THE MATHEMATICS AND
SCIENCE NEWSPAPER
Godalming College
CHRISTMAS 2014
EDITION 1
Table of contents
Introduction
3
The Peppered Moth
4
“Guess which part of your body ‘nose’ when death is imminent?”
5
Babylonian Mathematics
6
Abstract Algebra
8
‘Can Acceleration be Instantaneous?’
9
Maths: Real or Imagined?
21
Computer Vision
23
2 of 24
Introduction
Welcome to the first edition of the Godalming College Mathematics and Science
Newspaper. Our aim is to share with you the interests of Godalming students in an
accessible and entertaining way.
Our first edition covers a wide range of topics. We start off with ‘The Peppered Moth’, by
Abigail Baker, which discusses how evolution and natural selection occur using the
example of the Peppered Moth during the industrial revolution. Secondly, Holly Edwards
takes a look at a topic of research at the forefront of medicine in ‘Guess which part of
your body ‘nose’ when death is imminent’, looking at the reasoning behind why a person’s
sense of smell might be an indicator of their life expectancy.
Yan Yanchuk eases the transition into maths with his defence of Babylonian mathematics,
asking the question “Do we even need Pythagorus?”. Will Fourie then goes in completely
the opposite direction, delving straight into the modern, high level subject of Abstract
Algebra - expect this to be a hard read. Thankfully Scott Sussex brings us back down to
Earth afterwards with his article on acceleration, and whether or not it is meaningful for
it to be instantaneous. Our penultimate article- ‘Mathematics: Real or Imagined?’- by Kate
Stevenson, discusses the nature of mathematics, and whether it is something that we can
actively discover, or something that we have created to fit around the laws of the
universe.
Finally, we turn from pen and paper to the ambitious realm of ‘Computer Vision’, with
Lawrence Burvill, to look at research in how computers might be able to replicate the
human ability of sight.
Like all firsts, producing this issue has been a result of real organisation and dedication
from everyone involved. At the same time, it has been extremely rewarding to bring
together a group with incredibly diverse interests, and be able to learn from each other’s
very different skill sets in producing our articles. If you are interested in getting involved,
email [email protected]. We look forward to welcoming any new contributors for
future issues.
3 of 24
The Peppered Moth
By Abigail Baker
The peppered moth has been used as an example of natural selection in action. The original version
of the peppered moth was the variety seen below.
Figure 1 http://www.mothscount.org/
text/63/
peppered_moth_and_natural_selection.h
tml
Figure 2 http://www.simplybirdsandmoths.co.uk/sbam/index.php/
1931-peppered-moth-geometridae-biston-betularia/
These moths camouflage very well against the
colour of the lichen on tree bark.
However during the industrial revolution (1800s) the soot in the air killed the lichen and blackened
tree trunks causing the white moths to become much more visible to predators such as birds when
resting on the bark. This caused white peppered moth populations to decline.
At this point scientists observed a phenomenon where a
variation of the peppered moth the population of which
had previously been only been very small, increased.
The Black peppered moth, until this point had been
predated on much more than the White peppered moth
as they were unable to camouflage against the lichen
on the trees.
With the trees blackened by soot, the black peppered
moth was more suited to its environment than the
white peppered moth. The Black moth was predated by
birds less, so more survived to pass on their genes, and
the population of black moths increased.
Figure 3 http://
www.simplybirdsandmoths.co.uk/sbam/
index.php/1931-peppered-moth-geometridae-
The University of Liverpool has recently discovered that
“a single ancestor causes increased dark pigment” and
identified chromosome 17 as responsible for the colour
Bibliography
http://news.liv.ac.uk/2011/05/03/liverpool-team-shed-light-on-the-peppered-moths-dark-past/
http://www.simplybirdsandmoths.co.uk/sbam/index.php/1931-peppered-moth-geometridae-biston-betularia/
http://www.mothscount.org/text/63/peppered_moth_and_natural_selection.html
http://www.simplybirdsandmoths.co.uk/sbam/index.php/1931-peppered-moth-geometridae-biston-betularia/
4 of 24
change.
“Guess which part of your body ‘nose’ when death is imminent?”
By Holly Edwards
In a study carried out by the University of Chicago with the
National Social Life, Health and Ageing project, it was
found that the nose can adequately indicate when death is
imminent within 5 years (Jayant M. Pinto, 2014).
Over 3,000 participants aged between 57 and 85 were
asked to identify five common scents of rose, leather, fish,
orange and peppermint. The number of scents incorrectly
identified created a score to signify smell lost. Of the
12.5% of participants identified 5 years after the study who
had died, only 10% had a healthy sense of smell. As a
result, the participants who incorrectly identified all the
scents were four times more likely to die within the 5
years. Even considering race, sex, mental health and
Olfactory system http://79.170.44.108/
socioeconomic status, smell loss was an associated
coffeereal.co.uk/?m=201209
indicator of impending death. So it is concluded that
anosmic adults, those who lack a sense of smell, have over
three times the odds of death compared to normosmic adults, who have a sense of smell.
Researchers suggest this may be due to the olfactory nerve, which contains small receptors and requires
cellular regeneration. The olfactory nerve relays sensory data from the nasal cavity to the brain and
hence means we have a sense of smell (Healthline, n.d.). Being the only part of the nervous system that
is continuously regenerated by stem cells, if these stem cells are not repaired, sense of smell will
decline. As regeneration through stem cells reduces with age and body state, the inability to identify
scents could suggest the body is unable to continue repairing itself – hence indicating when death is
imminent.
A contrary reasoning could be that the
olfactory nerve is also unique in that
it’s the only part of the nervous system
exposed to open air. This means it can
relay poisons and pathogens to the
brain, which could cause death due to
its direct effects.
