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ALGEBRA A: CHAPTER ZERO
THE NATURE OF MATHEMATICS
DR MARK. V. LAWSON
1. What they didn’t tell you in school
The goal of this section is to provide some guidance on the transition from school mathematics to university mathematics. They are
very different, but the failure to understand those differences can cause
problems.
Landau’s koan “Please forget everything you have learned
in school; for you haven’t learned it.”
This quote, from the ‘preface to the student’ of Edmund Landau’s
book Foundations of analysis first published in Germany in 1930, has
scandalized generations of mathematics students. In fact, Landau was
trying to do something that the Zen masters used to do with their new
students: to disconcert them so that he could get them to think in
new ways. This is the key point. To be successful in university mathematics — that is to enjoy it, see why it is as it is, apply it, develop
intellectually, and even pass exams in it, if you are that way inclined
— you have to think in new ways. University Mathematics is not
just School Mathematics with harder sums and fancier notation, it is
different, fundamentally different, from what you did at school; it is
related to it only in the same way that a butterfly is related to the
grub it develops from.
Koan: From wikipedia: a story, dialogue, question, or statement which
is used in Zen-practice to provoke great doubt, and test a student’s
progress in Zen practice.
Mathematics and games
Suppose you wanted to play chess. What would you need to do first?
Of course, first of all you would need to understand the rules of chess
and know your pawns from your rooks. But simply knowing the rules
of a game is not enough. In addition, you would have to practise by
playing many games against many different opponents, in addition, you
would have to study the games of great players to learn how to play
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DR MARK. V. LAWSON
the game tactically. What is true of a game like chess is even more
true of a complex subject like mathematics. To do well in mathematics
you have to know the rules of the game, you have to practise, and you
have to read books written by experts so that you can learn tactics and
strategy from them — the tricks of the mathematical trade, if you like.
There are no shortcuts. There are no quick fixes in mathematics.
Most of the mathematics you learnt at school was
invented before 1800.
Here is a very rough chronology. You can find out more at [6]
2000 BCE: Solving quadratic equations.
400 BCE: The existence of irrational numbers.
300 BCE: Euclidean geometry and basic number theory.
200 BCE: Conics.
1550: Formulae for solving cubic and quartic equations.
1590: Logarithms.
1630: Analytic geometry.
1675: Calculus.
1700: Probability.
1750: Trigonometry.
1795: Complex numbers.
Only matrices (1850) and vectors (1880) were introduced more recently. The antiquity of school mathematics is not entirely surprising.
Mathematics is a cumulative subject and has to be learnt in an organized way, building on what went before. Furthermore, unlike any
other subject, what is true in mathematics does not change in time,
although what the centre of interest is does. However, if you think of
all the developments in physics since 1800 —- black holes, the big bang
theory, parallel universes, quantum — then you might suspect that
there have also been big developments in mathematics. There have,
but you would be forgiven for not knowing about them because they
are not promoted in the media or taught in school.
In much of school mathematics, you learn methods for solving specific problems. Often, you just
learn formulae.
A method for solving a problem that requires little thought in its
application is called an algorithm. Computer programs, such as apps,
are the supreme examples of algorithms. It is certainly true that finding algorithms for solving specific problems is an important part of
mathematics, but it is by no means the only part. What you might
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not have appreciated from school mathematics is the important role
played in mathematics by ideas. An idea is a tool to help you think.
Furthermore, mathematics is about problem solving, and problems do
not come neatly labelled with the methods needed for their solution.
Frequently, you have to invent new methods or at least adapt old ones
to apply to unforseen situations.
Mathematics at school is often taught without
reasons being given for why the methods work.
This is the fundamental difference between school mathematics and
university mathematics. A reason why something is true is called a
proof and proofs are the essence of mathematics. It is important to
understand two key facts about mathematics. First, it is not an experimental science. Second, it is not based on opinion. It is based solely
on the concept of a proof. I shall say more about proofs later.
School mathematics is based on the narrow syllabuses proscribed by the examining boards; it
gives little idea of the scope or nature of modern
mathematics.
