, this circle will become an
ellipse
...

The conics can also be described from the way points move in the plane
...
If a point moves so that its distance from one fixed point is
always the same, we get a circle
...
The ellipse was the key to
the motion of the planets
...


126

The parabola

Not so obvious is the point which moves so that its distance from a point (the
focus F) is the same as its perpendicular distance from a given line (the directrix)
...
The parabola has a host of useful properties
...

On the other hand, if TV signals are sent out by a satellite and hit a parabolashaped receiving dish, they are gathered together at the focus and are fed into
the TV set
...
Pythagoras loved the spiral and much later
Leonardo da Vinci spent ten years of his life studying their different types, while
René Descartes wrote a treatise on them
...


127

The logarithmic spiral

Jacob Bernoulli of the famed mathematical clan from Switzerland was so
enamoured with the logarithmic spiral that he wanted it carved on his tomb in
Basle
...
A three-dimensional spiral which winds itself around a
cylinder is called a helix
...

There are many classical curves, such as the limaçon, the lemniscate and the
various ovals
...
The
catenary curve was the subject of research in the 18th century and it was
identified as the curve formed by a chain hanging between two points
...


128

Three-bar motion

One aspect of 19th-century research on curves was on those curves that were
generated by mechanical rods
...
In the steam age
this was a significant step forward
...
If the ‘coupler bar’ PQ
moves in any which way, the locus of a point on it turns out to be a curve of
degree six, a ‘sextic curve’
...
For example, the circle of radius 1 has the
equation x2 + y2 = 1, which is an equation of the second degree, as all conics
are
...

In a major study Isaac Newton classified curves described by algebraic
equations of degree three, or cubic curves
...
The explosion of the number of
different types continues for quartic curves, with so many different types that the
129

full classification has never been carried out
...
Many curves
such as catenarys, cycloids (curves traced out by a point on a revolving wheel)
and spirals are not easisly expressible as algebraic equations
...
Camille Jordan proposed a theory of curves built on the
definition of a curve in terms of variable points
...
If we let x = t2 and y = 2t then, for different values of t,
we get many different points that we can write as coordinates (x, y)
...

If we plot these points on the x–y axes and ‘join the dots’ we will get a parabola
...
For him this was the
definition of a curve
...

Jordan’s celebrated theorem has meaning
...
Its apparent ‘obviousness’ is a deception
...
He could organize
the points on a square so that they could all be ‘traced out’ and at the same time
conform to Jordan’s definition
...

Examples of space-filling curves and other pathological examples caused
mathematicians to go back to the drawing board once more and think about the
foundations of curve theory
...
At the start of the 20th century this task took mathematics
into the new field of topology
...
High on
the agenda are qualities which do not change when shapes are transformed into other
shapes
...
Topologists are people
who cannot tell the difference between a donut and a coffee cup!

A donut is a surface with a single hole in it
...
Here’s how a donut can be transformed
into a coffee cup
...
An example of a polyhedron is a cube, with 6
square faces, 8 vertices (points at the junction of the faces) and 12 edges (the
lines joining the vertices)
...


Tetrahedron

Cube

Octahedron

133

Dodecahedron

Icosahedron

Truncated icosahedron

Topology is a relatively new subject, but it can still be traced back to the
Greeks, and indeed the culminating result of Euclid’s Elements is to show that
there are exactly five regular polyhedra
...


If we drop the condition that each face be the same, we are in the realm of the
134

Archimedean solids which are semi-regular
...
If we slice off (truncate) some corners of the icosahedron we
have the shape used as the design for the modern soccer ball
...
There are 90
edges and 60 vertices
...
These ‘bucky balls’ are a newly discovered form of carbon, C60, with a
carbon atom found at each vertex
...
This theorem
actually challenges the very notion of a polyhedron
...
Euler’s formula does not
work
...
Alternatively, the formula could be generalized
to include this peculiarity
...

There is no way of transforming the donut into a ball since the donut has a hole
but the ball does not
...

So a way of classifying surfaces is by the number of holes they contain
...
Once this is done, we can count the
number of vertices, edges, and faces
...
In the case of one hole (r = 1), as
was the case with the cube with a tunnel, V – E + F = 0
...
The outside of a ball is different from
the inside and the only way to cross from one side to the other is to drill a hole
in the ball – a cutting operation which is not allowed in topology (you can stretch
but you cannot cut)
...
The only place where one side meets the other side is along the bounding
curve formed by the edges of the paper
...
Nevertheless, a famous one
was discovered by the German mathematician and astronomer August Möbius in
the 19th century
...
The result is a ‘Möbius strip’, a
one-sided surface with a boundary curve
...
Before long you are back where you started!
It is even possible to have a one-sided surface that does not have a boundary
curve
...
What’s particularly impressive about this bottle is that it does not intersect
itself
...

Both these surfaces are examples of what topologists call ‘manifolds’ –
geometrical surfaces that look like pieces of two-dimensional paper when small
portions are viewed by themselves
...


The Poincaré conjecture
For more than a century, an outstanding problem in topology was the
celebrated Poincaré conjecture, named after Henri Poincaré
...

The part of the conjecture that remained unsolved until recently applied to
closed 3-manifolds
...
Poincaré conjectured that certain closed 3-manifolds which had
all the algebraic hallmarks of being three-dimensional spheres actually had to be
spheres
...

No one could prove the Poincaré conjecture for 3-manifolds
...
There were many false proofs, until in 2002 when it was
recognized that Grigori Perelman of the Steklov Institute in St Petersburg had
finally proved it
...


the condensed idea
From donuts to coffee cups

138

24

Dimension

Leonardo da Vinci wrote in his notebook: ‘The science of painting begins with the point,
then comes the line, the plane comes third, and the fourth the body in its vesture of
planes
...
What could be more
obvious? It is the way the point, line, plane and solid geometry had been propagated by
the Greek geometer Euclid, and Leonardo was following Euclid’s presentation
...
In
physical space we can move out of this page along the x-axis, or across it
horizontally along the y-axis or vertically up the z-axis, or any combination of
these
...


The space of three dimensions

A cube plainly has these three dimensions and so does everything else which
has solidity
...

Around the beginning of the 19th century, mathematicians began to dabble in
139

four dimensions and in even higher n-dimensional mathematics
...


Higher physical dimensions
Many leading mathematicians in the past thought that four dimensions could
not be imagined
...

A common way to explain why four dimensions could be possible was to fall
back to two dimensions
...
They could not see triangles, squares or circles which existed
in Flatland because they could not go out into the third dimension to view them
...
They had the same problems thinking about a
third dimension that we do thinking of a fourth
...

The need to contemplate the actual existence of a four-dimensional space
became more urgent when Einstein came on the scene
...
Unlike Newton, Einstein conceived time as
bound together with space in a four-dimensional space–time continuum
...

Nowadays the four-dimensional Einsteinian world seems quite tame and
matter of fact
...
In
this theory, the familiar subatomic particles like electrons are the manifestations
of extremely tiny vibrating strings
...

Current research suggests that the dimension of the accommodating space–time
continuum for string theory should be either 10, 11 or 26, depending on further
assumptions and differing points of view
...
It
is intended to uncover the structure of matter and, as a by-product, may point to
140

a better theory and the ‘correct’ answer on dimensionality
...


Hyperspace
Unlike higher physical dimensions, there is absolutely no problem with a
mathematical space of more than three dimensions
...
Since the early 19th century mathematicians have
habitually used n variables in their work
...
L
...
There
seemed no good reason to limit the mathematics and everything to be gained in
elegance and clarity
...
A circle in two
dimensions has an equation x2 + y2 = 1, a sphere in three dimensions has an
equation x2 + y2 + z2 = 1, so why not a hypersphere in four dimensions with
equation x2 + y2 + z2 + w2 = 1
...
The cube has six faces each
of which is a square and there are 2 × 2 × 2 = 8 corners
...
So there are 2 × 2 × 2 × 2 = 16 possible
corners for the four-dimensional cube, and eight faces, each of which is a cube
...
This shows a projection of the fourdimensional cube which exists in the mathematician’s imagination
...

A mathematical space of many dimensions is quite a common occurrence for
pure mathematicians
...
In the great problem of the
classification of groups, for instance (see page 155), the ‘monster group’ is a way
of measuring symmetry in a mathematical space of 196,883 dimensions
...

The mathematician’s concern for dimension is entirely separate from the
meaning the physicist attaches to dimensional analysis
...
So, using their
dimensional analysis a physicist can check whether equations make sense since
both sides of an equation must have the same dimensions
...
A dimensional analysis gives velocity as
metres per second so it has dimension of length divided by time or L/T, which
we write as LT −1
...

Coordinated people
Human beings themselves are many dimensioned things
...
We could use (a, b, c, d, e, f, g, h), for age, height, weight, gender, shoe
size, eye colour, hair colour, nationality, and so on
...
If we limit ourselves to this eight-dimensional ‘space’ of people, John Doe might have
coordinates like (43 years, 165 cm, 83 kg, male, 9, blue, blond, Danish) and Mary Smith’s
coordinates might be (26 years, 157 cm, 56 kg, female, 4, brown, brunette, British)
...
Other concepts of dimension
can be defined independently in terms of abstract mathematical spaces
...
Leading figures in many branches
of mathematics have delved into the meaning of dimension including Henri
Lebesgue, L
...
J
...

The pivotal book on the subject was Dimension Theory
...


Dimension in all its forms
From the three dimensions introduced by the Greeks the concept of dimension
has been critically analysed and extended
...
There have been forays into fractional dimensions with fractal
shapes (see page 100) with several different measures being studied
...
Dimension is so much more than the one,
two, three of Euclidean geometry
...
It dutifully struck dots in curious places on a white page, and when it
had stopped its clatter the result looked like a handful of dust smudged across the
sheet
...
He saw it was important, but what
was it? The image that slowly appeared before him was like the black and white print
emerging from a photographic developing bath
...


This was experimental mathematics par excellence, an approach to the subject
in which mathematicians had their laboratory benches just like the physicists and
chemists
...
New vistas opened up – literally
...

The downside of this experimental approach was that the visual images
preceded a theoretical underpinning
...
Although Mandelbrot coined the word ‘fractals’, what were they? Could
there be a precise definition for them in the usual way of mathematics? In the
beginning, Mandelbrot didn’t want to do this
...
He felt the notion of a fractal, ‘like a good wine – demanded a bit of
aging before being “bottled”
...
They were playing with the simplest of formulae
...
The formula which generated the Mandelbrot set was simply x2 + c
...
Let’s choose c = 0
...
Starting with
x = 0 we substitute into the formula x2 + 0
...
This first calculation gives 0
...
We now use this as x, substituting it into x2 + 0
...
5)2 + 0
...
75
...
75)2 + 0
...
0625
...
Carrying on we find that the answer gets bigger and bigger
...
5
...
5 to give – 0
...
Carrying on we get – 0
...
3660
...
5 the sequence starting at x = 0 zooms off to infinity,
but by choosing c = – 0
...
3660
...


The Mandelbrot set

This is not the whole story because so far we have only considered the onedimensional real numbers – giving a one-dimensional Mandelbrot set so we
wouldn’t see much
...
This will give
us a two-dimensional Mandelbrot set
...
In the Mandelbrot set we see another key property of fractals,
that of self-similarity
...


Before Mandelbrot
Like most things in mathematics, discoveries are rarely brand new
...

Unfortunately they did not have the computing power to investigate matters
further
...
As they were so pathological they had been
locked up in the mathematician’s cupboard and given little attention
...
With the popularity of fractals, other mathematicians
whose work was resurrected were Gaston Julia and Pierre Fatou who worked on
fractal-like structures in the complex plane in the years following the First World
War
...


The generating element of the Koch snowflake

Other famous fractals
The famous Koch curve is named after the Swedish mathematician Niels
Fabian Helge von Koch
...

It is generated from the side of the triangle treated as an element, splitting it into
three parts each of length ⅓ and adding a triangle in the middle position
...
It is a curve which encloses a finite area but has an ‘infinite’
circumference!
Another famous fractal is named after the Polish mathematician Wacław
Sierpinski
...


The Sierpin ski gasket

Fractional dimension
The way Felix Hausdorff looked at dimension was innovative
...
If a line is scaled up by a factor of 3 it is 3 times longer than it was
previously
...
If a solid square
147

is scaled up by a factor of 3 its area is 9 times its previous value or 32 and so the
dimension is 2
...
These values of the Hausdorff dimension
all coincide with our expectations for a line, square, or cube
...
Following the scheme described, the Hausdorff dimension is
the value of D for which 4 = 3D
...
262
...

The Hausdorff dimension informed Mandelbrot’s definition of a fractal – a set
of points whose value of D is not a whole number
...


The applications of fractals
The potential for the applications of fractals is wide
...

Fractals have already been applied to the growth of marine organisms such as
corals and sponges
...
In medicine they have found application in the
modelling of brain activity
...

Mandelbrot’s work opened up a new vista and there is much still to be
discovered
...
While Napoleon was advancing on Moscow, his
compatriot the Marquis Pierre-Simon de Laplace published an essay on the deterministic
universe: if at one particular instant, the positions and velocities of all objects in the
universe were known, and the forces acting on them, then these quantities could be
calculated exactly for all future times
...
Chaos theory shows us that the world is more intricate that
that
...
This was
reasonable, for surely sprinters who started a tenth of a second after the gun had
fired would break the tape only a tenth of a second off their usual time
...
Chaos theory exploded this idea
...
If
fine weather is predicted for a day in Europe, but a butterfly flaps its wings in
South America then this could actually presage storms on the other side of the
world – because the flapping of the wings changes the air pressure very slightly
causing a weather pattern completely different from the one originally forecast
...
If you drop a
ball-bearing through the opening in the top of a pinboard box it will progress
downwards, being deflected one way or the other by the different pins it
encounters on route until it reaches a finishing slot at the bottom
...
If you could do this exactly then the Marquis de
Laplace would be correct and the path followed by the ball would be exactly the
same
...


Pinboard box experiment

But of course you cannot let the ball go from exactly the same position with
exactly the same velocity and force
...
The result is the ballbearing may take a very different route to the bottom and probably end up in a
different slot
...
As
the pendulum swings back and forth, it gradually loses energy
...