To conclude, the odds for 5 year
mortality for olfactory dysfunction
compared to the most common causes
of death is much greater and more
distinguishable. This means there is
adequately strong evidence to show
that olfactory dysfunction is an
independent risk factor for death within
5 years.
10.1371/journal.pone.0107541.g002 1
Bibliography
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0107541
http://www.theguardian.com/science/neurophilosophy/2014/oct/01/your-noseknows-death-is-imminent
5 of 24
Babylonian Mathematics
“Why do we even need Pythagoras?”
By Yan Yanchuk
From our early school years, teachers have always liked to
glorify the mighty Pythagorus, and drill into our heads his
theorem, that for a right angled triangle a2 + b2 = c2.
As mathematicians we seem to encounter this theorem
almost on a daily basis; whether it’s as simple as proving a
triangle to be right-angled or as complex as to resolve
vectors in up to infinite dimensions, a2 + b2 = c2 appears
almost anywhere you look! Even Fermat’s Last Theorem, as
proved by Andrew Wiles in 1995, is just a variation of the
aforementioned theorem, with the powers of a, b and c
being raised to n>2 in the positive integers.
However Pythagoras did not come up with his Andrew Wiles, with “Fermat’s Last Theorem” written on the
theorem. In reality, it was the ancient race called the blackboard.
Babylonians (an empire which prospered between
1900-1500BC in what is now Iraq) who knew and used a2 + b2 = c2 over 1500 years previously. The
evidence comes from some 400 clay tablets unearthed in the mid-1850s, the most famous one
stating the method for finding one of the sides of a right-angled triangle – which gives evidence
that the Babylonians had their own version of Pythagorean mathematics.
In fact some fields of Babylonian Mathematics
could be seen to be much more advanced than
that of Pythagoras’ cult. For example, unlike the
Babylonians, Pythagoras downright refused to use
irrational numbers such as surds. There is even a
rumour of him drowning one of his students,
Hippasus, for presenting him with a right angled
triangle with two sides of 1 unit length, asking
him to find the hypotenuse. Since the answer is
the square root of two, there was no way of
expressing it as a rational number (i.e. as p/q
where p and q are integers) which infuriated Pythagoras, unleashing a mathematical rage. Thus
Hippasus is sometimes credited with the discovery of irrational numbers.
“4 is the length and 5 the diagonal. What is the
breadth?
Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.”
As mentioned before, the Babylonians have always suspected the existence of irrational numbers,
even going so far as to estimate the value of root-2 correct to seven significant figures. Amongst
irrational numbers and the Pythagorean theorem, their achievements included the use of π
(although much like the Bible, it was estimated to be just 3), the
creation of their own mile (equal to just over 7 modern miles), and most
famously for their geometrical applications to Astronomy.
The fascinating thing about Babylonian Astronomy is that in order for it
to be accurate, what is known as Euclidian Geometry (i.e. geometry that
we all use every day – wherein angles in a triangle add up to 180°)
cannot be used. Instead, a different type of geometry was used called
Spherical Geometry. In this type of geometry, angles in triangles do not
add up to 180°. An extreme example of this would be a triangle with 3
right angles.
6 of 24
To see how this is possible, what is needed is to imagine the usual plane instead as a perfect sphere
(in this example the Earth). The first point of the triangle is on the North Pole (exact top of the
sphere); the second is directly south of the first point, on the equator, the line forming a right
angle with the equator. The third point of the triangle is directly west of the second point going a
quarter of the way around the Earth. Then the second point is connected to the first point by going
directly north, forming another right angle with the equator. In addition, because the second and
third points encompass a quarter of the Earth, the lines between 1st and 2nd points, and the 3rd and
1st points form a right angle. As a result, if the triangle is then drawn on a map/plane, then it
would have three right angles (so even though its vertices are curved on a plane, they are in theory
straight on a sphere; and the sum of the angles is therefore 270°).
Another feat of the Babylonians was
their use of the hexadecimal base
numeral system, in other words they
used base sixty numbering rather
than the base ten system we use.
For example: in the decimal system
the number 123 is the same as 1x102
+ 2x101 + 3x100, however using the
hexadecimal notation,123 would
mean 1x602 + 2x601 + 3x600 (which
in the decimal system would be
written as 3723). The decimal
system uses the Indo-Arabic
notation (1,2,3,4,5,6,7,8,9,0), the
Babylonians developed their own notation, as shown on the right.
So even though Pythagoras is still famous for his theorem (even though it was not his invention),
the Babylonians were clearly a much more advanced in their knowledge and understanding of the
field of mathematics. Unfortunately, our knowledge of Babylonian Mathematics was only unearthed
in the mid-19th century - yet Pythagoras was known at least since the 11th century - so without
Pythagoras our society would be much less advanced. Subsequently, no matter how seemingly
obsolete Pythagorean mathematics seems to be – it would seem that the world owes Pythagoras for
his input into the world of mathematics for without it - we would be still be as ignorant as those
who came before us in terms of this immense field!
7 of 24
Abstract Algebra
Abstract Algebra
William Fourie
2014
1
1.1
Basic Concepts
Introduction
University mathematics is not like A-level maths. It is barely close.
At A-level, specifically in the core modules, maths is distilled down to you
mostly in the form of calculus and algebraic manipulation. You are, for the most
part, simply taught how to manipulate symbols - very little logical thought is
required apart from the rules of doing so. This changes, entirely, at university.
The pure maths that one would study there relies entirely on the concept of
logical proof, on the idea of building up what are known as theorems from a
series of logical steps, starting from assumed truths, or axioms.
Pure mathematics is about proof.
If one were to read popular books on the subject, such as “Fermat’s Last
Theorem” by Simon Singh, then they would have an idea of how this works, and
realise, I hope, that mathematics is not as boring as GCSE’s, A-levels, et-cetera,
would lead them to believe. However such books rarely delve into the maths
itself, they rarely show you what is going on! I intend to do so.