Let me give some examples of what I mean. The Mathematics
Subject Classification divides mathematics into nearly 100 areas with
dozens of further subdivisions within. This covers nearly 50 pages of
small print. Mathematics is therefore a vast subject.
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To make it more manageable, however, it could be summarized, very
roughly, into the following seven broad areas:
(1) Algebra.
(2) Analysis.
(3) Combinatorics.
(4) Geometry.
(5) Logic.
(6) Probability and statistics.
(7) Mathematical physics.
In this course, I shall discuss Algebra, Geometry and Combinatorics
and we shall use elements of Logic throughout.
What are algebra, geometry and combinatorics?
Let’s start with algebra since most of this course is algebra. Algebra
started as the study of equations. The simplest kinds of equations are
ones like
3x − 3 = 0
where there is only one unknown x and that unknown occurs to the
power 1. This means we have x alone and not, say, x1000 . It is easy to
solve this specific equation. Add 3 to both sides to get
3x = 3
and then divide both sides by 3 to get
x = 1.
This is the solution to my original equation and, to make sure, we check
our answer by calculating
3·1−3
and observing that we really do get 0 as required.
To carry out these calculations, I had to know what rules the numbers
and symbols obeyed. You probably applied them without thinking. This
is a point I shall return to later.
The method used for the specific example above can be applied to
any equation of the form
ax + b = 0
as long as a 6= 0. Here a, b are specific numbers, probably real numbers,
and x is the real number I am trying to find. This equation is an
example of a linear equation in one unknown. If x occurs to the power
2 then we get
ax2 + bx + c = 0
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DR MARK. V. LAWSON
where a 6= 0. This is an example of a quadratic equation in one unknown. You have learnt a formula to solve such equations and, I hope,
it was explained to you why it always works. If not, don’t worry, because I will later on. There is no reason to stop at 2. If x occurs to the
power 3 we get a cubic equation in one unknown
ax3 + bx2 + cx + d = 0
where a 6= 0. Solving such equations is much harder than solving
quadratics but there is a complicated formula for the roots. But there
is no reason to stop at cubics. We could look at equations of degree
4 or degree 5 or degree a million. You might expect there to be formulae for the roots of such equations. There aren’t. For equations of
degree 5 and more, there are no formulae which enable you to solve the
equations. I don’t mean that no formulae have yet been discovered, I
mean that someone has proved that such a formula is impossible. This
was proved by the young French mathematician Evariste Galois (1811–
1832). Galois’s work meant the end of the view that algebra was about
finding formulae to solve equations. We shall not study Galois’s work
in this course but it has had a huge impact on algebra. It is one of
the reasons why the algebra you study later in your university careers
will look so very different from the algebra you studied at school. In
this course, we shall still study how to solve equations but from a more
practical point of view.
I have talked about solving equations where there is one unknown.
There is no reason to stop there. We can also study equations where
there are any finite number of unknowns and those unknowns occur to
any powers. The best place to start is where we have any number of
unknowns but each unknown can only occur to the first power and no
products of unknowns are allowed. This means we are studying linear
equations like
x + 2y + 3z = 4.
Our goal is to find all the values of x, y and z that satisfy this equation.
Thus the solutions are ordered triples (x, y, z). For example, both
(0, 2, 0) and (2, 1, 0) are solutions whereas (1, 1, 1) is not a solution. It
is unusual to have just one linear equation to solve. Usually we have
two or more such as
x + 2y + 3z = 4 and x + y + z = 0.
Our goal then is to find all the triples (x, y, z) that satisfy both equations simultaneously. In fact, as you should check, all the triples
(λ − 4, 4 − 2λ, λ)
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where λ is any number satisfy both equations. For this reason, we often speak about simultaneous linear equations. It turns out that solving systems of linear equations never becomes difficult however many
unknowns there are. The modern way of studying systems of linear
equations uses matrix theory.