151

The free pendulum

The movement of the bob can be plotted in a phase diagram
...
The point of release is plotted at the point A on the
positive horizontal axis
...
As the bob moves through the vertical axis (where the displacement is
zero) the velocity is at a maximum, and this is plotted on the phase diagram at B
...
The bob then swings back through D (where it
is moving in the opposite direction so its velocity is negative) and completes one
swing at E
...
As the
pendulum swings less and less this phase portrait spirals into the origin until
eventually the pendulum comes to rest
...
If the displacement is small the motion of the double
pendulum is similar to the simple pendulum, but if the displacement is large the
bob swings, rotates, and lurches about and the displacement about the
intermediate joint is seemingly random
...


Chaotic motion
The characteristic of chaos is that a deterministic system may appear to
generate random behaviour
...
The value of a must be somewhere
between 0 and 4 to guarantee that the value of p stays in the range from 0 to 1
...
If we pick a starting value of, say p =
0
...
3 into a
× p × (1 – p) to give 0
...
Using only a handheld calculator we can repeat this
operation, this time with p = 0
...
4872)
...
In this case, the
population quickly settles down to p = 0
...
This settling down always takes place
for values of a less than 3
...
9, a value near the maximum permissible, and use
the same initial population p = 0
...
This is because the value of a is in the ‘chaotic region’, that is, a
is a number greater than 3
...
Moreover, if we choose a different initial
population, p = 0
...
3, the population growth shadows the
previous growth pattern for the first few steps but then starts to diverge from it
completely
...


Population changing over time for a = 3
...
Over just a few days forecasting the
weather still gives us nasty surprises
...

From meteorology to mathematics
The discovery of the butterfly effect happened by chance around 1961
...
He had been aiming to
recapture some interesting weather plots but found the new graph unrecognizable
...
Was it time to trade in his old computer and get something more
reliable?
After some thought he did spot a difference in the way he had entered the initial
values: before he had used six decimal places but on the rerun he only bothered with
three
...
After this discovery
his intellectual interests migrated to mathematics
...
The Navier–Stokes
equations that resulted are of intense interest to scientists
...
Applied to the problem of fluid flow, much is known about the
steady movements of the upper atmosphere
...

While a lot is known about the theory of linear systems of equations, the
Navier–Stokes equations contain nonlinear terms which make them intractable
...


155

Strange attractors
Dynamic systems can be thought of possessing ‘attractors’ in their phase
diagrams
...
With the double pendulum it’s
more complicated, but even here the phase portrait will display some regularity
and be attracted to a set of points in the phase diagram
...
So all is not lost
...


the condensed idea
The wildness of regularity

156

27

The parallel postulate

This dramatic story begins with a simple geometric scenario
...
How many lines can we draw through P parallel to the line l? It appears
obvious that there is exactly one line through P which will never meet l no matter how
far it is extended in either direction
...
Euclid of Alexandria included a variant of it as one of his postulates
in that foundation of geometry, the Elements
...
We shall see whether Euclid’s
assumption makes mathematical sense
...
One of the most influential mathematics texts ever written, Greek
mathematicians constantly referred to it as the first systematic codification of
geometry
...

The Elements percolated down to school level and readings from the ‘sacred
book’ became the way geometry was taught
...
As the poet A
...
Hilton quipped: ‘though they wrote it
all by rote, they did not write it right’
...
In English schools, it reached the zenith of its influence as a subject in
the curriculum during the 19th century but it remains a touchstone for
mathematicians today
...
Euclid’s postulates are an excellent example, and one that set the
model for later axiomatic systems
...
A straight line can be drawn from any point to any point
...
A finite straight line can be extended continuously in a straight line
...
A circle can be constructed with any centre and any radius
...
All right angles are equal to each other
...
If a straight line falling on two straight lines makes the interior angles on
the same side less than two right angles, the two straight lines, if extended
indefinitely, meet on that side on which the angles are less than two right angles
...
Sherlock
Holmes would have admired its deductive system which advanced logically from
the clearly stated postulates and may have castigated Dr Watson for not seeing it
as a ‘cold unemotional system’
...
Euclid added ‘definitions’ and
‘common notions’
...
Common notions include
such items as ‘the whole is greater than the part’ and ‘things which are equal to
the same thing are also equal to one another’
...


The fifth postulate
It is Euclid’s fifth postulate that caused controversy over 2000 years after the
Elements first appeared
...
Euclid himself was unhappy with it but he needed it to
prove propositions and had to include it
...


158

Later mathematicians either tried to prove it or replace it by a simpler
postulate
...
Around the same time, Adrien Marie Legendre substituted another
equivalent version when he asserted the existence of a triangle whose angles add
up to 180 degrees
...
They were more acceptable than the
cumbersome version given by Euclid
...

This exerted a powerful attraction on its adherents
...

Unfortunately attempts to do this turned out to be excellent examples of circular
reasoning, arguments which assume the very thing they are trying to prove
...
Gauss did not publish his work, but it seems
clear he reached his conclusions in 1817
...
There is no doubting the brilliance of all these men
...
By adding
its negation to the other four postulates, they showed a consistent system was
possible
...
How can this be?
Surely the dotted lines meet l
...
The diagram is therefore a confidence trick, for what Bolyai
and Lobachevsky were proposing was a new sort of geometry which does not
conform to the commonsense one of Euclid
...


The shortest paths between the points on a pseudosphere play the same role
as straight lines in Euclid’s geometry
...
This
geometry is called hyperbolic geometry
...
Put a different way, there are no lines through P which are ‘parallel’ to
l
...
One model for it is the geometry on the surface of a
sphere
...
In this
non-Euclidean geometry the sum of the angles in a triangle is greater than 180
degrees
...

The geometry of Euclid which had been thought to be the one true geometry
– according to Immanuel Kant, the geometry ‘hard-wired into man’ – had been
knocked off its pedestal
...
The different versions
were unified under one umbrella by Felix Klein in 1872
...
It is the
general theory of relativity which demands a new kind of geometry – the
geometry of curved space–time, or Riemannian geometry
...
The presence of massive objects in
space, like the Earth and the Sun cause space–time to be curved
...

This curvature measured by Riemannian geometry predicts how light beams
bend in the presence of massive space objects
...
One
reason is that Euclidean space is flat – there is no curvature
...
Underlying Riemannian space–time is a concept of curvature which varies
continuously – just as the curvature of a rumpled piece of cloth varies from point
to point
...

No wonder that Gauss was so impressed by young Riemann in the 1850s and
even suggested then that the ‘metaphysics’ of space would be revolutionized by
his insights
...

In ordinary geometry there are continuous lines and solid shapes to investigate, both of
which can be thought of as being composed of points ‘next to’ each other
...

Discrete geometry can involve a finite number of points and lines or lattices of points –
the continuous is replaced by the isolated
...
This geometry poses interesting problems and has applications in such
disparate areas as coding theory and the design of scientific experiments
...
Imagine the light ray
starts at the origin O and sweeps between the horizontal and the vertical
...

The equation of the ray through the origin is y = mx
...
If the ray is y = 2x then
it will hit the point with coordinates x = 1 and y = 2 because these values satisfy
the equation
...
Consequently if m is not a genuine fraction (it may be √2, for
example) the light ray will miss all the lattice points
...
We could imagine ourselves at the origin identifying the
points that can be seen from there, and those that are obscured
...
These are points
with coordinates, such as x = 2 and y = 3, where no number other than 1
divides both x and y
...


Pick’s theorem
The Austrian mathematician Georg Pick has two claims to fame
...
The other is that he
wrote a short paper, published in 1899, on ‘reticular’ geometry
...
This is pinball mathematics
...
In our example, the
number of points on the boundary is b = 22 and the number of interior points is
c = 7
...
The area is 17 square units
...
Pick’s theorem can be applied to any shape which joins
discrete points with whole number coordinates, the only condition being that the
boundary does not cross itself
...
Named after the
Italian mathematician Gino Fano, who pioneered the study of finite geometry, the
Fano plane is the simplest example of a ‘projective’ geometry
...

The seven points are labelled A, B, C, D, E, F and G
...
This is no problem since lines in
discrete geometry do not have to be ‘straight’ in the conventional sense
...


These two properties illustrate the remarkable duality which occurs in
geometries of this kind
...

If, in any true statement, we swap the two words and make small adjustments
to correct the language, we get another true statement
...
Not so Euclidean geometry
...
We can quite happily speak
of the concept of parallelism in Euclidean geometry
...
In projective geometry all pairs of lines meet in a point
...


The Fano plane made Euclidean

If we remove one line and its points from the Fano plane we are once more
back in the realm of unsymmetrical Euclidean geometry and the existence of
parallel lines
...

With one line fewer there are now six lines: AB, AC, AE, BC, BE and CE
...
Lines are parallel in this sense if they have no points in common – like
the lines AB and CE
...
It is one key to Thomas Kirkman’s
167

schoolgirl problem (see page 167)
...
Given a
finite number of n objects an STS is a way of dividing them into blocks of three
so that every pair taken from the n objects is in exactly one block
...


Pascal’s theorem

A pair of theorems
Pascal’s theorem and Brianchon’s theorem lie on the boundary between
continuous and discrete geometry
...

Pascal’s theorem was discovered by Blaise Pascal in 1639 when he was only 16
years old
...
We’ll call P
the point where the line A 1B2 intersects A 2B1; Q the point where the line A 1C2
intersects A 2C1; and R the point where the line B1C2 intersects B2C1
...

Pascal’s theorem is true whatever the positions of the different points around
the ellipse
...


Brianchon’s theorem

Brianchon’s theorem was discovered much later by the French mathematician
and chemist Charles-Julien Brianchon
...
Next
we can define three diagonals, the lines p, q and r, by the meeting of lines, so
that: p is the line between the points where a1 meets b2 and where a2 meets b1; q
is the line between the points where a1 meets c2 and a2 meets c1; and r is the line
between the points where b1 meets c2 and b2 meets c1
...

These two theorems are dual to each other, and it is another instance of the
theorems of projective geometry occurring in pairs
...
At school we draw curves which show
the relationship between variables x and y
...


Königsberg is a city in East Prussia famous for the seven bridges which cross
the River Pregel
...


In the 18th century a curious question was posed: was it possible to set off
and walk around Königsberg crossing each bridge exactly once? The walk does
not require us to finish where we started – only that we cross each bridge once
...
In our semiabstract diagram, the island in the middle of the river is labelled I and the banks
of the river by A, B and C
...
The key step is to peel
away the semi-abstractness and progress to complete abstraction
...
The land is represented by ‘points’ and the
bridges joining them are represented by ‘lines’
...
These things are unimportant
...

170

Euler made an observation about a successful walk
...

Translating this thought into the abstract picture, we may say that lines
meeting at a point must occur in pairs
...

The number of lines meeting at a point is called the ‘degree’ of the point
...


171

Looking at the graph representing Königsberg, every point is of odd degree
...
If the bridge setup were changed then such a walk may become
possible
...
This means we could begin a walk on A and end on B
having walked over every bridge exactly once
...


The hand-shaking theorem
If we were asked to draw a graph that contained three points of odd degree,
we would have a problem
...
It cannot be done because
In any graph the number of points with odd degree must be an even
number
...
In any
graph every line has a beginning and an end, or in other words it takes two
people to shake hands
...
Next we say there are x points with
odd degree and y points with even degree
...
So we have Nx + Ny = N, and therefore Nx = N – Ny
...
But x itself cannot be odd because the addition of an odd
172

number of odd degrees would be an odd number
...


Non-planar graphs
The utilities problem is an old puzzle
...
We have to connect each of the houses to each of the
utilities, but there’s a catch – the connections must not cross
...
The graph described by connecting three points to another three points
in all possible ways (with only nine lines) cannot be drawn in the plane without
crossings
...
This utilities graph, along with the
graph made by all lines connecting five points, has a special place in graph
theory
...


Trees
173

A ‘tree’ is a particular kind of graph, very different from the utitlities graph or
the Königsberg graph
...
Such
a route from a point and back to itself is called a cycle
...


A familiar example of a tree graph is the way directories are arranged in
computers
...
Because there are no cycles there is no way to cross
from one branch other than through the root directory – a familiar manoeuvre
for computer users
...
For example, there are exactly three different tree types with five
points:

174

Cayley was able to count the number of different tree types for small numbers
of points
...

Since then the calculations have advanced as far as trees with as many as 22
points
...

Even in its own time, Cayley’s research had practical applications
...
Compounds with the
same number of atoms but with different arrangements have different chemical
properties
...


the condensed idea
Across the bridges and into the
trees

175

30

The four-colour problem

Who might have given young Tiny Tim a Christmas present of four coloured wax crayons
and a blank county map of England? It could have been the cartographer neighbour who
occasionally sent in small gifts, or that odd mathematician Augustus De Morgan, who
lived nearby and passed the time of day with Tim’s father
...

The Cratchits lived in a drab terrace house in Bayham Street, Camden Town just north of
the newly opened University College, where De Morgan was professor
...


De Morgan had definite ideas on how this should be done: ‘you are to colour
the map so that two counties with a common border have different colours’
...
De Morgan would
have smiled and left him to the task
...
The problem stirred De Morgan’s
mathematical imagination
...
Three colours, though, are not
enough
...
If only blue, green and
red were available we could start off by colouring Nevada and Idaho
...
So far so good
...
This means that
both Oregon and Idaho are coloured green so cannot be distinguished
...
Would these four colours – blue, green, red
and yellow be sufficient for any map? This question is known as the four-colour
problem
...
In
the 1860s, Charles Sanders Peirce, an American mathematician and philosopher,
thought he had proved it but there is no trace of his argument
...
He saw publicity value in it and inveigled
177

the eminent Cambridge mathematician Arthur Cayley to write a paper on it in
1878
...
This contribution spurred on his student Alfred
Bray Kempe to attempt a solution
...
Cayley heartily congratulated him, his proof was published, and
he gained election to the Royal Society of London
...
There was a
surprise ten years later when Durhambased Percy Heawood found an example of
a map which exposed a flaw in Kempe’s argument
...
It would be back to the drawing boards for mathematicians and a
chance for some new tyro to make their mark
...
This would have been a great result if someone could
construct a map that could not be coloured with four colours
...
What about maps drawn on a surface like a donut – a surface
more interesting to mathematicians for its shape than its taste
...
He even proved a result for a multi-holed donut (with a
number, h, holes) in which he counted the number of colours that guaranteed
any map could be coloured – though he had not proved these were the minimum
number of colours
...
The square brackets indicate that
we only take the whole number part of the term within them
...
3107
...
Heaward’s formula was derived on the strict
understanding that the number of holes is greater than zero
...