One fundamental topic that is used by mathematicians today is that of
Abstract Algebra. This is not at all similar to the type of algebra you would
have seen before, for although you could say that that sort of algebra is pretty abstract, it doesn’t come close to this. Abstract algebra is the study of structures.
It is the study of what lies behind the equations, essentially, and so can be
extremely powerful as a tool in answering questions about them.
A good example of such a question would be: “Could there exist a formula
using radicals that can solve the general quintic equation?” What this is asking
is, is there a formula, similar to that of the quadratic formula, that one can
simply plug in the coefficient of any quintic and be given all of the solutions?
It seems perfectly reasonable to think the answer is yes, because such a formula
exists for quadratics, cubics and quartic polynomials. If you start with the
general quintic and try to algebraically manipulate it to get “x” on one side of
the equals sign, however, you will fail every time, but you will not be able to
prove it is impossible just by doing that. What you need is abstract algebra, to
study how the roots of a polynomial are related to it’s solubility. Specifically
what is needed for this question is Galois Theory, which is where two other sub
1
8 of 24
disciplines, group theory and field theory, intersect. The story of Evariste Galois
is quite a tragic one, of a prodigy, dead before even twenty one years old, and I
would recommend you find out as much as you can about his life if you are at all
interested. Before one can possibly move on to study Galois Theory - covered
only in third year university courses - one needs to first understand both group
and field theory, and without too much more introduction, I shall move on to
the first of those soon.
1.2
Sets
One is barred from almost any way of entry into modern mathematics without
knowledge of set theory. It underpins all of abstract algebra and is responsible
for much of the notation you’ll have to learn in these pages. It is a branch of
mathematics which was formally created by a man named Georg Cantor, in
the 1870s. As seems to be the case with many of these pioneers, his tale was
quite a sad one, for his ideas about infinity, namely that there are di↵erent
types, some larger than others, caused him to be hounded by the mathematical
community for several years, resulting in deep depression. Later on, however,
he was accepted, so much so that one of the giants of the 20th century, David
Hilbert, proclaimed: “No one will drive us from the paradise which Cantor
created for us”.
A set is a collection of elements, there is no simpler definition. Anything
could be an element of a set, be it numbers, shapes, symmetries, types of fruit,
even sets themselves. Examples of sets are the integers, Z, the rational numbers,
Q, the real numbers, R, and many other collections of numbers such as the more
exotic complex numbers, C, in which 1 has a square root, i.
There are several notations that one needs to be familiar with, the first of
which is 2, which means “is a member of” or “in”. One can write 2 2 Z, or
3 2 Z but not 32 2 Z as 32 is not a member of the integers, as it is not a whole
number. We write 32 2
/Z
Another piece of notation is ⇢. One thing you might notice about some sets,
such as the integers and the rationals, is that in some sense one is contained
within the other - every integer is a rational number, too, for example. If every
member of one set is also in another set, then it can be said to be a subset of
that other set. So Z ⇢ Q means that the integers are a subset of the rationals.
Another sign ✓ is used to represent that one set is either a subset or is equal
to another, so if for sets A and B, A ✓ B and B ✓ A, then A = B, i.e. the two
sets are exactly the same.
[ and \ should be familiar to those who have studied statistics, they denote
the union and intersection of two sets. A [ B = {a | a 2 A or a 2 B}, so
both sets are combined. A \ B denotes the set that consists of elements which
lie both in A and B, so A \ B = {a | a 2 A and 2 B}.
A \ B is essentially the opposite of A \ B, it denotes the elements of A which
are not in B, so Q \ Z denotes the set of fractions which are not integers. It is
the di↵erence of two sets.
2
9 of 24
8 and 9, though notation I will try to avoid, are simply abbreviations for
“for all” and “there exists”, similarly ) and () are used to shorten “implies”
and “if and only if”.
Note: A if and only if B means that if A is true, then B is true, and if
B is true, then A is true. A if B means that if B is true, then A is true, but
if A is true, then B is not necessarily true also. A and B in this case represent
logical statements such as “bananas are yellow” or “they are ripe”.
The last topic that must briefly be covered on sets for now is that of a
relation and an equivalence. If one denotes the set of pairs of elements (a, b)
where a, b 2 A for some set A by A ⇥ A, then a relation is defined as a subset
of A ⇥ A, given the symbol ⌘, so one can write:
a⌘b
to mean
(a, b) is an element of ⌘
with a 6⌘ b meaning that (a, b) is not in ⌘. One defines an equivalence as a
particular type of relation, in which the relation is reflexive, transitive and
symmetric by which I mean
(1) a ⌘ a 8a 2 A,
(2) If a ⌘ b and b ⌘ c then a ⌘ c (this could be written a ⌘ b, b ⌘ c ) a ⌘ c),
(3) a ⌘ b () b ⌘ a.
In e↵ect, equivalence is a weaker form of equality, for though equality is an
equivalence, not all equivalences are equalities (there will soon be an example
of one which is not). An equivalence class of an element a is defined as the
set of all elements equivalent to a, written as [a], so
[a] = {x 2 A|x ⌘ a},
and so if the equivalence is equality, [a] = {a}, the set containing only a,
known as the singleton.
More could be said on sets, but for now I shall leave the topic as it is so I
can move on to some number theory.
1.3
Divisibility and Integers Modulo n
Divisibility is something which you should be familiar with, the concepts of
fractions, multiples, divisors and remainders were learnt in primary school. Integers modulo n, on the other hand, are something you may never have heard
of. They are an example of a set (or rather many sets) which was first thought
up by a man who was known as the “Prince of mathematicians”, Carl Friedrich
Gauss. Nowadays modular arithmetic is used primarily in codes - the type of
encryption used on your phone, your computer, and in fact for the internet in
general, is likely the type which uses modular integers, known as RSA.