That leaves studying equations where there are at least 2 unknowns
and where there are no constraints on the powers of the unknowns and
the extent to which they may be multiplied together. This is much
more complicated. If you only allow squares such as x2 or products
of at most two unknowns, such as xy, then there are relatively simple
methods for solving them. But, even here, strange things happen. For
example, the solutions to
x2 + y 2 = 1
can be written (x, y) = (sin θ, cos θ). If you allow cubes or products
of more than two unknowns then you enter the world of subjects like
algebraic geometry and connect with current research.
The subject of geometry is understood to be about the properties of
shapes in the plane like triangles and circles. In this course, we shall
be interested in geometry in space. The problem is how we should
get a handle on studying that geometry. In this course, we shall use
vectors and their algebra. This will enable us to convert questions
about geometry into algebraic questions. But the algebra will be very
different from the algebra of numbers that lies behind solving equations.
The final topic combinatorics may be unfamiliar to you. It is about
counting arrangements and although my use of the word counting may
make it sound very easy, it is in fact very difficult in general. Counting
lies behind probability theory, for example. In addition, we don’t just
want to count finite sets, we also want to be able to count infinite
sets. The discovery that there are infinite numbers of different sizes
was made towards the end of the nineteenth century.
Mathematics is useful.
In everyday life, mathematics is almost invisible but in fact it lies
behind all of the technology that makes modern life possible. Here
is one example, where you might not have thought mathematics was
used. Conventional X-ray images provide just a shadow photograph of
your innards but in tomography cross-sectional images are obtained by
moving the X-ray beam around the patient. But the X-ray machine
produces only the data of X-ray intensities. To convert that data into
an actual image requires mathematics. This mathematics is called the
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DR MARK. V. LAWSON
radon transform and was invented by the Austrian mathematician Johann Radon in 1917.
Mathematics is surprising.
A solid the size of a pea may be cut into a finite number of pieces
and then stuck back together again in such a way as to form a solid
the size of the sun. This is known as the Banach-Tarski Paradox (1924).
Mathematics is difficult.
The Millennium Problems is a list of seven outstanding problems
posed by the Clay Institute in the year 2000. A correct solution to
any one of them carries a one million dollar prize. To date, only one
has been solved, the Poincaré conjecture, by Grigori Perelman in 2010,
who declined to take the prize money.
2. What are proofs?
I explained above that the most fundamental difference between
school mathematics and university mathematics lies in proofs. In this
section, I shall begin to explain some of the workings of proofs. However, it is important to understand that you are not going to be able to
do proofs after a few minutes. It takes years to get to grip with them.
My goal is simply to get you started.
2.1. Three fundamental assumptions. If you are going to understand how mathematical proofs work, there are three simple, but fundamental, assumptions you have to understand about the way mathematics works.
I. Mathematics only deals in statements that are
capable of being either true or false.
Mathematics does not deal in statements which are ‘sometimes true’
or ‘mostly false’. There are no approximations to the truth in mathematics and no grey areas. Either a statement is true or a statement is
false, though we might not know which.
In everyday life, we say things which contain a grain of truth. For
example, if I say ‘It always rains in North Wales’ then as it stands
this sentence is false. For this to be true, the rain would have to rain
everyday. It doesn’t: even in North Wales the sun shines sometimes.
On the other hand, if someone is visiting North Wales then telling them
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that it rains everyday warns them to pack a raincoat and not to expect
unbroken balmy weather.
Mathematics also doesn’t deal in statements that are neither true
nor false like ‘Out damned spot!’, which is the exclamation Lady Macbeth made when she was washing her hands, or questions like ‘To be
or not to be?’ reflecting Hamlet’s existential doubts.
II. If a statement is true then its negation is false,
and if a statement is false then its negation is
true.
In natural languages, negating a sentence is achieved in different
ways. In English, the negation of ‘It is raining’ is ‘It is not raining’. In
French, the negation of ‘Il pleut’ is obtained by wrapping the verb in ‘ne
. . . pas’ to get ‘It ne pleut pas’. To avoid grammatical idiosyncracies,
we can use the formal phrase ‘it is not the case that’ and plonk it in
front of any sentence to negate it. So, ‘It is not the case that it is
raining’.