The problem solved?
After 50 years, the problem which had surfaced in 1852 remained unproved
...

Some progress was made and one mathematician proved that four colours
were enough for up to 27 countries on a map, another bettered this with 31
countries and one came in with 35 countries
...
In fact the observations made by Kempe and Cayley in their
very early papers provided a better way forward, and mathematicians found that
they had only to check certain map configurations to guarantee that four colours
were enough
...
This
checking could not be done by hand but luckily the German mathematician
Wolfgang Haken, who had worked on the problem for many years, was able to
enlist the services of the American mathematician and computer expert Kenneth
Appel
...
By late June 1976, after many sleepless nights, the job was done and in
partnership with their trusty IBM 370 computer, they had cracked the great
problem
...
They replaced their ‘largest discovered prime’ postage stamp with the
news that ‘four colours suffice
...

The applause was patchy
...
The trouble was that it was a computer-based
proof and this stepped right outside the traditional form of a mathematical proof
...
It raised the issue
of ‘checkability’
...
Errors in computer coding can surely be
made
...

That was not all
...
How could
anyone read through the proof and appreciate the subtlety of the problem, or
experience the crucial part of the argument, the aha moment
...
He thought that a computer
proof had as much credibility as a proof by a reputable fortune teller
...
They might well have done pre Appel and
Haken, but not afterwards
...
This said,
the mathematical world still awaits a shorter proof along traditional lines
...


the condensed idea
Four colours will be enough
180

Four colours will be enough

181

31

Probability

What is the chance of it snowing tomorrow? What is the likelihood that I will catch the
early train? What is the probability of you winning the lottery? Probability, likelihood,
chance are all words we use every day when we want to know the answers
...


Probability theory is important
...
But how can a theory involving uncertainty
be quantified? After all, isn’t mathematics an exact science?
The real problem is to quantify probability
...

What is the probability of getting a head? We might rush in and say the answer is
½ (sometimes expressed as 0
...
Looking at the coin we make the
assumption it is a fair coin, which means that the chance of getting a head equals
the chance of getting a tail, and therefore the probability of a head is ½
...
There are two main theories in the assignment of
probabilities
...
Another is the relative frequency approach, where we conduct the
experiment a large number of times and count the number of heads
...

But what about coming to a sensible measure of the probability of it snowing
tomorrow? There will again be two outcomes: either it snows or it does not
snow, but it is not at all clear that they are equally likely as it was for the coin
...
But even
then it is not possible to pinpoint an exact number for this probability
...
In mathematics,
probability is measured on a scale from 0 to 1
...
A probability of 0
...
9 would signify a high probability
...
They found a
simple game puzzling
...
This view was shattered
when the probabilities were analysed
...

Because the results of the throws do not affect each other, they are ‘independent’
and we can multiply the probabilities
...
517746
...

The probability of at least one double six is therefore
1 − (35/36)24 = 0
...


We can take this example a little further
...
When two distinguishable dice (red and blue) are
thrown there are 36 possible outcomes and these may be recorded as pairs (x,y)
and displayed as 36 dots against a set of x/y axes – this is called the ‘sample
space’
...

There are 6 combinations that each add up to 7, so we can describe the event by
A = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

and ring it on the diagram
...
If we let B be the event of getting the sum on the
dice equal to 11 we have the event B = {(5,6), (6,5)} and Pr(B) = 2/36 = 1/18
...
You win at the first throw if either the event A or B occurs – this is
called a ‘natural’
...
You lose at the first stage if you throw a 2, 3
or a 12 (this is called ‘craps’)
...
If a sum of either 4, 5, 6, 8, 9 or 10 is thrown,
you go onto a second stage and the probability of doing this is 24/36 = 2/3
...
In craps,
for every 36 games you play, on average you will win at the first throw 8 times
and not win 28 times so the odds against winning on the first throw are 28 to 8,
which is the same as 3
...


184

The monkey on a typewriter
Alfred is a monkey who lives in the local zoo
...
He sits in a
corner filled with literary ambition, but his method of writing is curious – he hits
the keys at random
...
More
than this, there is a chance (albeit smaller) he will follow this with a translation
into French, and then Spanish, and then German
...
The chance of all this is minute, but it is certainly not zero
...
Let’s see how long he will take to type the soliloquy in Hamlet,
starting off with the opening ‘To be or’
...

The number of possibilities for the first position is 30, for the second is 30,
and so on
...
The chance of Alfred getting as far as ‘To be or’ is 1
chance in 6
...
If Alfred hits the typewriter once every second there is
an expectation he will have typed ‘To be or’ in about 20,000 years, and proved
himself a particularly long-lived primate
...
Alfred will produce nonsense like ‘xo,h?yt?’ for a great
deal of the time
...
In 1933, Andrey
Nikolaevich Kolmogorov was instrumental in defining probability on an axiomatic
basis – much like the way the principles of geometry were defined two millennia
before
...
the probability of all occurrences is 1
2
...
when occurrences cannot coincide their probabilities can be added

From these axioms, dressed in technical language, the mathematical
properties of probability can be deduced
...
Much of modern life cannot do without it
...
Who’d have thought the gambling problems that kick-started these ideas
in the 17th century would spawn such an enormous discipline? What were the
chances of that?

the condensed idea
The gambler’s secret system

186

32

Bayes’s theory

The early years of the Rev
...
Born in the southeast of
England, probably in 1702, he became a nonconformist minister of religion, but also
gained a reputation as a mathematician and was elected to the Royal Society of London
in 1742
...
It gave a formula for finding inverse
probability, the probability ‘the other way around’, and it helped create a concept
central to Bayesian philosophy – conditional probability
...
The
frequentists adopt a view of probability based on hard numerical data
...


Conditional probability
Imagine that the dashing Dr Why has the task of diagnosing measles in his
patients
...
A patient may have measles without having spots and
some patients may have spots without having measles
...

Bayesians use a vertical line in their formulae to mean ‘given’, so if we write
prob(a patient has spots | the patient has measles)

it means the probability that a patient has spots given that they have measles
...
In relation to each other,
one is the probability the other way around
...
Mathematicians like nothing better than using
notation to stand for things
...
The symbol is the event of a patient not
having spots and the event of not having measles
...

187

Venn diagram showing the logical structure of the appearance of spots and measles

This tells Dr Why that there are x patients who have measles and spots, m
patients who have measles, while the total number of patients overall is N
...
The
conditional probability, the probability that someone has spots given that they
have measles, written prob(S|M), is x/m
...
Dr Why will have
a good idea about prob(S|M), the probability that if a patient has measles, they
have spots
...
Finding this out is the inverse problem and the kind of problem
188

addressed by Bayes in his essay
...
These will be subjective but what is important is to see how they
combine
...
9 and if the patient does not have measles, the
probability of them having spots prob(S| ) will be low, say 0
...
In both these
situations Dr Why will have a good idea of the values of these probabilities
...
This is expressed as prob(M) = 0
...
The
only other piece of information we need is prob(S), the percentage of people in
the population who have spots
...
From our key relations,
prob(S) = 0
...
2 + 0
...
8 = 0
...
Substituting these values into Bayes’s
formula gives:

The conclusion is that from all the patients with spots that the doctor sees he
correctly detects measles in 60% of his cases
...
9 to 0
...
15 to 0
...
How does this change improve his
rate of measles detection? What is the new prob(M|S)? With this new
information, prob(S) = 0
...
2 + 0
...
8 = 0
...
2 divided by prob(S) = 0
...
95,
which comes to 0
...
So Dr Why can now detect 70% of cases with this
improved information
...
99, and 0
...
961 so his chance of a
correct diagnosis in this case would be 96%
...
The contentious sticking point is
189

interpreting probability as degrees of belief or, as it is sometimes defined
subjective probability
...
Strictly speaking this criterion only applies to civil
cases but we shall imagine a scenario where it applies to criminal cases as well
...
Not so the Bayesian who does
not mind taking feelings on board
...
Here is a possible scenario
...
During deliberations in the jury room the
jury is called back into court to hear further evidence from the prosecution
...
95 if the prisoner is
guilty, but if he is innocent the probability of finding the weapon would be only
0
...
The probability of finding a weapon in the prisoner’s house is therefore
much higher if the prisoner is guilty than if they are innocent
...
The juror has
made an initial assessment that prob(G) = 1/100 or 0
...
This probability is
called the prior probability
...
Bayes’s formula in the form

shows the idea of the prior probability being updated to the posterior
probability prob(G | E)
...
If the prosecution had made the
greater claim that the probability of finding the incriminating weapon was as high
as 0
...
01, then repeating the Bayes’s formula calculation the juror
would have to revise their opinion from 1% to 50%
...
The
leading thrust has been on how one arrives at the prior probability
...
The Bayesian method has applications
in areas as diverse as science, weather forecasting and criminal justice
...
It has a lot going for it
...
As it is likely
that all the passengers are independent of each other, we may safely assume that their
birthdays are randomly scattered throughout the year
...
It is not many, but enough to claim there is a better than even
chance that two passengers share a birthday
...
Even a seasoned expert in probability, William Feller, thought it
astounding
...
How many people must gather in the room so that it
i s certain that two people share the same birthday? There are 365 days in a
standard year (and we’ll ignore leap years just to make things simpler) so if there
were 366 people in the room, at least one pair would definitely have the same
birthday
...

This is the pigeonhole principle: if there are n + 1 pigeons who occupy n
pigeonholes, one hole must contain more than one pigeon
...
However, if you take 365
people at random this would be extremely unlikely and the probability of two
people not sharing a birthday would be minuscule
...
5% chance that two people share a birthday
...
We find that 23 people is the number for which the probability
is just greater than ½ and for 22 people the probability that a birthday is shared
is just less than ½
...
While the answer to the
classic birthday problem is surprising it is not a paradox
...
The probability
that another person has the same birthday as this person is 1/365 and so the
probability these two do not share a birthday is one minus this (or 364/365)
...
The probability of none of
these three sharing a birthday is the multiplication of these two probabilities, or
(364/365) × (363/365) which is 0
...

Continuing this line of thought for 4, 5, 6,
...
When we get as far as 23 people with our pocket calculator we
get the answer 0
...

The negation of ‘none of them sharing a birthday’ is ‘at least two people share a
birthday’ and the probability of this is 1 – 0
...
5073, just greater than the
crucial ½
...
4757, which is
less than ½
...
The birthday result makes a statement about two people
sharing a birthday, but it does not tell us which two people they are
...
If Mr Trevor Thomson whose birthday is on 8
March is in the room, a different question might be asked
...
The probability of Mr Thomson
not sharing his birthday with another person is 364/365 so that the probability
that he does not share his birthday with any of the other n – 1 people in the
room is (364/365)n – 1
...

If we compute this for n = 23 this probability is only 0
...
If we increase the value of n, this
probability will increase
...
For n = 254, its
value is 0
...
This is the cutoff point because n = 253 will give the value
0
...
There will have to be a gathering of 254 people in
the room for a chance greater than ½ that Mr Thomson shares his birthday with
someone else
...

193

Other birthday problems
The birthday problem has been generalized in many ways
...
In this case 88 people would be
required before there is a better than even chance that three people will share the
same birthday
...
are required to share a birthday
...

Other forays into the birthday problem have inquired into near birthdays
...
It turns out that a mere 14 people in
a room will give a greater than even chance of two people having a birthday in
common or having a birthday within a day of each other
...
This can be compared with 23 in the classic birthday problem
...
Suppose we have a long queue forming outside a Bob Dylan
concert and people join it randomly
...
As the fans enter
they are asked their birthdays
...
Experimental results show this is not exactly true (more are
born during the summer months) but it is close enough for the solution to be
applicable
...
In the birthday problem, the
number of cells is 365 (these are identified with possible birthdays) and the balls
to be placed at random in the cells are the people
...
For the boysand-girls problem, the balls are of two colours
...

Satyendra Nath Bose was attracted to Albert Einstein’s theory of light based on
photons
...
For him, the cells were not
days of the year as in the birthday problem but energy levels of the photons
...
There are many applications of occupancy problems in
other sciences
...

The world is full of amazing coincidences but only mathematics gives us the
way of calculating their probability
...


the condensed idea
Calculating coincidences

195

196

34

Distributions

Ladislaus J
...
Not for him a gloomy topic,
they were a field of enduring scientific enquiry
...
Then there was
Frank Benford, an electrical engineer who counted the first digits of different types of
numerical data to see how many were ones, twos and so on
...


All these examples involve measuring the probabilities of events
...
It is also a discrete distribution because
the values of x only take isolated values – there are gaps between the values of
interest
...
As we’ll see, in the case of the Benford distribution we
are only interested in the appearance of digits 1, 2, 3,
...
23
...
He looked at the number of deaths (this was what
mathematicians call the variable) and the number of corps-years when this
number of deaths occurred
...
At the
barracks, Corp C (say) in one particular year experienced four deaths
...
Bortkiewicz
obtained the following data:

197

The Poisson formula

Thankfully, being killed by a horse-kick is a rare event
...
With this technique, could Bortkiewicz
have predicted the results without visiting the stables? The theoretical Poisson
distribution says that the probability that the number of deaths (which we’ll call
X) has the value x is given by the Poisson formula, where e is the special number
discussed earlier that’s associated with growth (see page 24) and the exclamation
mark means the factorial, the number multiplied by all the other whole numbers
between it and 1 (see page 26)
...
We need to find this average over our 200 corpsyears so we multiply 0 deaths by 109 corps-years (giving 0), 1 death by 65
corps-years (giving 65), 2 deaths by 22 corps-years (giving 44), 3 deaths by 3
corps-years (giving 9) and 4 deaths by 1 corps-year (giving 4) and then we add
all of these together (giving 122) and divide by 200
...
61
...
The results are:

It looks as though the theoretical distribution is a good fit for the experimental
data gathered by Bortkiewicz
...
, 9 to be uniformly
distributed
...
In 1938 the electrical engineer Frank Benford found that this was not
true for the first digits of some sets of data
...