3
10 of 24
1.3.1
Divisibility
2
If one writes 14
3 = 4 3 then they are using the fact that 10 = 4 · 3 + 2. 4 is said to
be the quotient and 2 is said the be the remainder of the division. In general
if n and d are integers, then there exist integers q and r, with 0  r < d such
that n = qd + r. This representation of n is unique, because if qd + r = q1 d + r1
with 0  r1 < d and r1 r then (q q1 )d = r1 r is an integral multiple of d
which is a nonnegative integer less than d, which can only occur if r = r1 and
so q = q1 .
If r = 0 then n = qd and d is said to be a divisor of n, denoted d|n. Note
that 1|n and n|0 are true for all n. The following properties of divisors will be
used frequently:
Theorem 1: Let m, n and d be integers, then:
(1) n|n for all n.
(2) If d|m and m|n then d|n.
(3) If d|n and n|d then n = ±d.
(4) If d|n and d|m then d|(xn + ym) for all x, y 2 Z.
Proof: (1) and (2) are quite clear from the definition of divisibility.
(3) If d = qn and n = q1 d then d = qq1 d, which implies that qq1 = 1 and so
either both q, q1 = 1 or both are equal to 1.
(4) If n = qd and m = q1 d then xn + ym = xqd + yq1 d = (xq + yq1 )d and so
that d is a divisor is clear, as (xq + yq1 ) 2 Z.
Expressions such as xn + ym are called linear combinations of n and m.
An integer d which divides both m and n is known as a common divisor of
m and n. An integer, d is known as the greatest common divisor of m and
n if:
(1) d is a common divisor of m and n.
(2) d 1
(3) If k is a common divisor of m, n then k|d.
The greatest common divisor of m and n is written gcd(m, n), and the
following holds:
Theorem 2: If d = gcd(m, n) then d = xm + yn for x, y 2 Z.
Proof: Let X be the set
X = {sm + tn|s, t 2 Z, sm + tn
1}
of all positive linear combinations of m and n. Then let d be the smallest
member of this set (every set of positive integers must have a smallest member),
so d 1 and d = xm + yn for integers x, y, which shows that if k is a common
divisor of m, n, then k|d by Theorem 1. It remains to prove that d is a common
4
11 of 24
divisor of n, m. To show this we need only show that d|n, for the proof of d|m
is the same. As n = qd + r, 0  r < d, we have:
r = n qd = n q(xm + yn) = ( qx)m + (1 qy)n
and so r is a linear combination of n and m, non-negative by its definition.
As d is defined as the smallest member of X and r < d, r must equal 0, and so
n = qd. Thus the three criteria for d = gcd(m, n) are fulfilled.
This leads onto the more important theorem:
Theorem 3: Let m, n 2 Z, not both zero. Then gcd(m, n) = 1 if and only
if 1 = xm + yn for x, y 2 Z.
Proof: If 1 = gcd(n, m) then 1 = xm + yn by Theorem 2. Conversely if
1 = xm + yn then any common divisor of m and n must divide 1, implying that
gcd(m, n) = 1.
If gcd(m, n) =1, then m and n are said to be relatively prime.
1.3.2
Integers modulo n
If a, b and n are integers, then we can say that a and b are congruent modulo n
if n|(a b), this is written as a ⌘ b(modn) and we refer to n as the modulus.
Thus 10 ⌘ 1(mod3), 18 ⌘ 4(mod7) etc. The fact that ⌘ is used implies that
we can in fact say that 18 and 4 are equivalent modulo n, this is true, and
summarised in the theorem below.
Theorem 4: Congruence modulo n is an equivalence on Z, so:
(1) a ⌘ a(modn) 8a 2 Z
(2) If a ⌘ b(modn) then b ⌘ a(modn)
(3) If a ⌘ b(modn) and b ⌘ c(modn) then a ⌘ c(modn)
The proof of this is left as a question at the end of the chapter.
If a is an integer, its equivalence class [a] for congruence modulo n is known
as its residue class modulo n, which is written ā for convenience.
Thus ā = b̄ () a ⌘ b(modn).
The set of all possible residue classes for a particular modulus is denoted
Zn . This gives:
Zn = {0̄, 1̄, 2̄ . . . n
1}
because n̄ = 0̄ and because a = qn + r, 0  r < n, is true for all a, which
implies that a ⌘ r(modn). Thus one can think of Zn as the set of all possible
remainders when integers are divided by n. We define ā+ b̄ = a + b and ā· b̄ = ab.
This is the reason that modular arithmetic is sometimes called clock arithmetic,
5
12 of 24
because if one is working in Z12 then 11 + 4 ⌘ 3(mod12), just as on a clock
3:00pm is four hours after 11:00am.
The final property of modular integers which must be proved is that of
inverses. b̄ is said to be the inverse of ā if ba = ab = 1̄. The next theorem
summarises when inverses of elements of Zn exist.
Theorem 5: Let a and n be integers with n
and only if gcd(a, n) = 1.
2, then ā has an inverse if
Proof: If 1 = gcd(a, n) then 1 = ba+cn for b, c 2 Z, thus 1̄ = ab. Conversely
if b̄ is the inverse of ā then ba ⌘ 1( mod n) and so n|(1 ba), say 1 ba = cn for
some integer c, then 1 = cn + ab and so by Theorem 3 gcd(m, n) = 1.
Thus if n is prime, every element except 0̄ of Zn has an inverse.