In some languages, and French is one of them, adding negatives is
used for emphasis. This used to be the case in older forms of English
and is often the case in informal English. In formal English, we are
taught that two negatives make a positive which is actually the rule
taken from mathematics above where it is true. In fact, negating negatives in natural languages is more complex than this. For example,
‘not unhappy’ isn’t quite the same as ‘happy’.
III. Mathematics is free of contradictions.
A contradiction is where both a statement and its negation are true.
This is impossible by (II) above.
The nonextistence of contradictions in mathematics is the basis for
a method of proving results that can seem quite disconcerting when
first encountered. Suppose I want to show that a statement S is true.
Then I can try to do this by showing that the statement ‘It is not true
that S’ is false.
Let me put this in more concrete terms. Suppose you are confronted
with two politicians whom we shall call Alice and Bob. One of these
politicians always lies and the other always tells the truth. Suppose
you ask Bob the question: is it true that 2 + 2 = 5? If he replies
‘yes’ then you know Bob is lying. Without further ado, you know that
Alice is that paragon of politicians and always tells the truth. We can
therefore deduce something about Alice without Alice being anywhere
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in the vicinity.
It’s possible that some of you may never have seen a proof before.
So, I want to begin with five examples to show you what they look like
and I shall also discuss some of the issues that arise in dealing with
proofs.
2.2. Proof 1: The square of an even number is even, and the
square of an odd number is odd. The terms odd and even are
only used of whole numbers such as 0, 1, 2, 3, 4, . . .. These numbers are
called the natural numbers and they are the first kinds of numbers we
learn about as children.
By definition, a natural number n is even if it is exactly divisible
by 2. This means that n = 2m for some natural number m. So, the
even numbers are 0, 2, 4, 6, . . .. Notice that 0 is an even number because
0 = 2 × 0. In other words, 0 is exactly divisible by 2. Remember, you
cannot divide by 0 but you can certainly divide into 0.
You might have been told that a number is even if its last digit is one
of the digits 0, 2, 4, 6, 8. In fact, this is a consequence of our definition
rather than a definition itself. I shall ask you to prove this result in
the exercises.
A number is odd if it is not even. This is not a very useful definition
so we shall describe a better one. If you attempt to divide a number
by 2 then there are two possibilities: either it goes exactly, in which
case the number is even, or it goes so many times plus a remainder of
1, in which case the number is odd. It follows that a better way of
defining an odd number n is one that can be written n = 2m + 1 for
some natural number m.
The next point is that we are making a claim about all even numbers.
If you pick a few even numbers at random and square them then you
will find in every case that the result is even. But this does not prove
our claim. Even if you checked a trillion even numbers and squared
them it wouldn’t prove the claim. This is because the claim is about all
even numbers. This means that, in effect, we have to prove an infinite
number of statements: 02 is even, and 22 is even, and 42 is even . . . I
cannot therefore prove my claim by picking a specific even number, like
12, and checking that its square is even. This simply verifies one of the
infinitely many statements above. As a result, the starting point for
my proof cannot be a specific even number. It has to be a general even
number. We now prove our claims.
First, we prove that the square of an even number is even.
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(1) Let n be an even number. This is our first move. The word let
simply means that an even number is on the table. Notice that
it is not a specific even number. We want to prove something
for all even numbers so we cannot argue with a specific one.
(2) By definition n = 2m for some natural number m. We are
using the mathematical term even number so we had better
write down what it means.
(3) Square both sides of the equation in (2) to get n2 = 4m2 .
(4) Now rewrite this equation as n2 = 2(2m2 ).
(5) Since 2m2 is a natural number, it follows that n2 is even using
our definition of an even number.
This proves our claim.
Second, we prove that the square of an odd number is odd.
(1)
(2)
(3)
(4)
(5)
Let n be an odd number.
By definition n = 2m + 1 for some natural number m.