198

Yesterday I conducted a little experiment
...
There were exchange rates like
2
...
119 to buy £1 sterling
...
59 to buy £1 sterling and (Hong Kong dollar) HK $15
...
Reviewing the results of the data and recording the number of
appearances by first digit, gave the following table:

These results support Benford’s law, which says that for some classes of data,
the number 1 appears as the first digit in about 30% of the data, the number 2
in 18% of the data and so on
...

It is not obvious why so many data sets do follow Benford’s law
...

Instances where Benford’s distribution can be detected include scores in
sporting events, stock market data, house numbers, populations of countries,
and the lengths of rivers
...
Benford’s law has
practical applications
...


Words
One of G
...
Zipf’s wide interests was the unusual practice of counting words
...
The most common word was given rank 1, the next
rank 2, and so on
...

It is not surprising that ‘the’ is the most common, and ‘of’ is second
...
We shall only consider the top ten words
...
The surprising fact is that the ranks have a bearing on the actual
number of appearances of the words in a text
...
The actual
number is given by a well-known formula
...
The theoretical Zipf’s law says that the percentage
of occurrences of the word ranked r is given by

where the number k depends only on the size of the author’s vocabulary
...
0694
...
94% of
all words in a text
...
47% of the words
...

For writers with only 20,000 words at their command, the value of k rises to
0
...
The smaller the vocabulary, the more often you will see ‘the’
appearing
...
We may not be able to predict a dead cert but knowing how the
probabilities distribute themselves is much better than taking a shot in the dark
...


200

the condensed idea
Predicting how many

201

35

The normal curve

The ‘normal’ curve plays a pivotal role in statistics
...
It certainly has important mathematical properties but
if we set to work analysing a block of raw data we would rarely find that it followed a
normal curve exactly
...
The significance of the normal curve lies less in nature and more in
theory, and in this it has a long pedigree
...
Pierre Simon Laplace published results
about it and Carl Friedrich Gauss used it in astronomy, where it is sometimes
referred to as the Gaussian law of error
...
In other experiments he measured the heights of French conscripts
and the chest measurements of Scottish soldiers and assumed these followed the
normal curve
...


The cocktail party
Let’s suppose that Georgina went to a cocktail party and the host, Sebastian,
asked her if she had come far? She realized afterwards it was a very useful
question for cocktail parties – it applies to everyone and invites a response
...

The next day, slightly hungover, Georgina travelled to the office wondering if
her colleagues had come far to work
...
She took advantage of the fact that she was the Human Resources
Manager of a very large company to tack a question on the end of her annual
employee questionnaire: ‘how far have you travelled to work today?’ She wanted
202

to work out the average distance of travel of the company’s staff
...


Georgina’s histogram of distance travelled by her colleagues to work

This average turned out to be 20 miles
...
The variability in the population
is denoted by the Greek letter sigma, written σ, which is sometimes called the
standard deviation
...
The company’s
marketing analyst, who had trained as a statistician, showed Georgina that she
might have got around the same value of 20 by sampling
...
This estimation technique depends on the Central Limit
Theorem
...
The
larger the sample the better, but 30 employees will do nicely
...
When we calculate the average distance for
our sample, the effect of the longer distances will average out the shorter
distances
...
In Georgina’s case, it is most likely that the value of will be near 20, the
average of the population
...


203

How the sample average is distributed

The Central Limit Theorem is one reason why the normal curve is important
to statisticians
...
What does this
mean? In Georgina’s case, x represents the distance from the workplace and is
the average of a sample
...

This is why we can use the average of a sample as an estimate of the
population average μ
...
If the variability of the x values is the standard deviation σ, the variability
of is σ/√n where n is the size of the sample we select
...


Other normal curves
Let’s do a simple experiment
...
The chance of
throwing a head each time is p = ½
...
Altogether there are 16 possible outcomes
...
There are in fact four possible outcomes
giving three heads (the others are HTHH, HHTH, HHHT) so the probability of
three heads is 4/16 = 0
...

With a small number of throws, the probabilities are easily calculated and
placed in a table, and we can also calculate how the probabilities are distributed
...
These probabilities may be
represented by a diagram in which both the heights and areas describe them
...
What happens if we throw it a
large number, say 100, times? The binomial distribution of probabilities can be
applied where n = 100, but it can usefully be approximated by the normal bellshaped curve with mean μ = 50 (as we would expect 50 heads when tossing a
coin 100 times) and variability (standard deviation) of σ = 5
...


205

Distribution of the probability for the number of heads in 100 throws of a coin

For large values of n, the variable x which measures the number of successes
fits the normal curve increasingly well
...
Now let’s say we
want to know the probability of throwing between 40 and 60 heads
...
To find the
actual numerical value we need to use precalculated mathematical tables, and
once this has been done, we find prob(40 ≤ (x ≤ 60) = 0
...
This shows that
getting between 40 and 60 heads in 100 tosses of a coin is 95
...

The area left over is 1 – 0
...
0455
...
This is just 2
...
If you visit Las Vegas this would be a bet
to leave well alone
...
Correlation and regression go together like a horse and carriage, but
like this pairing, they are different and have their own jobs to do
...

Regression can be used to predict the values of one property (say weight) from the
other (in this case, height)
...
He
originally termed it ‘co-relation’, a better word for explaining its meaning
...
The Pearson correlation coefficient, named after
Galton’s biographer and protégé Karl Pearson, is measured on a scale between
minus one and plus one
...
9, there is said to
be a strong correlation between the variables
...
If it is near to zero the
correlation is practically non-existent
...
Let’s take the example of the sales of
sunglasses and see how this relates to the sales of ice creams
...
If we plot points on a graph where the x (horizontal)
coordinate represents sales of sunglasses and the y (vertical) coordinate gives the
sales of ice creams, each month we will have a data point (x, y) representing
both pieces of data
...
We can plot the monthly data points (x, y) for a whole year on
a scatter diagram
...
9 indicating a strong correlation
...
It is positive because the straight line has a
positive gradient – it is pointing in a northeasterly direction
...
There may be a cause and effect relation between the
two variables but this cannot be claimed on the basis of numerical evidence
alone
...

In the sunglasses and ice cream example, there is a strong correlation
between the sales of sunglasses and that of ice cream
...
It would be ludicrous
to claim that the expenditure on sunglasses caused more ice creams to be sold
...
For
example, the expenditure on sunglasses and on ice creams is linked together as a
result of seasonal effects (hot weather in the summer months, cool weather in
the winter)
...
There may be a high
correlation between variables but no logical or scientific connection at all
...


Spearman’s correlation
Correlation can be put to other uses
...

Occasionally we have only the ranks as data
...
It will be a subjective evaluation
...
If
Albert and Zac ranked them in exactly the same way, that would be fine but life
is not like that
...
The reality is that the
rankings would be in between these two extremes
...
Zac rated Ellie the best, followed by Beth, Ann, Dorothy and Charlotte
...


Spearman’s formula

How can we measure the level of agreement between the judges? Spearman’s
correlation coefficient is the instrument mathematicians use to do this for ordered
209

data
...
6 which indicates a limited measure of agreement
between Albert and Zac
...

The formula for this correlation coefficient was developed in 1904 by the
psychologist Charles Spearman who, like Pearson, was influenced by Francis
Galton
...
If we were all shorter than our parents then
the population would gradually diminish in height and this is equally unlikely
...

Francis Galton conducted experiments in the 1880s in which he compared the
heights of mature young adults with the heights of their parents
...

We are talking about a practical scientist here, so out came the pencils and sheets
of paper divided into squares on which he plotted the data
...
4 cm) which value he called the mediocrity
...
In other words, the children’s heights
regressed towards the mediocrity
...
His batting average in an exceptional
season is likely to be followed by an inferior average in the next, yet overall
would still be better than the average for all players in the league
...

Regression is a powerful technique and is widely applicable
...
The research team observes the
number of staff employed in each
...


211

Let’s plot this on a graph, where we’ll make the x coordinate the number of
customers (we call this the explanatory variable) while the number of staff is
plotted as the y coordinate (called the response variable)
...
The average number of customers in the stores is plotted as 6 (i
...
6000
customers) and the average number of staff in the stores is 40
...
There are formulae
for calculating the regression line, the line which best fits the data (also known as
the line of least squares)
...
8 + 3
...
2 and is positive (going up from left to right)
...
8
...
So if we want to know how many staff should be employed in a store
that receives 5000 customers a month we could substitute the value x = 5 into
the regression equation and obtain the estimate ŷ = 37 staff showing how
regression has a very practical purpose
...
The problems of genetics
require mathematics but genetics has also suggested new branches of algebra
...

Hereditary characteristics such as eye colour, hair colour, colourblindness, left/righthandedness and blood group types are all determined by factors (alleles) of a gene
...


So how could eye-colour factor be transmitted to the next generation? In the
basic model there are two factors, b and B:
b is the blue eyes factor
B is the brown eyes factor

Population representing the proportions 1:1:3 of genotypes bb, bB and BB

In individuals, the factors appear in pairs giving rise to possible genotypes bb,
bB and BB (because bB is the same as Bb)
...
For example, a population could
consist of a fifth of people with the genotype bb, another fifth with the genotype
bB and the remaining three-fifths with the genotype BB
...
This can
be represented by a diagram showing these proportions of genotypes
...
A person with a pure genes genotype
BB will have brown eyes, but so too will a person with mixed factors, that is,
those with a hybrid genotype bB because B is dominant
...

214

A burning question in the field of biology arose at the beginning of the 19th
century
...


The Hardy–Weinberg law
This was explained by the Hardy–Weinberg law, an application of basic
mathematics to genetics
...

G
...
Hardy was an English mathematician who prided himself on the nonapplicability of mathematics
...
Wilhelm Weinberg came from a very different background
...
He
discovered the law at the same time as Hardy, around 1908
...

There are no preferred pairings so that, for instance, blue-eyed people do not
prefer to mate with blue-eyed people
...
For example, a hybrid genotype bB mating with a hybrid bB
can produce any one of bb, bB, BB, but a bb mating with a BB can only produce
a hybrid bB
...
The transmission probability of a b-factor being included in the
genotype of a child is therefore 3/10 or 0
...
The transmission probability of a Bfactor being included is 7/10 or 0
...
The probability of the genotype bb being
included in the next generation, for example, is therefore 0
...
3 = 0
...
The
complete set of probabilities is summarized in the table
...
21 + 0
...
42
...
Because
B is the dominant factor, 42% + 49% = 91% of the first generation will have
brown eyes
...

The initial distribution of genotypes was 20%, 20% and 60% and in the new
generation the distribution of genotypes is 9%, 42% and 49%
...
The proportion of b-factors is 0
...
42 = 0
...
42 + 0
...
7
...
The distribution of
genotypes bb, bB and BB in the further generation is therefore the same as for
the previous generation, and in particular the genotype bb which gives blue eyes
does not die out but remains stable at 9% of the population
...
→ 9%, 42%, 49%

This is in accordance with the Hardy–Weinberg law: after one generation the
genotype proportions remain constant from generation to generation, and the
transmission probabilities are constant too
...

Hardy begins with the initial distribution of genotypes bb, bB and BB as p, 2r
and q and the transmission probabilities p + r and r + q
...
2, 2r = 0
...
6
...
2 + 0
...
3 and r + q = 0
...
6 = 0
...
What if there were a different initial distribution of the genotypes
bb, bB and BB and we started with, say, 10%, 60% and 30%? How would the
Hardy–Weinberg law work in this case? Here we would have p = 0
...
6
an d q = 0
...
4 and r + q = 0
...
So the distribution of next generation
of genotypes is 16%, 48% and 36%
...
4 and 0
...
With these figures 16% of the population will have blue eyes
and 48% + 36% = 84% will have brown eyes because B is dominant in the
genotype bB
...
→ 16%, 48%, 36%

So the Hardy–Weinberg law implies that these proportions of genotypes bb,
bB and BB will remain constant from generation to generation whatever the initial
distribution of factors in the population
...

Hardy stressed that his model was only approximate
...
In the
model the probability of gene mutation or changes in the genes themselves has
been discounted, and the consequence of the transmission proportions being
constant means it has nothing to say about evolution
...
This
will cause variations in the overall proportions and new species will evolve
...
It
awaited the genius of R
...
Fisher to reconcile the Mendelian theory of inheritance
with the continuous theory where characteristics evolve
...
Then there was a dramatic advance
contributed by Francis Crick, James Watson, Maurice Wilkins and Rosalind
Franklin
...
Mathematics is needed
to model the famous double helix (or a pair of spirals wrapped around a
cylinder)
...

Mathematics is indispensable in studying genetics
...
The science of genetics has also repaid the compliment to
mathematics by suggesting new branches of abstract algebra of interest for their
217

intriguing mathematical properties
...
These involved the theory of groups, mathematical
constructs that can be used to quantify symmetry
...
Group theory is the glue which binds the ‘everything’ together
...
Greek vases have it, snow crystals have it,
buildings often have it and some letters of our alphabet have it
...
We’ll just look at two-dimensional symmetry – all our objects of study
live on the flat surface of this page
...
A tripod has mirror symmetry, but the triskelion (tripod with feet) does
not
...


Rotational symmetry
We can also ask whether there is an axis perpendicular to the page so that the
object can be rotated in the page through an angle and be brought back to its
original position
...

The triskelion, meaning ‘three legs’, is an interesting shape
...

If we rotate it through 120 degrees or 240 degrees the rotated figure will
coincide with itself; if you closed your eyes before rotating it you would see the
same triskelion when you opened them again after rotation
...
Objects for which the image in the mirror is distinct from the object in front
of the mirror are called chiral – they look similar but are not the same
...

220

This is the case with the compound limosene which in one form tastes like
lemons and in the other like oranges
...


Measuring symmetry
In the case of our triskelion the basic symmetry operations are the (clockwise)
rotations R through 120 degrees and S through 240 degrees
...
We can create a table based on the combinations of these
rotations, in the same way we might create a multiplication table
...
According to the most widely used convention, the
multiplication R ° S means first rotate the triskelion clockwise through 240
degrees with S and then by 120 degrees with R, the result being a rotation by
360 degrees, as if you did nothing at all
...