1.4
Groups
The notion of a group is generally accepted to have been created by Evariste
Galois around about 1830, he at least was the first to use the term groupe while
writing his paper (Galois was French). The ideas behind group theory did spread
further back than that, however, for one important theorem of group theory is
Lagrange’s Theorem, proven by Lagrange in the 1770s. Group theory is still an
active area of research in modern times, only in 1982, for example, the so called
monster group was discovered, which was the symmetry group of a 196,883 dimensional object - consisting of 808,017,424,794,512,875,886,459,904,961,710,757,
005,754,368,000,000,000 symmetries!
A group is a type of set in which there is an operation associated with it
that is used to combine elements, so for example with integers that could be
addition. We say that the integers under addition are a group. However a group
is more than just a set that has an operation, it must also obey four axioms, or
rules. These are:
1. The group must be closed, if a 2 G, where G represents some group,
and b 2 G, then ab 2 G. By ab or a · b I just mean a and b, combined under
the operation. With this notation ab could even mean a + b. This is just the
notation I prefer to use, it is valid to use addition as notation, too, if you want.
2. The operation must be associative, so (ab)c = a(bc), i.e. brackets do
not matter. This is not the same as commutativity, that ab = ba, that does
not have to hold, and doesn’t for some operations of groups, such as matrix
multiplication.
3. There exists an identity, denoted 1 (or ✏), such that 1a = a1 = a. The
identity is the “do nothing” element, an example would be 0 in additive integers
because a + 0 = a. There is only ever one identity for a specific operation,
6
13 of 24
something which can be easily proved. If 1 and e were both identities then
1 = 1 · e, but 1 · e = e, so 1 = e.
4. Every element g has an inverse g 1 , such that g 1 g = gg 1 = 1. The
example in additive integers is that every integer has a negative, such that
a + ( a) = ( a) + a = 0. For every element there is only 1 inverse, which
could even be the element itself, in which case it is said to be self inverse
(or of order two). This is, again, simple to prove, as if a 1 = b or b1 , then
ab = 1 = ba and b1 a = 1 = ab1 , so b1 = b1 1 = b1 (ab) = 1b = b.
Were you to check, you would see that Z under addition is a group, for it
obeys all four axioms but Z⇤ , integers under multiplication, is not a group. Z⇤
obeys axioms 1, 2 and 3, but not 4, as if you multiply two integers, a and b,
together, you can never get the answer 1 (unless, of course, a and b are both
1). One of the numbers must be a rational number, say b = a1 = a 1 . This
seems to imply that Q⇤ is a group, but it is not unless you exclude 0, for 0 has
no inverse 10 . This applies to R⇤ and C⇤ , too, so generally Q⇤ , C⇤ , and R⇤ are
taken to mean without 0.
Here are a few more notations associated with groups:
If g 2 G, then ggg=g 3 , or if n-many g’s, g n . The rules for exponents in
groups are the same as with numbers, so g a g b = g a+b and (g a )b = g ab , a, b 2 Z.
In the familiar integers, we know that if a+b = a+c, then b=c. This is
known as a (one-sided) cancellation law, and holds for groups in general, as
shown by the following proof: If gh = gf , then by multiplying by g 1 on the
lef t side, g 1 gh = g 1 gf , so 1h = 1f = h = f . Therefore h and f are the same
element. You can do similarly if hg = f g but cannot do anything if gh = f g
unless G is said to be an abelian group, i.e. the operation is commutative.
Figure 1: Triangle Symmetries
One good example of a group that does not involve numbers is the group
of symmetries of a triangle (fig1). A triangle has three reflectional symmetries,
7
14 of 24
which we shall denote ⌧a , ⌧b and ⌧c , with a, b, c representing the lines of symmetry - the letters are in order, clockwise round the triangle at the vertices. A
triangle can also be rotated clockwise 120 degrees 3 times, denoted 120 , 240 ,
360 , with each rotation placing the triangle perfectly onto itself. Therefore if we
take S3 to be the set of triangle symmetries, then this set contains six elements,
with 360 = ✏. Hopefully you can see, however, that it needn’t be so complex,
for the other rotations can be made by simply repeating the first, so in fact we
can just take 120 = and the other two rotations as
= 2 and
= 3
= ✏.
Similarly, though maybe less obvious, if you take ⌧a = ⌧ , then ⌧b = ⌧ 2 and
⌧c = ⌧ (strictly speaking, all the products should be the other way round, as
these are essentially combining mappings, but for ease of reading right now they
read left to right). Thus, S3 = {✏, , 2 , ⌧ , ⌧ , ⌧ 2 }.
1
Other useful things to notice are that ⌧ = ⌧ , and so as
= 2, ⌧
2
2
2
= ⌧
= ⌧ , and similarly ⌧ = ⌧ . Also, all elements containing ⌧ (in
simplest form) are self inverse. If you were to draw up a table of elements and
their combinations, you would see that the group is closed. It would also be
possible to verify that the operation is associative. Every element has an inverse,
and clearly 1 exists. Thus S3 is a group.
It is now possible to represent S3 as an abstract group, which means that
all that is defined is how many elements there are, and the rules for combining
them, essentially a template which the group follows. So for S3 the abstract
group, called D3 - the dihedral group of order 3 - for reasons you’ll later see,
is defined as D3 = {1, a, a2 , b, ba, ba2 } aba = b. Abstract groups are a useful
concept as if one can prove a theorem about an abstract group (or better yet,
an abstract group f amily) then that theorem will be true for all groups that
have that abstract group. You’ll notice that there are 3!=6 elements in D3 , so
this abstract group could also represent the number of di↵erent ways to arrange,
or permute, 3 objects. P ermutation groups are some of the most important
groups in group theory, as a theorem known as Cayley’s Theorem will later
show.