Square both sides of the equation in (2) to get n2 = 4m2 +4m+1.
Now rewrite the equation in (3) as n2 = 2(2m2 + 2m) + 1.
Since 2m2 + 2m is a natural number, it follows that n2 is odd
using our definition of an odd number.
2.3. Proof 2: If the square of a number is even then that
number is even, and if the square of a number is odd then
that number is odd. At first reading, you might think that I am
simply repeating what I proved above but it is different. In Proof 1, I
proved
‘if n is even then n2 is even’.
Here, I want to prove
‘if n2 is even then n is even’.
Our assumptions in each case are different and our conclusions in each
case are different. We prove the first claim.
(1) Suppose that n2 is even.
(2) Now it is very tempting to try and use the definition of even
here and write n2 = 2m for some natural number m. But this
turns out to be a deadend. Just like playing a game such as
chess not every possible move is a good one.
(3) So we make a different move. We know that n is either odd or
even. Our goal is to prove that it must be even.
(4) Could n be odd? The answer is no, because if n is odd then, as
we showed above, n2 is odd.
(5) Therefore n is not odd.
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(6) But a number that is not odd must be even. It follows that n
is even.
We use a similar strategy to prove the second claim.
The proofs here were more subtle than in our first example. They
work as follows: there are two possibilities exactly one of which is true.
We rule out one of those possibilities and so deduce that the other
possibility must be true.
2.4. Proof 3: The sum of the angles in a triangle add up to
180◦ . This is a famous result that everyone knows. You might have
learnt about it at school by drawing lots of triangles and measuring
their angles. The proof we give is very old and occurs in Euclid’s
book the Elements written about 300 BCE which I will say a little
more about later. To follow this proof, you will need to refer to the
diagrams below
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Draw a triangle and call its three angles α, β and γ respectively. Our
goal is to prove that
α + β + γ = 180◦ .
In fact, we shall show that the three angles add up to a straight line
which is the same thing. With the triangle as drawn, draw a line
through the point P parallel to the base of the triangle. Then extend
the two sides of the triangle that meet at the point P as shown. As
a result, we get three angles that I have called α0 , β 0 and γ 0 . I claim
first that β 0 = β because the angles are opposite each other in a pair of
intersecting straight line; I claim that α0 = α because these two angles
are formed from a straight line cutting two parallel lines; I claim that
γ 0 = γ for the same reason. But since α0 and β 0 and γ 0 add up to give
a straight line, we have proved the claim.
Of course, we have proved our result on the basis of three other
results:
(1) That given a line l and a point P not on that line I may draw
a line through the point P and parallel to l.
(2) If two line intersect, then opposite angles are equal.
(3) If a line l cuts two parallel lines l1 and l2 the angle l makes with
l1 is the same as the angle it makes with l2 .
How do we know they are true? One of them, number (2), can be
readily proved. The other two are more subtle and lead to an important
insight that can be traced back to the Ancient Greeks over two and a
half thousand years ago.
I said that everything must be proved but in terms of what? In
terms of results that are known to be true. And how do we know they
are true? By proving them. The problem is that there is a danger here
of infinite regress: to prove A, I need to assume B, and to prove B I
need to assume C and so on. In addition, I have to make sure that I
don’t argue in circles and, at some point, assume what it is I am trying
to prove. At some point, therefore, we have to stop and the unproved
assumptions we are left with are called axioms.
An axiom is a basic assumption which underpins a branch of mathematics. You should think of axioms as being like the rules of the game.
If you want to play the game of Euclidean geometry you have to play
by Euclid’s rules, the axioms of geometry that he laid out in his book.
Euclid uses five axioms for geometry, one of which is equivalent to (1)
above. It turns out that (3) can be proved from the axioms. I have
listed the axioms below just for information. They are taken from [3].
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All of the results you learnt in school about triangles and circles can
be proved from these axioms1.
1Actually,
Euclid made a few mistakes but they were all sorted out by David
Hilbert in his book The foundations of geometry (1899).