The symmetry group of the triskelion is made up of I, R and S and the
multiplication table of how to combine them
...
The table is also called a Cayley table
(named after the mathematician Arthur Cayley, distant cousin to Sir George
Cayley a pioneer of flight)
...
But it also
has mirror symmetry and therefore has a larger symmetry group
...

The larger symmetry group of the tripod, which is of order six, is composed
of the six transformations I, R, S, U, V and W and has the multiplication table
shown
...
This is actually a rotation of the tripod through 120
degrees, in symbols U ° W = R
...
In particular U ° W ≠ W ° U
...


Reflections of a tripod

A group in which the order of combining the elements is immaterial is called
an abelian group, named after the Norwegian mathematician Niels Abel
...


Abstract groups
The trend in algebra in the 20th century had been towards abstract algebra, in
which a group is defined by some basic rules known as axioms
...
There are systems in algebra that are more basic than a group
and require fewer axioms; other systems that are more complex require more
axioms
...
It is remarkable that from so few axioms such a large
body of knowledge has emerged
...

A feature of group theory is that there may be small groups sitting inside
bigger ones
...
J
...
Lagrange proved a basic
fact about subgroups
...
So we automatically know the
symmetry group of the tripod has no subgroups of order four or five
...
There is no need to list them all because some groups are built up from
basic ones, and it is the basic ones that are needed
...
The symmetry
group of the tripod of six elements is a ‘compound’ being built up from the group
of rotations (of order three) and reflections (of order two)
...
There is an element 1 in G so that 1 ° = a ° 1 = a for all elements a in the
group G (the special element 1 is called the identity element)
...
For each element a in G there is an element ā in G with ā ° a = a ° ā = 1
(the element ā is called the inverse element of a)
...
For all elements a, b and c in G it is true that a ° (b ° c) = (a ° b) ° c (this
is called the associative law)
...
The complete
classification, called ‘the enormous theorem’, was announced by Daniel
Gorenstein in 1983 and was arrived at through the accumulated work of 30
years’ worth of research and publications by mathematicians
...
The basic groups fall into one of four main types, yet 26 groups
have been found that do not fall into any category
...

The sporadic groups are mavericks and are typically of large order
...
The smallest sporadic group is of
order 7920 = 24 × 32 × 5 × 11 but at the upper end are the ‘baby monster’ and
the plain ‘monster’ which has order 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19
× 23 × 29 × 31 × 41 × 47 × 59 × 71 which in decimal speak is around 8 ×
1053 or, if you like, 8 with 53 trailing zeros – a very large number indeed
...

Although snappy proofs and shortness are much sought after in mathematics,
the proof of the classification of finite groups is something like 10,000 pages of
closely argued symbolics
...


the condensed idea
Measuring symmetry

224

39

Matrices

This is the story of ‘extraordinary algebra’ – a revolution in mathematics which took
place in the middle of the 19th century
...


Ordinary algebra is the traditional algebra in which symbols such as a, b, c, x
and y represent single numbers
...
In comparison, ‘extraordinary
algebra’ generated a seismic shift
...


Multiple dimensioned numbers
In ordinary algebra a might represent a number such as 7, and we would
write a = 7, but in matrix theory a matrix A would be a ‘multiple dimensioned
number’ for example the block

This matrix has three rows and four columns (it’s a ‘3 by 4’ matrix), but in
principle we can have matrices with any number of rows and columns – even a
giant ‘100 by 200’ matrix with 100 rows and 200 columns
...
More than this, we can manipulate these blocks of
numbers simply and efficiently
...


225

A practical example
Suppose the matrix A represents the output of the AJAX company in one
week
...
In our example, the quantities, tallying with matrix A
opposite, are:

In the next week the production schedule might be different, but it could be
written as another matrix B
...
Sadly, matrix multiplication is less obvious
...
We
can certainly compute the overall profit for Factory 1 with outputs 7, 5, 0, 1 of its
four products
...

But instead of dealing with just one factory we can just as easily compute the
total profits T for all the factories

226

Look carefully and you’ll see the row by column multiplication, an essential
feature of matrix multiplication
...
Matrix theory is very powerful
...
With matrix algebra the
calculations, and our understanding, are fairly immediate, without having to
worry about the details which are all taken care of
...
The most celebrated difference occurs in the multiplication of matrices
...

Another difference occurs with inverses
...
If a= 7 its inverse is 1/7 because it has the property that 1/7 × 7 =
1
...

An example in matrix theory is

and we can verify that

because

where
is called the identity matrix and is the matrix counterpart of 1
in ordinary algebra
...


Travel plans
228

Another example of using matrices is in the analysis of a flight network for
airlines
...
In practice this
may involve hundreds of destinations – here we’ll look at a small example: the
hubs London (L) and Paris (P), and smaller airports Edinburgh (E) , Bordeaux
(B), and Toulouse (T) and the network showing possible direct flights
...
If there
is a direct flight between airports a 1 is recorded at the intersection of the row
and column labelled by these airports (like from London to Edinburgh)
...


The lower submatrix (marked out by the dotted lines) shows there are no
direct links between the three smaller airports
...
So, for example, there
are 3 possible roundtrips to Paris via other cities but no trips from London to
Edinburgh which involve stopovers
...
This is another
example of the ability of matrices to capture the essence of a vast amount of data
under the umbrella of a single calculation
...
From an applied
perspective, matrix theory was very much a ‘solution looking for a problem’
...
An early
application occurred in the 1920s when Werner Heisenberg investigated ‘matrix
mechanics’, a part of quantum theory
...
When asked
how she discovered the subject she replied that it was the other way around,
matrix theory had found her
...


the condensed idea
Combining blocks of numbers

230

40

Codes

What does Julius Caesar have in common with the transmission of modern digital
signals? The short answer is codes and coding
...

Caesar used codes to communicate with his generals and kept his messages secret by
changing around the letters of his message according to a key which only he and they
knew
...
Caesar also wanted to keep his codes to himself as
do the cable and satellite broadcasting television companies who only want
paying subscribers to be able make sense of their signals
...
Human error or ‘noise along the line’ can always
occur, and must be dealt with
...


Error detection and correction
One of the first binary coding systems was the Morse code which makes use of
two symbols, dots • and dashes –
...
B
...
It was a code designed for the electric telegraph of the mid-19th century
with little thought to an efficient design
...
A telegraph operator sending ‘CAB’ would send the string – • – • / • – /
– •••
...
If the Morse code operator wished to send ‘CAB’, but mistyped
a dot for a dash in C, forgot the dash in A and noise on the wire substituted a
dash for a dot in B, the receiver getting •• – • / • / –– ••, would see nothing
wrong and interpret it as ‘FEZ’
...
Suppose an army
commander has to transmit a message to his troops which is either ‘invade’ or
231

‘do not invade’
...
If a 1 or a 0 was incorrectly transmitted the receiver would
never know – and the wrong instruction would be given, with disastrous
consequences
...
If this time we
code the ‘invade’ instruction by 11 and the ‘do not invade’ by 00, this is better
...
As only 11 or 00
are legitimate code words, the receiver would certainly know that an error had
been made
...
If 01 were received, how would we
know whether 00 or 11 should have been sent?
The way to a better system is to combine design with longer code words
...
If we knew that at most one
error could be made (a reasonable assumption since the chance of two errors in
one code word is small), the correction could actually be made by the receiver
...
With our rules, it could not be 000 since this code word is two errors away
from 110
...

The same principle is used when word processing is in autocorrect mode
...
The English language is not fully correcting though
because if we type ‘lomp’ there is no unique nearest word; the words, lamp,
limp, lump, pomp and romp are all equidistant in terms of single errors from
lomp
...
By choosing the legitimate code words far enough apart, both
detection and correction are possible
...
Long code words with high performance
in terms of error correction take longer to transmit so there is a tradeoff between
length and speed of transmission
...

232

Making messages secret
Julius Caesar kept his messages secret by changing around the letters of his
message according to a key that only he and his generals knew
...
In
medieval times, Mary Queen of Scots sent secret messages in code from her
prison cell
...
More sophisticated than the Roman
method of rotating all letters by a key, her codes were based on substitutions but
ones whose key could be uncovered by analysing the frequency of letters and
symbols used
...
In this case it was a formidable challenge but
the code was always vulnerable because the key was transmitted as part of the
message
...
Running counter to everything that had been previously believed, it said
that the secret key could be broadcast to all and yet the message could remain
entirely safe
...
The method depends on a
200 year old theorem in a branch of mathematics glorified for being the most
useless of all
...
What he does next is rather curious
...
Receiver
...
This information is available to all and sundry,
and it is all the information John Sender requires to encrypt his message, which
for simplicity is his calling card, J
...

Sender encrypts 74 by calculating 745 (modulo 247), that is, he wants to know
the remainder on dividing 745 by 247
...
Sender’s
encrypted message is 120 and he transmits this to Receiver
...
But
not everyone could decrypt it
...
Receiver has more information up his sleeve
...
In this case he obtained the number 247 by multiplying p = 13 and q
= 19, but only he knows this
...
Dr R
...
What is a so that dividing 5 × a by 12 × 18 = 216
leaves remainder 1? Skipping the actual calculation he finds a = 173
...
With it he works out
the remainder when he divides the huge number 120173 by 247
...

The answer is 74, as Euler knew two hundred years ago
...

You might say, surely a hacker could discover the fact that 247 = 13 × 19 and
the code could be cracked
...
But the encryption and
decryption principle is the same if Dr Receiver had used another number instead
of 247
...

Finding the two prime factors of a very large number is virtually impossible –
what are the factors of 24,812,789,922,307 for example? But numbers much
larger than this could also be chosen
...
In
the end it is considerably easier for Dr Receiver to ‘mix boxes of black sand and
white sand together’ than for any hacker to unmix them
...
It is not about adding up a column of figures in your head
...
This makes
combinatorial problems attractive
...


A tale from St Ives
Children can start combinatorics at a tender age
...

Kits, cats, sacks and wives,
How many were going to St Ives?
The last line contains the trick question
...
Some
people exclude the narrator and for them the answer would be ‘none’
...
We could ask: how many were coming from St Ives? Again
interpretation is important
...

We’ll assume the entourage was coming along the single road away from the
Cornish seaside town and that the ‘kits, cats, sacks and wives’ were all present
...

236

In 1858 Alexander Rhind a Scottish antiquarian visiting Luxor came across a 5
metre long papyrus filled with Egyptian mathematics from the period 1800 BC
...
A few years later it was acquired by the British Museum and its
hieroglyphics translated
...
Both involve powers of 7 and the same kind of analysis
...


Factorial numbers
The problem of queues introduces us to the first weapon in the combinatorial
armoury – the factorial number
...
By swapping the people around other queues are formed; how
many different queues are possible?
The art of counting in this problem depends on choice
...
When we come to
the last position there is no choice at all as it can only be filled by the person left
over
...
If we
started with 6 people, the number of different queues would be 6 × 5 × 4 × 3 ×
2 × 1 = 720 and for 7 people there would be 7 × 6 × 5 × 4 × 3 × 2 × 1 =
5040 possible queues
...
These occur so often in mathematics that they are written using
the notation 5! (read ‘5 factorial’) instead of 5 × 4 × 3 × 2 × 1
...
Straightaway, we see that
237

quite ‘small’ configurations give rise to ‘large’ factorial numbers
...


If we’re still interested in forming queues of 5 people, but can now draw on a
pool of 8 people A,B, C, D, E, F, G, and H, the analysis is almost the same
...
But this time there are 4 choices for the last slot
...
The two queues number factorial
C E B A D

D A C E B

are made from the same letters but are different queues
...
If we’re interested in
counting the ways of selecting 5 people from 8 immaterial of order we must
divide 8 × 7 × 6 × 5 × 4 = 6720 by 5!
...


Kirkman’s problem
Combinatorics is a wide field and, though old, it has rapidly developed over
the past 40 years, due to its relevance to computer science
...

The essence of combinatorics is captured by a master of the subject, Rev
...
He made many original contributions to discrete
geometry, group theory and combinatorics but never had a university
appointment
...
In 1850 Kirkman
introduced the ‘15 schoolgirls problem’, in which schoolgirls walk to church in 5
rows of 3 on each day of the week
...
We need to organize a daily schedule so that no two walk together
more than once
...

There are actually seven distinct solutions to Kirkman’s problem, and the one
we’ll give is ‘cyclic’ – it is generated by ‘going around’
...


It is called cyclic since on each subsequent day the walking schedule is
changed from a to b, b to c, down to g to a
...

The underlying reason for the choice of notation is that the rows correspond
to lines in the Fano geometry (see page 115)
...


the condensed idea
How many combinations?

240

42

Magic squares

‘A Mathematician’, wrote G
...
Hardy, ‘like a painter or a poet, is a maker of patterns
...
They lie on
the border between heavily symbolled mathematics and the fascinating patterns loved
by puzzlesmiths
...


Squares with just one row and one column are technically magic squares but
are very boring so we’ll forget them
...
If there were it would have the form shown
...
This means b = c, contradicting the fact that all the entries must be
distinct
...
We’ll start with a normal magic square, one where the grid is
filled out with the consecutive numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9
...

241

If we add up all the numbers in the grid we have
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

and this total would have to be the same as adding the totals of 3 rows
...
Now let’s
look at the middle cell – we’ll call this c
...
If we add the numbers in these four lines
together we get 15 +15 +15 +15 = 60 and this must equal all the numbers
added together plus 3 extra lots of c
...
Other facts can also be learned such as not being able to place
a 1 in a corner cell
...
Try it!

A solution for the 3×3 square by the Siamese method

Of course we’d like a totally systematic method for constructing magic
squares
...
Loubère took an interest in Chinese
mathematics and wrote down a method for constructing magic squares that have
an odd number of rows and columns
...
If blocked the next number beneath the
current number is used
...
Every other 3×3 magic square can be obtained from this one by
rotating numbers about the middle and/or reflecting numbers of the square in
the middle column or middle row
...
Legend says that it was first seen on the back of a turtle
emerging from the Lo river
...