1.5
Subgroups
The concept of a subset is quite easy to understand, and subgroups are simply
an extension of that concept. A subgroup of a group is a subset of that group
that is also a group, i.e. it obeys the four axioms. An example would be the
subset {1, a, a2 } in D3 , as D3 ’s operation is associative then so is this subset’s,
this subset contains 1, every element has an inverse, and it is closed as a3 =1,
thus this set is a group, called C3 - the cyclic group of order 3. As every subset
will have the same associative operation, to test if a subset is a subgroup one
only needs to test the other three axioms. There is a simpler test for subgroups
8
15 of 24
of finite groups - those with a finite number of elements - which is that you
simply need to check if the subset is closed.
Theorem 1: A subset H of a finite group G is a subgroup if and only if H
is closed.
Assume H is closed, then let us take h 2 H. As H is closed every power,
h, h2 , h3 , etc is also in H, and as there are only so many elements the powers
can’t all be distinct. So for some integers n and m, hn = hn+m . This implies
that hm = 1. So 1 is a member of H. Similarly hm 1 is the inverse of h (to see
why, multiply by h). Thus, H is a subgroup as the choice of h was arbitrary.
The converse is clear by the definition of a group.
I’m also going to prove the existence of so called conjugate subgroups.
Theorem 2: Let H be a subgroup of a group G, then if g 2 G, gHg
also a subgroup of G. gHg 1 = {ghg 1 |g 2 G, h 2 H}.
1
is
Proof: 1 is a member of gHg 1 , as g1g 1 = gg 1 = 1. Also, if h 2 H,
then (ghg 1 ) 1 is just gh 1 g 1 as ghg 1 gh 1 g 1 = ghh 1 g 1 = gg 1 = 1,
so every element has an inverse. gHg 1 is closed as if h and h1 are in H,
ghg 1 gh1 g 1 = ghh1 g 1 , which as hh1 2 H, is also a member of gHg 1 . Thus,
gHg 1 is a subgroup.
If for a specific g, g is also in H, then as the subgroup H is closed, gHg 1
= H, just rearranged. If gHg 1 = H is true for every g 2 G, even those not in
H, then H is said to be a normal subgroup of G. This is a very important type
of subgroup which I will later revisit more deeply. Long before that, however,
will be a “case study” of an important family of groups. Cyclic groups.
1.6
Cyclic Groups
Cyclic groups are some of the most useful groups in abstract algebra, for they
can be used to prove a myriad of properties about abelian (commutative) groups.
The reason for this is because, though this will not be proven until much later,
every single abelian group is actually the same as a so called product of cyclic
groups. Exactly what I mean by “the same” and “product” is again something
which will have to wait.
Cyclic groups are those of the form {. . . a 1 , 1, a, a2 , a3 , a4 . . .}, and are denoted Cn , with n being the number of elements in the group, or alternatively
hai. If you look at the additive integers, they are an example of a cyclic group,
with a = 1 (the number, not the identity), and 12 meaning 1+1. Another example is the group of rotational symmetries of a triangle - or any regular polygon.
Cyclic groups consist of the powers of an element.
Here’s the first important theorem about these types of groups:
9
16 of 24
Theorem 1: Let hgi be the cyclic group generated by g:
If | hgi | (the size of the group) is finite, with g n = 1 (n > 0 and is the
smallest such integer) then:
(1) g k = 1 if and only if n|k,
(2) g k = g m if and only if k ⌘ m(modn),
(3) hgi = {1, g, g 2 , . . . , g n 1 } and all of these powers up to n are distinct.
If | hgi | is infinite, then:
(4) g k = 1 if and only if k = 0,
(5) g k = g m if and only if k = m,
(6) hgi = {. . . g 2 , g 1 , 1, g, g 2 . . .} and every single power is distinct.
Proof:
(1) If n|k, then k = qn, q 2 Z, thus g k = (g n )q = 1q = 1. Conversely, if
k
g = 1, then k = qn + r, 0  r < n. r = k qn, so g r = g k qn = g k (g n ) q = 1.
But r < n, and we said that n is the smallest such integer, so r = 0, thus
k = qn.
(2) g k = g m if and only if g k m = 1, therefore k m = qn so k ⌘ n mod n.
(3) Clearly {1, g, g 2 , . . . , g n 1 } is at least a subset of hgi. To prove it equals
hgi take some g k , with k = qn + r (0  r < n), then g k = g r (g n )q , = g r 1 = g r ,
so every member of hgi is also a member of {1, g, g 2 , . . . , g n 1 } (as g r is), so as the two are subsets of each other - the sets equal each other.
(4) Clearly g k = 1 if k = 0. If g k = 1 when k 6= 0, then g k = (g k ) 1 = 1,
too. Thus g n = 1 for some n > 0. This implies that | hgi | is finite, and thus
can’t be true as we said that | hgi | is infinite.
(5) g k = g m if and only if g k m = 1, thus k m = 0 by (4), so k = m.
(6) hgi = {g k |k 2 Z} by definition, so all you need to do is show that the
powers are all distinct, but this was proven in (5).
Due to the fact that powers of an element in a group will always form a
cyclic group, we can define the order of an element g, |g |, as the order (number
of elements) of the cyclic group formed by that element, | hgi |, or by the lowest
power n such that g n = 1. I’ll now move on to prove several more theorems
about these groups:
Theorem 2: Every cyclic group is abelian, i.e. it’s operation is commutative.
Proof: Let G = hgi be a cyclic group generated by g. If x, y 2 G, let
x = g k and y = g m , with k, m 2 Z. The exponent laws show that xy = g k g m =
g k+m = g m+k = g m g k = yx, so G is an abelian group.
Theorem 3: Every subgroup of a cyclic group is cyclic.
10
17 of 24
Proof: Suppose that G = hgi is cyclic and let H be a subgroup of G. If H
= {1}, i.e. it only contains the identity element, then H = h1i and so is cyclic.