If there is one 3×3 magic square, how many distinct 4×4 magic squares are
there? The staggering answer is that there are 880 different ones (and be
prepared, there are 2,202,441,792 magic squares of order 5)
...

242

Dürer and Franklin’s squares
The Lo Shu magic square is well known for its age and uniqueness but one
4×4 magic square has become iconic for its association with a famous artist
...
This is the 4×4 square in Albrecht Dürer’s
engraving of Melancholia, which he made in the year 1514
...
Dürer even
managed to ‘sign’ his masterpiece with the date of its completion in the middle of
the lowest row
...
He was adept at this,
and to this day mathematicians have little idea how he did it; large magic squares
cannot be constructed by serendipity
...
Here’s one he discovered in his youth
...
All the rows,
columns and diagonals add up to 260, as do the ‘bent rows’, one of which we’ve
highlighted
...
Look closely
and you’ll find an interesting result for every 2×2 square
...

The problem of constructing these was posed by the French mathematician
Edouard Lucas in 1876
...


All rows and columns and one diagonal of this square add up to the magic
sum 21,609 but the other diagonal fails since 1272 + 1132 + 972 = 38,307
...
5 × 1025 so there’s little point in looking
for a square with small numbers! This is serious mathematics which has a
connection with elliptic curves, the topic used to prove Fermat’s Last Theorem
...

The search for squared squares has, however, been successful for larger
squares
...
In 1770 Euler produced
an example without showing his method of construction
...


Exotic magic squares
Large magic squares may have spectacular properties
...
In 2001 a 1024×1024
square was produced in which all powers of elements up to the fifth power make
magic squares
...

We can create a whole variety of other magic squares if the requirements are
relaxed
...
Removing the
condition that the sum of the diagonal elements must equal the sums of the
rows, and of the columns, ushers in a plethora of specialized results
...
By going into
higher dimensions we are led to consider magic cubes and hypercubes
...
We also find that the number of letters of the magic sums of both
3×3 squares (21 and 45) is 9 and fittingly 3 × 3 = 9
...
Across the land, pens and pencils are
chewed waiting for the right inspiration for the number to put in that box
...
Commuters emerge from their trains in the mornings having expended
more mental effort than they will for the rest of the day
...
Is it 5, 4, or maybe 7? All these people are playing with Latin squares
– they are being mathematicians
...
The object is
to fill in the rest using the given numbers as clues
...

It is believed that Sudoku (meaning ‘single digits’) was invented in the late
1970s
...
The appeal of the puzzle is that, unlike crosswords, you don’t
have to be widely read to attempt them but, like crosswords, they can be
247

compelling
...


3×3 Latin squares
A square array containing exactly one symbol in each row and each column is
called a Latin square
...
Can we fill out a blank 3×3 grid so that each row and column
contains exactly one of the symbols a, b and c? If we can, this would be a Latin
square of order 3
...
Unlike magic squares, however, Latin squares are not
concerned with arithmetic and the symbols do not have to be numbers
...


A 3×3 Latin square can be easily written down
...
Team One is made up of Larry, Mary and Nancy and Team Two of Ross,
Sophie and Tom
...

The Latin square arrangement ensures a meeting takes place between each pair
of team members and there is no clash of dates
...
If we interpret A, B and C as
topics discussed at the meetings between Team One and Team Two, we can
produce a Latin square which ensures each person discusses a different topic with
a member of the other team
...

But when should the discussions take place, between who, and on what topic?
What would be the schedule for this complex organization? Fortunately the two
Latin squares can be combined symbol by symbol to produce a composite Latin
Square in which each of the possible nine pairs of days and topics occurs in
exactly one position
...
Latin squares which combine in this way
are called ‘orthogonal’
...
This is something
Euler discovered
...
In 1782 Euler posed the same problem for ‘36 officers’
...
He couldn’t
find them and conjectured there were no pairs of orthogonal Latin squares of
orders 6, 10, 14, 18, 22 … Could this be proved?
Along came Gaston Tarry, an amateur mathematician who worked as a civil
servant in Algeria
...

Mathematicians naturally assumed Euler was correct in the other cases 10, 14,
18, 22 …
In 1960, the combined efforts of three mathematicians stunned the
mathematical world by proving Euler wrong in all the other cases
...

We’ve seen that there are two mutually orthogonal order 3 Latin squares
...
It can be shown that there are never more than n − 1 mutually orthogonal
Latin squares of order n, so for n = 10, for example, there cannot be more than
nine mutually orthogonal squares
...
To date,
no one has been able to even produce three Latin squares of order 10 that are
mutually orthogonal to each other
...
A
...
He
used them to revolutionize agricultural methods during his time at Rothamsted
Research Station in Hertfordshire, UK
...
Ideally we would want to plant crops in identical soil conditions so that soil
quality wasn’t an unwanted factor influencing crop yield
...
The only way of ensuring identical soil conditions would be to use the
same soil – but it is impractical to keep digging up and replanting crops
...

A way round this is to use Latin squares
...

250

If we mark out a square field into 16 plots we can envisage the Latin square as a
description of the field where the soil quality varies ‘vertically’ and ‘horizontally’
...
If we suspect another factor might influence crop
yield, we could deal with this too
...
Label four time zones during the day as A, B, C
and D and use orthogonal Latin squares as the design for a scheme to gather
data
...

The design for the experiment would be:

Other factors can be screened out by going on to create even more elaborate
Latin square designs
...


the condensed idea
Sudoku revealed

251

44

Money mathematics

Norman is a super salesperson when it comes to bikes
...
The customer pays for it with a cheque for £150, and
as the banks are closed, Norman asks his neighbour to cash it
...
Calamity follows
...
The bike originally cost him £79, but how much did Norman lose
altogether?

The concept of this little conundrum was proposed by the great puzzlesmith
Henry Dudeney
...
It shows how money depends on time and that inflation
is alive and well
...
A way to combat inflation is through the interest on money
...


Compound interest
There are two sorts of interest, known as simple and compound
...

Their father gives them each £1000, which they both place in a bank
...

Historically, compound interest was identified with usury and frowned upon
...
Compound interest is interest compounded on interest, and that is why
Charlie likes it
...
Simon can understand it easily, as the
principal earns the same amount of interest each year
...
That the formula for compound interest
has a greater immediacy than his E = mc2 is undeniable
...

What do the symbols stand for? The term P stands for principal (the money you
save or borrow), i is the percentage interest rate divided by 100 and n is the
number of time periods
...
How
much will accrue in three years? Here P = 1000, i = 0
...
The symbol
A represents the accrued amount and by the compound interest formula
A=£1225
...

Simon’s account pays the same interest rate, 7%, as simple interest
...
He would
therefore have 3 × £70 interest giving a total accrued amount of £1210
...

Sums of money that grow by compounding can increase very rapidly
...
A key component of
compound interest is the period at which the compounding takes place
...
How
much would he stand to gain with this scheme?
Simon thinks he knows the answer: he suggests we multiply the interest rate
1% by 52 (the number of weeks in the year) to obtain an annual percentage rate
of 52%
...

Charlie reminds him, however, of the magic of compound interest and the
compound interest formula
...
01 and n = 52, Charlie
calculates the accrual to be £1000 × (1
...
Using his calculator he finds this is
£1677
...
Charlie’s
equivalent annual percentage rate is 67
...

Simon is impressed but his money is already in the bank under the simple
interest regime
...

This gives 14
...
It is a long time to wait
...
This is a
little more complicated but a friend tells him about the rule of 72
...
Though Charlie is
interested in years the rule of 72 applies to days or month as well
...
The
calculation is 72/7 = 10
...
The rule is an
approximation but it is useful where quick decisions have to be made
...
Charlie
is very excited
...
Charlie is not so
happy
...
He goes to
his bank and promises them the £100,000 in ten years time
...
The bank has to estimate the size of investment now that would realize
£100,000 in ten years
...
The
bank believes that a growth rate of 12% would give them a healthy profit
...

This time we are given A = £100,000 and have to calculate P, the present value
of A
...
12, the bank will be prepared to advance Charlie the
amount 100,000/1
...
32
...


How can regular payments be handled?
254

Now that Charlie’s father has promised to give £100,000 to his son in ten
years time, he has to save the money up
...
By
the end of this period he will then be able to hand over the money to Charlie on
the day he has promised, and Charlie can hand the money to the bank to pay off
the loan
...
He
gives Charlie the task of working out the annual payments
...
If regular
payments R are made at the end of each year in an environment where the
interest rate is i, the amount saved after n years can be calculated by the regular
payments formula
...
08 and calculates that R =
£6902
...

Now that Charlie has his brand new Porsche, courtesy of the bank, he needs a
garage to put it in
...
He recognizes this as a problem in which the £300,000 is the
present value of a stream of payments to be made and he calculates his annual
payments with ease
...
He has just been given a retirement lump sum of £150,000 and wants
to purchase an annuity
...
Instead of the mortgage company advancing me
money that I repay in regular instalments, you are giving them the money and
they are making the regular payments to you
...


255

the condensed idea
Compound interest works best

256

45

The diet problem

Tanya Smith takes her athletics very seriously
...
Tanya makes her way in the world by taking part-time jobs
and has to watch where the money goes
...
The amounts have been
determined by her coach
...
To
make sure she follows this regime Tanya relies on two food supplements
...
Her problem is to decide how much of each she should purchase each
month to satisfy her coach
...
It was a prototype for problems in linear programming, a subject
developed in the 1940s that is now used in a wide range of applications
...
On a back of a packet of Solido she finds out it contains 2
mg vitamins and 10 mg minerals, while a carton of Liquex contains 3 mg
vitamins and 50 mg minerals
...
As she
proceeds towards the checkout she wonders if she has the right amount
...
In the 30 packets of Solido
she has 2 × 30 = 60 mg vitamins and in the Liquex, 3 × 5 = 15
...
Repeating the calculation for minerals,
she has 10 × 30 + 50 × 5 = 550 mg minerals
...
Tanya’s problem is
juggling the right amounts of Solido and Liquex with the vitamin and mineral
requirements
...
She now has 40 packets and 15
257

cartons
...
Now Tanya
certainly satisfies her coach’s recommendation and has even exceeded the
required amounts
...
This is
called a possible combination, or a ‘feasible’ solution
...

Tanya has many more options
...
If
she did this she would need to buy at least 88 packets
...
If she
bought only Liquex she would need at least 40 cartons, the feasible solution (0,
40) satisfies both vitamin and mineral requirements, because 2 × 0 + 3 × 40 =
120 mg vitamins and 10 × 0 + 50 × 40 = 2000 mg minerals
...


Optimum solutions
Money is now brought into the situation
...
She notes that the packets and cartons are equally
priced at £5 each
...
This will
be the least cost purchase and the dietary requirement will be achieved
...
On the spur of the moment Tanya has
tried various combinations of Solido and Liquex and figured out the cost in these
cases only
...

Linear programming problemsTanya’s always been coached to visualize
her goals
...
This is possible
because she is only considering two foods
...
The
combinations above this line have more than 120 mg vitamins
...
The
combinations of foods that are above both these lines is the feasible region and
represents all the feasible combinations Tanya could buy
...
The word ‘programming’ means a procedure (its usage
before it became synonymous with computers) while ‘linear’ refers to the use of
straight lines
...
Tanya has discovered a new
feasible solution at the point B with coordinates (48, 8) which means that she
could purchase 48 packets of Solido and 8 cartons of Liquex
...
At £5 for both a packet and a carton this
combination would cost her £280
...

The optimum combination ultimately depends on the relative costs of the
supplements
...

The best purchase for Tanya with these prices is 48 packets of Solido and 8
cartons of Liquex, with a bill of £152
...
It was so successful that Dantzig became known in the West
as the father of linear programming
...
In 1975, Kantorovich and the Dutch mathematician Tjalling
Koopmans were awarded the Nobel Prize for Economics for work on the
allocation of resources, which included linear programming techniques
...
When Dantzig
found his method there were few computers but there was the Mathematical
Tables Project – a decade-long job creation scheme which began in New York in
1938
...

While the simplex method and its variants have been phenomenally successful,
other methods have also been tried
...

The basic linear programming model has been applied to many situations
other than choosing a diet
...
It is of a
special structure and has become a field in its own right
...
In some linear programming
problems the objective is to maximize (like maximizing profit)
...

It remains to be seen whether Tanya Smith wins her gold medal at the
Olympic Games
...


The condensed idea
Keeping healthy at least cost

261

46

The travelling salesperson

James Cook, based in Bismarck (North Dakota, USA), is a super salesperson for the
Electra company, a manufacturer of carpet cleaners
...
His sales area
takes in the cities of Albuquerque, Chigaco, Dallas and El Paso, and he visits each of
them in a round trip once a month
...
It is the classical travelling
salesperson problem
...
For
example, the distance between Bismarck and Dallas is 1020 miles, found at the
intersection (shaded) of the Bismarck column with the Dallas row
...
One route he often takes starts from Bismarck
travelling to Chicago, Albuquerque, Dallas and El Paso in turn before returning to
Bismarck
...
Can he do better?
Making a plan of the sales area should not disguise the fact that James is not
in the mood for detailed planning – he wants to get out there and sell
...
It is 706
miles away as against 883 to Albuquerque, 1020 miles to Dallas, and 1100 miles
to El Paso
...
When he
gets to Chicago and completes his business there he looks for the nearest city to
go to
...


Once in Dallas he has notched up 706 + 785 miles
...
He chooses Albuquerque because it is closer
...
His total mileage is 706 + 785 + 580 +
236 + 1100 = 3407 miles
...

This way of thinking is often called the greedy method of finding a short
route
...
With this method he
never attempts to look forward by more than one step at a time
...
The fact that he
finished in El Paso meant he was forced to take a long route back to Bismarck
...

James sees how he can take advantage of there being only five cities involved
...
With five cities there are only 24 routes to examine or just 12 if we
count a route and its reverse as equivalent
...
The method serves James Cook well and he learns that
the route BAEDCB (or its reverse BCDEAB) is actually optimum, being only 3199
miles long
...
It is not the
263

distance he wants to save, but time
...


When the problem was focused on mileages, James knew that the sum of the
distances along two sides of a triangle is always greater than the length of the
third side; in this case the graph is called Euclidean and much is known about
solution methods
...
Flying
on main routes is often faster than side routes and James Cook notes that in
going from El Paso to Chicago it is quicker to fly via Dallas
...

The greedy method applied to the time-problem produces a total time of 22
hours on the route BCDEAB, whereas there are two distinct optimum routes
BCADEB and BCDAEB totalling 14 hours each
...
James Cook is happy that by choosing
BCDAEB he has saved the most
...


From seconds to centuries
The real difficulty associated with the travelling salesperson problem occurs
when there is a large number of cities
...
He now has
to visit 13 cities from Bismarck instead of the previous 4
...

He sets out to list the possible routes for his 13 cities
...
1 x 109 routes to examine
...
A problem with 100 cities as input would tie up the computer for
millennia
...
Exact methods have been given which apply to 5000 cities or less and
one has even successfully dealt with a particular problem of 33,810 cities, though
the computer power required in this case was colossal
...
Methods of this type have the advantage of being able to handle
problems with millions of cities
...
Simply listing all possible routes is a
worst case scenario
...
If we threw in an extra 2 cities
the time would go up to over 20,000 years!
Of course these estimates will depend on the actual computer used, but for n
cities the time taken rises in line with n factorial (the number you get by
multiplying together all whole numbers from 1 to n)
...
1 × 10 9
routes for 13 cities
...

Other methods are available for attacking the problem in which the time for n
cities rises with 2n (2 multiplied by itself n times) so for 13 cities this would
involve the order of 8192 decisions (8 times more than for 10 cities)
...
The holy grail of
these ‘combinatorial optimization problems’ is to find an algorithm which
depends not on the nth power of 2, but on a fixed power of n
...
A method of this ‘complexity’ is said to be conducted
i n polynomial time – problems solved this way are ‘quick problems’ and could
take 3 minutes, rather than centuries
...
We don’t know if the travelling salesperson problem is one of
these
...

A wider class denoted by NP consists of problems whose solutions can be
265

verified in polynomial time
...
You just add the distances along
the given route and compare it with the given number
...

Is every problem verifiable in polynomial time able to be solved in polynomial
time? If this were true the two classes P and NP would be identical and we could
write P = NP
...
More than half the profession think this is not true: they believe there
are problems out there that can be checked in polynomial time but cannot be
solved in polynomial time
...


The condensed idea
Finding the best route

266

Game theory

47

Some said Johnny was the smartest person alive
...
When people heard that he arrived at
a meeting in a taxi having just scribbled out his ‘minimax theorem’ in game theory, they
just nodded
...
He made contributions to
quantum mechanics, logic, algebra, so why should game theory escape his eye? It didn’t
– with Oskar Morgenstern he coauthored the influential Theory of Games and Economic
Behavior
...


Two-person zero-sum games
It sounds complicated, but a two-person zero-sum game is simply one ‘played’
by two people, companies, or teams, in which one side wins what the other
loses
...
There
is no point in A cooperating with B – it is pure competition with only winners and
losers
...
This is the origin of the term ‘zero-sum’
...
Each company must make a bid for
one country only and they will base their decision on the projected increased size
of their viewing audiences
...
These are
conveniently set down in a ‘payoff table’ and measured in units of a million
viewers
...
The meaning of the minus
267

sign, as in the payoff −3, is that ATV will lose an audience of 3 million
...

We’ll assume the companies make their one-off decisions on the basis of the
payoff table and that they make their bids simultaneously by sealed bids
...

If ATV chooses Scotland the worst that could happen would be a loss of 3
million; if it bids for England, the worst would be a gain of 2 million
...
It couldn’t do
worse than gain 2 million viewers whatever BTV chooses
...

BTV is in a weaker position but it can still work out a strategy that limits its
potential losses and hope for a better payoff table next year
...
The safest strategy for
BTV would be to choose England (column 2) for it would rather lose an audience
of 4 million than 5 million
...

A beautiful mind
John F
...
1928) whose troubled life was portrayed in the 2001 movie A Beautiful
Mind won the Nobel Prize for Economics in 1994 for his contributions to game theory
...
The ‘Nash equilibrium’ (like a saddle point equilibrium) gave a much broader
perspective than that set down by von Neumann, resulting in a greater understanding
of economic situations
...


When is a game determined?
The following year, the two TV companies have an added option – to operate
in Wales
...


268

As before, the safe strategy for ATV is to choose the row which maximizes the
worst that can happen
...
The safe strategy for BTV is to choose the column which minimizes from
{+4, +5, +1}
...

By choosing Wales, ATV can guarantee to win no less than 1 million viewers
whatever BTV does, and by choosing England (column 3), BTV can guarantee to
lose no more than 1 million viewers whatever ATV does
...
In this game the
maximum of {+1, −1, −3} = minimum of {+4, +5, +1}

and both sides of the equation have the common value of +1
...


Repetitive games
The iconic repetitive game is the traditional game of ‘paper, scissors, stone’
...


In ‘paper, scissors, stone’, two players show either a hand, two fingers, or a
fist, each symbolizing paper, scissors or stone
...
If playing ‘paper’
the payoffs are therefore 0, −1, +1, which is the top row of our completed
payoff table
...

If a player always chooses the same action, say paper, the opponent will detect
this and simply choose scissors to win every time
...

According to the mathematics, players should choose randomly but overall the
choices of paper, scissors, stone should each be made a third of the time
...
They are
good at second-guessing their opponents
...
A famous example is the ‘prisoner’s dilemma’ designed by A
...

Tucker
...
The payoffs, in this case jail sentences, not only depend on their individual
responses to police questioning but on how they jointly respond
...
If A doesn’t confess but B does,
the sentences go the other way around
...


the condensed idea
Win–win mathematics

271

48

Relativity

When an object moves, its motion is measured relative to other objects
...
Yet we are both
travelling at 70 mph relative to the ground
...
The theory of relativity changed this
way of thinking
...
Einstein’s famous paper on special relativity revolutionized the study of
how objects move, reducing Newton’s classical theory, a magnificent
achievement in its time, to a special case
...
In our
example, Jim Diamond is on board a train travelling at 60 mph
...
His speed is 62
mph relative to the ground
...
This is what Newton’s theory tells us
...

Because all motion is relative, we talk about a ‘frame of reference’ as the
viewpoint from which a particular motion is measured
...
The zero position is determined by a point marked on the
platform and the time read from the station clock
...

There is also a reference frame on board the train
...
It is also possible to synchronize these two coordinate
systems
...
If Jim sets = 0 at this point, and puts = 0 on his
wristwatch, there is now a connection between these coordinates
...
We
can calculate how far he is from the station after five minutes
...
So in total Jim is a distance (x) which is 510/60 miles from the station
...

Turning the equation around to give the distance Jim has travelled relative to the
reference frame on the train, we get
The concept of time in the classical Newtonian theory is a one-dimensional
flow from the past to the future
...
Since it is an absolute quantity, Jim’s time on board the train is the same
for the station master on the platform t, so
These two formulae for
and , first derived by Galileo, are types of
equations called transformations, as they transform quantities from one reference
frame to another
...

By the 17th century people recognized that light had speed, and its
approximate value was measured in 1676 by the Danish astronomer Ole Römer
...
More than this, he became aware that
the transmission of light was very different from the transmission of sound
...
This
paradoxical result had to be explained
...
These transformations are very similar to the ones
we have already worked out but involve a (Lorentz) factor depending on v and
the speed of light, c
...

If Jim Diamond flicked a torch on and off while passing through the station on
his speeding train, firing the light beam down the carriage in the direction the
train was moving, he would measure its speed as c
...
Einstein also assumed a second principle:
One frame of reference moves with constant speed in relation to another
...
Sound waves
travel as vibrations of molecules in the medium through which the sound is being
carried
...
No one knew what it was, but they gave it a name – the luminiferous
aether
...
Instead, he deduced the Lorentz transformations from the two
274

simple principles of relativity and the whole theory unfolded
...
For the energy of a body at rest (when v = 0 and so = 1), this
leads to the iconic equation showing that mass and energy are equivalent:
E = mc2

Lorentz and Einstein were both proposed for the Nobel Prize in 1912
...
That was quite a year for the Swiss patent clerk
...

In these situations the relative speed v is so small compared with the speed of
light that the value of the Lorentz factor α is almost 1
...
So for
slow speeds Einstein and Newton would agree with each other
...
Even the record breaking French TGV train has not reached these
speeds yet and it will be a long time in the development of rail travel before we
would have to discard the Newtonian theory in favour of Einstein’s
...

The general theory of relativityEinstein published his general theory in
1915
...

Using the general theory Einstein was able to predict such physical
phenomena as the deflection of light beams by the gravitational fields of large
objects such as the Sun
...
This precession could not be fully explained by Newton’s
theory of gravitation and the force exerted on Mercury by the other planets
...

The appropriate frame of reference for the general theory is that of the fourdimensional space–time
...
It displaces the Newtonian force of gravity as the explanation for objects
being attracted to each other
...
In 1915 Einstein launched
another scientific revolution
...
For instance, 52 +
122 = 132
...
Fermat’s last theorem says that for any
four whole numbers, x, y, z and n, there are no solutions to the equation xn + y n = zn
when n is bigger than 2
...


Fermat’s last theorem is about a Diophantine equation, the kind of equation
which poses the stiffest of all challenges
...
They are named after Diophantus of Alexandria
whose Arithmetica became a milestone in the theory of numbers
...

A versatile mathematician, he enjoyed a high reputation in the theory of
numbers, and is most notably remembered for the statement of the last theorem,
his final contribution to mathematics
...
’
Fermat solved many outstanding problems, but it seems that Fermat’s last
theorem was not one of them
...
This
proof could not be written in any margin and the modern techniques required to
generate it throw extreme doubt on Fermat’s claim
...
Actually this makes the
equation x + y = z quite easy to solve
...
It is
as simple as that
...
We can also see that some values of x, y and z
are not solutions of the equation
...


The equation x2 + y2 = z2
We’ll now think about squares
...
If x = 3 then x2 = 3 × 3 =
9
...
For this z would have to be the square root of
58 (z = √58) which is approximately 7
...
We are certainly entitled to claim
that x = 3, y = 7 and z = √58 is a solution of x2 + y2 = z2 but unfortunately
Diophantine equations are primarily concerned with whole number solutions
...

The equation x2 + y2 = z2 is connected with triangles
...

Conversely, if x, y and z satisfy the equation then the angle between x and y is a
right angle
...


How can we find Pythagorean triples? This is where the local builder comes to
the rescue
...
The
278

values x = 3, y = 4 and z = 5 turn out to be a solution of the kind we are
looking for because 32 + 42 = 9 + 16 = 52
...
This is the mathematical fact
that the builder uses to build his walls at right angles
...


There are other whole number solutions x2 + y2 = z2
...
The builder’s solution x = 3, y = 4
and z = 5 holds pride of place since it is the smallest solution, and is the only
solution composed of consecutive whole numbers
...


From feast to famine
279

It looks like a small step to go from x2 + y2 = z2 to x3 + y3 = z3
...
The equation x2 + y2 =
z2 has an infinite number of different solutions but Fermat was unable to find
even one whole number example of x3 + y3 = z3
...

One way to approach the problem of proving this is to start on the low values
of n and move forward
...
The case n = 4
is actually simpler than n = 3 and it is likely Fermat had a proof in this case
...
Lamé initially
thought he had a proof of the general theorem but was unfortunately mistaken
...
The French Academy of Sciences offered a prize of 3000 francs for
a valid proof, eventually awarding it to Kummer for his worthy attempt
...
For example, he could not prove there were
no whole numbers which satisfied x67 + y67 = z67
...
This was
perhaps a greater contribution to mathematics than settling the question itself
...
In 1908 Paul Wolfskehl bequeathed a 100,000 marks prize to be
awarded to the first provider of a proof, a prize made available for 100 years
...


The proof
While the link with Pythagoras’s theorem only applies for n = 2, the link with
geometry proved the key to its eventual proof
...
In 1993 Andrew Wiles
gave a lecture on this theory at Cambridge and included his proof of Fermat’s
theorem
...

The similarly named French mathematician André Weil dismissed such
attempts
...
The pressure was on
...
Many thought Wiles
would join that throng of the nearly people
...
This time he convinced the experts and proved the
theorem
...
The tenyear-old boy sitting in a Cambridge public library reading about the problem
years before had come a long way
...

The Poincaré conjecture and Fermat’s last theorem have been conquered but not the
Riemann hypothesis
...


The story starts with the addition of fractions of the kind

The answer is 1⅚ (approximately 1
...
But what happens if we keep adding
smaller and smaller fractions, say up to ten of them?

Using only a handheld calculator, these fractions add up to approximately 2
...
A table shows how the total grows as more and more terms are
added
...
The harmonic label originates with the
Pythagoreans who believed that a musical string divided by a half, a third, a
quarter, gave the musical notes essential for harmony
...
If we add the first 8 terms (recognizing
that 8 = 2 × 2 × 2 = 23) for example

(where S stands for sum) and, because ⅓ is bigger than ¼ and ⅕ is bigger
than ⅛ (and so on), this is greater than

So we can say

and more generally

If we take k = 20, so that n = 220 = 1,048,576 (more than a million terms),
the sum of the series will only have exceeded 11 (see table)
...
The series is said to diverge to
infinity
...
Quite
dramatically the series converges to π2/6 = 1
...

283

In this last series the power of the terms is 2
...
If the
power increases by a minuscule amount to a number just above 1 the series
converges, but if the power decreases by a minuscule amount to a value just
below 1, the series diverges
...


The Riemann zeta function
The celebrated Riemann zeta function ζ(s) was actually known to Euler in the
18th century but Bernhard Riemann recognized its full importance
...
The value of ξ(2) is π2/6, the result
discovered by Euler
...
Roger Apéry proved the important result that ξ(3) is an irrational
number but his method did not extend to ξ(5), ξ(7), ξ(9), and so on
...
This enables the
powerful techniques of complex analysis to be applied to it
...
In a paper presented to the Berlin Academy of Sciences
in 1859, Riemann showed all the important zeros were complex numbers that lay
in the critical strip bounded by x = 0 and x = 1
...