Otherwise let g k 2 H, k 6= 0. As H is a subgroup (g k ) 1 = g k 2 H, so we
can assume k > 0 as if it is not we can just use it’s inverse instead to the same
e↵ect. Hence let m be the smallest positive integer such that g m 2 H. Then
hg m i ✓ H, at least. If for some g k 2 H, we have k = qm + r and r = 0, then
hg m i = H for every k is a multiple of m. We can prove this condition is true
for all subgroups of a cyclic group as g r = (g m ) q g k 2 H, which cannot be true
unless r = 0 as r < m and we said that m was the smallest positive integer for
which g m 2 H.
Theorem 4: Let G = hgi be a cyclic group, where |g | = n. Then G could
also be generated by other elements g k if and only if gcd(k, n) = 1.
Proof: If G = hg k i, then g 2 hg k i, say g = (g k )m , where m 2 Z. Thus
g = g km , so n divides 1 km. Thus 1 km = qn (q 2 Z), and so 1 = km + qn,
implying that gcd(k, n) = 1
Conversely, gcd(k, n) = 1 implies that 1 = xk + yn for some integers x and
y, hence g = g 1 = g xk+yn = g xk 1y = (g k )x 2 h g k i, so G = hg k i, too.
1
Now onto the last and most important theorem for the moment, The Fundamental
Theorem of Finite Cyclic Groups:
Theorem 5: Let G = hgi be a cyclic group of order n, then:
(1) If H is any subgroup of G, then H = hg d i for some d|n.
(2) If H is any subgroup of G with |H| = k, then k|n.
(3) Following (2), if k|n, then hg n/k i is the subgroup of G of order k.
Proof:
(1) If |H| = 1, then this is true as then H = {1}, {1} = hg n i and n|n. If the
order is not 1, then let H = hg m i for some m > 0. Let d = gcd(m,n), then as
d|n if we show that H = hg d i then it is proven. Well d|m, too, say m = qd, so
g m = (g d )q 2 hg d i, hence H ✓ hg d i. However d = xm + yn (again, by Euclid’s
Algorithm), with x, y 2 Z, so g d = (g m )x (g n )y = (g m )x 1y = (g m )x 2 hg m i = H.
Thus hg d i ✓ H, and therefore hg d i = H.
(2) By (1) let H = hg d i, where d|n, then nd = |H| = k. Thus k|n.
(3) Let K be a subgroup of G of order k. (1) shows that K = hg m i, with
n
m|n, thus k = |K| = |g m | = m
. Algebraic manipulation shows that nk = m, so
n/k
K = hg i.
Part (2) of theorem 5 is actually true for every single group, even those that
are not cyclic. This is a result known as Lagrange’s theorem, which you will see
after Cayley’s Theorem.
11
18 of 24
That concludes the first, and almost certainly longest, chapter of this text
on abstract algebra. Below are some questions for you to try out, involving
logical proof. Mathematics is not a spectator’s subject, you get better at maths
by doing it!
QUESTIONS
In order of difficulty:
1. Prove Theorem 4 of Section 1.3.
2. Show that |h | = |ghg
1
| for all g, h 2 G.
3. Show that |gh| = |hg| for all g, h 2 G.
4. Show that Zp is a group under multiplication for all prime p, and that
Zn is not a group under multiplication for any non-prime n. 0 is taken to be
absent from these sets.
5. If G = {1, g1 , g2 , g2 . . . gr } is an abelian group, show that the product
1g1 g2 g3 . . . gr is equal to the product of the elements of order 2 (the self inverse
ones).
6. Prove Wilson’s Theorem: If p is a prime number then (p
1(modp). [Hint: Consider the group Z⇤p from Q4, as well as Q5.]
1)! ⌘
To be clear, G refers to any group, thus you simply need to think using the
axioms, the theorems shown and whatever algebraic manipulation is possible.
Also, for those who haven’t encountered it yet, n! means n factorial, or 1 ⇥ 2 ⇥
3 ⇥ ... ⇥ (n 1) ⇥ n.
If you need help with what the questions are asking, or have an answer you
want checked but don’t want to visit the Maths Group to give it to me, then
just email me on Godalming online.
12
19 of 24
‘Can Acceleration be Instantaneous?’
By Scott Sussex
I’ve never had physical aspects of variable motion turn on up on my doorstep with
roses, so I’m going to tackle the question in a mathematical way. We’ll see if we
can answer the question using Newton’s laws of motion and rigid bodies, first.
To phrase the question formally: can a point mass moving under Newton’s laws of
motion along a piecewise smooth path (meaning the point can’t teleport)
experience an instantaneous change in acceleration?
Now an example of a solution. Take a point mass and place it on a flat surface with
which the mass will experience friction. Additionally, let’s add a spring and attach
it to the mass, as well as a wall to keep the spring fixed. We’ll start with the point
in any position such that the spring is either extended or compressed. Now we will
discuss all of the forces on the point which acts in a direction parallel to the
surface.
Friction is simple. A constant
force will act upon the point
in the direction opposite to
the point’s velocity.
The tension in the spring will
act away from the wall if
the spring is compressed or
towards the wall if the
spring is extended.
Now, as the point moves
away from the wall, and the
string is extended, the
frictional force will act
towards the wall and so will
the tension in the string. Once the velocity of the point reaches zero, the frictional
force will instantaneously change direction towards the wall. Meanwhile, the
tension in the spring remains a continuous function (this is known by Hooke’s law).
Therefore we must have an instantaneous change in acceleration, which occurs
every time the velocity reaches 0 until the system reaches equilibrium.
There are numerous other solutions to this problem, some of which can be
experienced in reality, such as the jerk when a car reaches a halt with the break
fully applied- in fact, this works in a very similar way to our Newtonian example.
20 of 24
Maths: Real or Imagined?