The first real step towards settling this hypothesis was made in 1896
independently by Charles de la Vallée-Poussin and Jacques Hadamard
...
In 1914, the English mathematician G
...
Hardy proved that an infinity of
zeros lie along the line x = ½ though this does not prevent there being an
infinity of zeros lying off it
...
While these
experimental results suggest that the conjecture is reasonable, there is still the
possibility that it may be false
...


Why is the Riemann hypothesis important?
There is an unexpected connection between the Riemann zeta function ξ (s)
and the theory of prime numbers (see page 36)
...
Using primes,
we can form the expression

and this turns out to be another way of writing ξ(s), the Riemann zeta
function
...

In 1900, David Hilbert set out his famous 23 problems for mathematicians to
solve
...
Before leaving port
he would send his friend a postcard with the claim that he had just proved the
Riemann hypothesis
...
If the boat sank he would
have the posthumous honour of solving the great problem
...

The person who can rigorously resolve the issue will win a prize of a million
dollars offered by the Clay Mathematics Institute
...


the condensed idea
The ultimate challenge

286

Glossary
Algebra Dealing with letters instead of numbers so as to extend arithmetic,
algebra is now a general method applicable to all mathematics and its
applications
...

Algorithm A mathematical recipe; a set routine for solving a problem
...

Axiom A statement, for which no justification is sought, that is used to define
a system
...

Base The basis of a number system
...

Binary number system A number system based on two symbols, 0 and 1,
fundamental for computer calculation
...
The cardinality of the set {a, b,
c, d, e} is 5, but cardinality can also be given meaning in the case of infinite sets
...

Commutative Multiplication in algebra is commutative if a × b = b × a, as in
ordinary arithmetic (e
...
2 × 3 = 3 × 2)
...
g
...

Conic section The collective name for the classical family of curves which
includes circles, straight lines, ellipses, parabolas and hyperbolas
...

Corollary A minor consequence of a theorem
...
The
statement ‘All swans are white’ is shown to be false by producing a black swan as
a counterexample
...
In the fraction 3/7, the number
7 is the denominator
...
For an expression describing how distance depends on time,
287

for example, the derivative represents the velocity
...

Diophantine equation An equation in which solutions have to be whole
numbers or perhaps fractions
...
AD 250)
...
There are gaps between
discrete values, such as the gaps between the whole numbers 1, 2, 3, 4,
...
For example, the Poisson distribution gives the probabilities of x
occurrences of a rare event happening for each value of x
...
The
number 2 is a divisor of 6 because 6 ÷ 2 = 3
...

Empty set The set with no objects in it
...

Exponent A notation used in arithmetic
...
The expression 5 × 5 × 5 is written 53, and so
on
...
Equivalent terms are power and index
...

Geometry Dealing with the properties of lines, shapes, and spaces, the
subject was formalized in Euclid’s Elements in the third century BC
...

Greatest common divisor, gcd T h e gcd of two numbers is the largest
number which divides into both exactly
...

Hexadecimal system A number system of base 16 based on 16 symbols, 0,
1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F
...

Hypothesis A tentative statement awaiting either proof or disproof
...

Imaginary numbers Numbers involving the ‘imaginary’ i = √ –1
...

Integration A basic operation in Calculus that measures area
...

Irrational numbers Numbers which cannot be expressed as a fraction (e
...

288

the square root of 2)
...
For example, starting with 3 and repeatedly adding 5 we have the
iterated sequence 3, 8, 13, 18, 23,
...

Matrix An array of numbers or symbols arranged in a square or rectangle
...

Numerator The top part of a fraction
...

One-to-one correspondence The nature of the relationship when each
object in one set corresponds to exactly one object in another set, and vice versa
...

This may be a solution that minimizes cost or maximizes profit, as occurs in
linear programming
...
In 73, the place value of 7 means ‘7 tens’ and of 3 means ‘3 units’
...
For example, a tetrahedron has
four triangular faces and a cube has six square faces
...
For
example, 7 is a prime number but 6 is not (because 6 ÷ 2 = 3)
...

Pythagoras’s theorem If the sides of a right-angled triangle have lengths x,
y and z then x2 + y2 = z2 where z is the length of the longest side (the
hypotenuse) opposite the right angle
...
R
...

Rational numbers Numbers that are either whole numbers or fractions
...
The number 17 divided by 3 gives 5 with
remainder 2
...

Series A row (possibly infinite) of numbers or symbols added together
...

289

Square number The result of multiplying a whole number by itself
...
The square numbers are 1, 4,
9, 16, 25, 36, 49, 64,
...
For example, 3 is the square root of 9 because 3 × 3 = 9
...
It cannot be done
...
If a shape can be rotated so that it fills
its original imprint it is said to have rotational symmetry
...

Theorem A term reserved for an established fact of some consequence
...
The number π is a transcendental number
...
For
example, the twins 11 and 13
...

Unit fraction Fractions with the top (numerator) equal to 1
...

Venn diagram A pictorial method (balloon diagram) used in set theory
...


290

Index
Bold page numbers indicate a glossary entry
A
Abel, Niels 58, 154
adding
fractions 13–14
imaginary numbers 33–4
matrices 156–7
zero 5, 7
algebra 56–9, 204
abstract groups 154
curves 90
Fermat’s last theorem 196–9
genetics 148
matrices 58, 156–9, 205
Pascal’s triangle 53
and topology 95
algorithms 60–3, 187, 204
amicable numbers 42
angles
Euclid’s postulates 109
measuring 8
trisecting 80–1
Archimedes of Syracuse 20–1
area
circle 21
greatest common divisor 61
polygons 113
triangle 87
under curve 79
Argand diagram 34–5, 204
Aristotle 64, 65
averages 141–2
axioms 59, 72, 74, 109, 127, 154, 155, 204
B
291

base 10 8, 10, 11, 204
base 60 8, 204
Bayes’s theory 128–31
Benford, Frank 136, 138
Bernoulli, Jacob 24, 89
binary system 11, 160–2, 204
birthday problem 132–5
Bourbaki, Nicholas 72
Brianchon’s theorem 115
bridges, Warren truss 87
bucky balls 93
buildings
golden ratio 50, 51
with triangles 87
butterfly effect 104–5, 107
C
Caesar, Julius 160, 162
calculus 76–9
Cantor, Georg 28–31, 72, 73, 75
cardinality 29–30, 74–5, 204
catenary curve 90
Cayley, Arthur 35, 98, 102, 119, 121, 153–4
central limit theorem 141, 142
chance 124, 128, 132
chaos theory 104–7, 204
Chinese remainder theorem 63
circle 88, 109, 111, 115
pi (π) 20–1
squaring 22, 81–2, 205
codes 160–3
coincidence 135
colour
four-colour problem 120–3
genetics 148–9
combinatorics 164–7, 187
commutative 204
complex numbers 32–5
compound interest 176–9
conic sections 88–90, 115, 204
constructions 80–3
292

continuum hypothesis 75
corollary 204
correlation 144–6
counterexample 69, 204
counting 4, 28–9, 30, 119, 164–7
cubed numbers 81, 93, 96, 98, 196
curves 88–91
algebraic 90
calculus 79
Koch snowflake 102, 103
normal 140–3
D
data, connecting 144–7
da Vinci, Leonardo 89, 96
decimal numbers 10–11
converting fractions 14–15
origins 8
deficient numbers 41
De Morgan, Augustus 66, 70, 73, 120
denominator 12, 204
Descartes, René 32, 42, 89, 90
diet problem 180–3
differentiation 76, 77–9, 204
dimensions 96–9
fractional 99, 103
Diophantine equations 62–3, 196, 197, 204
direct method 69–70
discrete 204
discrete geometry 112–15, 167
distributions 136–9, 140, 204
dividing
Euclid’s algorithm 61–3
zero 6, 7
divisor 204
DNA 89, 151
dodecahedron 93
donut 92, 94, 122
Dudeney, Henry 176, 179
Dürer, Albrecht 169–70

293

E
e 24–7
Egyptians 8, 15, 165
Einstein, Albert 97, 111, 113, 177, 192, 194–5
ellipse 88, 89, 115
empty set 7, 204
encryption 160–3
equations 56–8, 62–3, 90, 196
error detection, codes 161
Euclid of Alexandria
algorithm 60–3
constructing polygons 82, 83
perfect numbers 40, 43
postulates 108–11
prime numbers 38
QED 70
triangles 84, 85
Euler, Leonhard
e 26, 27
Euler line 85–6
Euler’s formula 93–4, 163
Fermat’s last theorem 198
graphs 116–17
Latin Square 172–3, 174
perfect numbers 43
pi (π) 21–2
squared squares 170
exponent 204
F
factorial numbers 165–6
Fano plane 114
Fermat, Pierre de
Fermat’s last theorem 170, 196–9
prime numbers 39, 83
probability 125
Fibonacci sequence 44–7, 54
four-colour problem 120–3
fractals 54, 99, 100–3, 107
fractional dimension 103
fractions 8, 12–15, 204
294

converting to decimals 14–15
counting 30
Riemann hypothesis 200–3
square roots 18–19
Franklin, Benjamin 170
fuzzy logic 67
G
Galileo 77, 78, 192, 193
Galton, Francis 121, 144, 146
gambling 125–6, 127, 143
game theory 188–91
Gauss, Carl Friedrich 37, 39, 83, 110, 140
Gelfond’s constant 26
genetics 148–51
geometry 204
dimension 96–9
discrete 112–15
elliptic 111
Euclidean 108–10, 114
hyperbolic 110
parallel postulate 75, 108–11
projective 114
topology 92–5, 99
Gödel’s theorem 74, 75
Goldbach conjecture 38–9
golden ratio (Φ) 46–7, 50
golden rectangles 48–51
graphs 116–19, 123
Grassmann, Hermann 59, 98
gravity 77–9, 195
greatest common divisor (gcd) 61–3, 204
group theory 59, 152–5, 167
H
Halmos, Paul 70, 123
Hamilton, Sir William Rowan 34–5, 58–9
hand-shaking theorem 117–18
Hardy, G
...
149–51, 168, 202, 203
Heawood, Percy 121, 122
295

hexadecimal system 204
hieroglyphs 15
highest common factor 61
Hilbert, David 75, 99, 203
Hindu-Arabic numbers 4, 8
hyperbola 88, 115
hyperspace 97–8
hypothesis 204
I
i 33–4
icosahedron 93
imaginary numbers 32–5, 204
indirect method 70
infinity (∞) 6, 28–31
integration 76, 79, 205
interest 25, 176–9
irrational numbers 19, 21, 25, 30–1, 205
iteration 100–1, 205
J
Jordan, Camille 90–1
K
Kirkman, Rev
...
189
Navier–Stokes equations 107
negative numbers 32–3, 54
Newton, Isaac 76, 78, 90, 193, 195
non-planar graphs 118
normal curve 140–3
297

number systems 8–11
numerator 12, 205
numerology 39, 42
O
octahedron 93
one-to-one correspondence 28, 30, 205
optimum solution 205
P
paper, sizes 48–9
parabola 17, 88, 89, 115
parallel lines 108, 114, 115
parallel postulate 75, 108–11
Pascal, Blaise
Pascal’s theorem 115
Pascal’s triangle 52–5, 142
probability 125
Pearson’s correlation 144–5
pendulums 105–6
perfect numbers 40–3
Pick’s theorem 113
pi (π) 20-3
place-value system 8, 205
Poincaré, Henri 95, 102
Poisson distribution 137, 139
polygons 16, 21, 82–3, 113
polyhedra 92–4, 205
polynomial time 187
predictions 106, 139
prime numbers 36–9, 43, 83, 200, 203, 205
prisoner’s dilemma 191
probability 124–7
Bayes’s theory 128–31
birthday problem 132–5
conditional 128–9
distributions 136–9
e 27
genetics 148–51
normal curve 142–3
298

proof 68–71, 85, 123
Pythagoras 16, 40, 41, 89, 200
theorem 18, 84–5, 205
Q
quadratic equations 57–8
quaternions 58–9, 205
R
rational numbers 13, 205
real numbers 30–1, 72
reasoning 64
rectangles, golden 48–51
regression 144, 146–7
relativity 111, 192–5
remainder 205
Riemann, Bernhard
elliptic geometry 111
Riemann hypothesis 200–3
Roman numbers 8–10
rotational symmetry 152–3
Russell, Bertrand 73–4
S
scientific notation 11
sequence 205
series 205
sets 7, 28–9, 67, 72–5, 205
Sierpinski gasket 54, 102
space–time 97, 99, 111, 195
Spearman’s correlation 145–6
spheres 94, 95, 98, 111
spiral, logarithmic 89
square numbers 16–17, 39, 170, 196–8, 205
square root 17–19, 205
of –1 32, 33
squares
Latin 167, 172–5
magic 168–71
squaring the circle 22, 81–2, 205
299

statistics
connecting data 144–7
normal curve 140–3
probability 128, 136–8
Steiner Triple System (STS) 115
Stokes, George Gabriel 107
string theory 97, 99
subtracting, zero 5
Sudoku 172
superabundant numbers 40–1
supergolden ratio/rectangle 47, 51
syllogism 65
symmetry 152–5, 205
T
tetrahedron 93
theorem 68, 70, 205
three-bar motion 90
topology 92–5, 99
torus 122
travelling salesperson 184–7
travel/transport 159, 183
trees 118, 119
triangles 84–7
constructing 82
elliptic geometry 111
Fano plane 114
Pascal’s 52–5
Sierpinski gasket 102
symmetry 154
triangular numbers 16–17
trigonometry 84, 86
tripod 152, 154
triskelion 152, 153, 154
twin primes 38–9, 205
U
unit fraction 205
unity 35

300

V
Venn diagram 72, 129, 205
von Lindemann, Ferdinand 22, 26, 82, 199
von Neumann, John 188
W
weather forecasts 104, 106–7
Weinberg, Wilhelm 149–51
Wiles, Andrew 196, 199
X
x–y axes 205
Z
Zermelo-Fraenkel axioms 74, 75
zero 4–7, 10, 202
zeta function 202–3

301

Quercus Publishing Plc
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London
WC1A 2NS
First published in 2007
Copyright © Tony Crilly 2007
Tony Crilly has asserted his right to be identified as author of this Work
...
No part of this publication may be reproduced, stored in a
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...

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UK and associated territories
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ISBN-13: 978 1 84724 008 8
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