Is mathematics real, or a creation of the human mind?
By Kate Stevenson
‘For the things in this world cannot be made known without knowledge of mathematics’
Roger Bacon
Mathematics surrounds us; its rules control the way we live, the money we use and the
nature that surrounds us. Even the most unpredictable things, such as the stock market
and the weather, have their roots set firmly in mathematical soil – chaos theory, for
example, is the study of extremely sensitive dynamic systems that have seemingly random
and ‘chaotic’ results. But it can still be a mathematical theory.
But for all the problems that maths provides us with, the concept of mathematics
itself is surprisingly elusive – what is maths, really? Is the study of maths the search and
discovery of the mathematical rules that make up the universe? Or are we just fitting our
man-made mathematical concepts to the things that we see (thus explaining why we can
only approximate things like the motion of a pendulum – but we cannot perfectly predict
it)?
*
Mathematical Platonism
One school of thought is based on the longstanding theory of the great philosopher Plato,
known as the Platonic Theory. Mathematical Platonism dictates that maths is the study of
mathematical objects which exist, but are abstract – they exist as an idea, as a concept,
but are not physically real. These objects are common-place in mathematics: numbers,
matrices, functions; geometry, with 2D, 3D and 4D shapes, vertices and lines; groups
within algebra. The list goes on.
Another important quality of these objects is that they are independent to us, in
that they have the same features as they would if we were unaware of them, or if our
rational language, thoughts or practices were different. One of the most convincing
arguments for the existence of such objects is this:
The language of mathematics purports to refer to and quantify over abstract
mathematical objects. And a great number of mathematical theorems are true. But a
sentence cannot be true unless its sub-expressions succeed in doing what they purport to
do. So there exist abstract mathematical objects that these expressions refer to and
quantify over.
Gottlob Frege, Foundations of Arithmetic, 1953
In short, mathematical objects must exist, as many of the mathematical expressions which
are dependent on such objects have been proven to be true. So, in accordance with this,
all parts of the expression must be seen to be true; including the abstract objects.
Therefore this argument sides with the view that mathematics is real - the objects, being
independent of us, are out there in the universe, waiting to be discovered.
*
21 of 24
Logicism
Platonism isn’t without opposition; the logistic theory, or Logicism, argues that maths is
simply an extension of human logic, and so mathematics is nothing but the logic it is
derived from. However, the theory stands more strongly when it is specified that all
mathematical truths are a variation of logic, as oppose to mathematical theorems – truths
are proved using logic, whereas theorems are proved using already accepted statements.
According to this argument, whole branches of maths can be reduced (when a complex
phenomena can be explained by its tiniest physical mechanisms) to logic by use of logical
definitions from the bare bones of maths, in other words the things that we assume.
One example of Logicism is the principle of noncontradiction: that statements that contradict
each other cannot both be true. Aristotle, another
great Greek philosopher and student of Plato,
stated that without the principle of noncontradiction, we could not surely know anything
that we do know. And this indeed works for maths,
as we can deduce mathematical truths with our
own eyes and minds, our own logic. For example,
using Fig 1 we can clearly see and understand that
some S can be P when all S are P; but all S cannot
be P when no S is P, as these statements are
contradictory. We see this using our natural sense
of human logic, and it is with this argument that
Logicism aims to show that maths is a device
rather than something to discover.
This said, the principle of non-contradiction
cannot be proved true or false, as to do so would
require the principle itself, and the laws of logic
which you are trying to prove or disprove.
Nevertheless, it is an interesting argument with
which to approach our question, and certainly
Fig 1: Visual example of the principle of
stands for the perspective that maths is of our
non-contradiction
own creation.
*
There is no way for us to surely define mathematics as a concept, however hard we try. As
it often is with philosophy, but infrequently in maths: there will be no clean answer; no
satisfying proof; no correct answer in the back of the textbook. But maybe that is what
really matters when it all comes down to it – maths produces results, credible and
incredible ones at that. There is so much of it for us to discover, so much to take in. So
whether we are discovering the mathematics of the universe or of our own minds, it plays
an important part in our lives, and we’re only part way down the vast iceberg of maths.
Maybe we’ll figure it out someday. In the meantime ho hum, back to the calculus.
22 of 24
Computer Vision
By Lawrence Burvill
I recently visited the University of York, where I got to hear an intriguing lecture about the
difficulties of computer vision. Here is a good summary of what I’d heard.
How is a Computer supposed to tell that one of these photos contains humans, and one contains
simply dummies? They have the same general body shape, they are wearing similar clothes and
each one is unique. To us it’s easy, but subconsciously we are using so much of our brain, in order to
realise this. So how might we make a computer see as we do? We can program a computer to
recognise the general human shape, two legs, two arms, a complete face, but what happens when
it “sees” a disabled person? How will it interpret this? Or a man who has gone blind in one eye and
has taken to wearing an eye patch? As humans we would not hesitate to identify both of these
people as other humans, but a computer might struggle. Computer Scientists are still working on
this problem, as face recognition software in cameras will attest.
23 of 24
Now if I asked you which of these letters, A or B, is a brighter shade, many of you would struggle to
answer, or have the firm opinion that A is brighter. A computer would not hesitate in pointing out
that they are the same shade, as this second picture will show.
So why can computers not be fooled by optical illusions? It is down to how we see, as humans, we
have top-down vision, which means everything we see is affected by its surroundings, as in the
picture, the shadow of the cylinder affects our view of B, even when we don’t want it to. It means
we can look upon a face in a crowd and see a human, when in fact it’s a dummy, and its
surroundings give it the cover it needs. Computers see bottom-up, they see each pixel in a photo
individually, and so will be unable to put them together to make an image. Research in this area is
still ongoing, and maybe we will soon have computers that see exactly like we do.
24 of 24