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But how do we know which axioms are the right ones? It is important to remember that mathematics is not physics. I could state it in
the following grandiloquent way: mathematics is about logically
consistent mathematical universes. Some of these mathematical
universes turn out to be so close to the real one that we can use our
mathematics to make predications about it. It turns out that the Euclidean mathematlcal universe is an excellent tool for calculating about
much of our real universe.
2.5. Proof 4: Pythagoras’ theorem. We refer to the diagram below.
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We want to prove that
a2 + b2 = c2 .
Look first at the shape on the left. It has been constructed from four
copies of our triangle and two squares of areas a2 and b2 , respectively.
I claim that this shape is actually a square. First, the sides all have the
same length a + b. Second, the angles at the corners are right angles by
Proof 3. Now look at the shape on the right. This is also a square with
sides a + b so it has the same area as the first square. Using Proof 3,
the shape in the middle really is a square with area c2 . It we subtract
the four copies of the original triangle from both squares, the shapes
that remain must have the same areas, and we have proved the claim.
√
2.6. Proof 5: 2 cannot be written as an exact fraction. If you
square each of the fractions in turn
3 7 17 41
, , , ,...
2 5 12 29
you will find that you get closer and closer to 2 and so each of these
numbers is an approximation to the square root of 2. This raises the
question: is it possible to find a fraction xy whose square is exactly 2?
In fact, it isn’t but that isn’t proved just because my attempts above
failed. Maybe, I just haven’t looked √
hard enough. So, I have to prove
that it is impossible. To prove that 2 is not an exact fraction, I am
actually going to begin by trying to show you that it is.
√
(1) Suppose that 2 = xy where x and y are positive whole numbers
and clearly y 6= 0.
(2) We may assume that xy is a fraction in its lowest terms so that
the only natural number that divides both x and y is 1. Keep
your eye on this assumption because it will come back to sting
us later.
2
(3) Square both sides of the equation in (2) to get 2 = xy2 .
(4) Multiply both sides of the equation in (3) by y 2 . This is allowed
since y 2 6= 0.
(5) We therefore get the equation 2y 2 = x2 .
(6) Since 2 divides the lefthand-side of this equation, it must divide
the righthand-side. This means that x2 is even.
(7) We now use Proof 2 to deduce that x is even.
(8) We may therefore write x = 2u for some natural number u.
(9) Substitute this value for x we have found in (5) to get 2y 2 = 4u2 .
(10) Divide both sides of the equation in (9) by 2 to get y 2 = 2u2 .
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(11) Since the righthand-side of the equation in (10) is even so is the
lefthandside. Thus y 2 is even.
(12) Since y 2 is even, it follows by Proof 2, that y is even.
(13) If (1) is true then we are led to the following two conclusions.
From (2), we have that the only natural number to divide both
x and y is 1. From (7) and (12), 2 divides both x and
√ y. This
is a contradiction. Thus (1) cannot be true. Hence 2 cannot
be written as an exact fraction.
2.7. Key points.
• One of the goals of this course is to introduce you to proofs.
This does not mean that you will afterwards be able to do
proofs. That takes time and practice.
• Your main goal should be to try to understand the proofs that
I give. To understand a proof means seeing why it is true and
being able to explain it yourself to someone else. It is much
easier to check that a proof is correct then it is to invent the
proof in the first place. However, be warned, it can also take a
long time to understand a proof.
• I shall ask you to find proofs for yourself but usually by adapting
a proof you have already read. But do not expect to be able
to find all the proofs I ask for or expect to find them in a few
minutes. Constructing proofs takes time, trial and error and,
yes, luck.
• If you don’t understand the words used in a statement then you
are not going to be able to prove that statement. Definitions
are vitally important in mathematics.
• Every statement that you make in a proof must be justified. If
it is a definition, say that it is a definition. If it is a result known
to be true, say that it is known to be true. If it is one of the
assumptions, say that it is one of the assumptions. If it is an
axiom, say that it is an axiom. In constructing a proof it might
be helpful to think in terms of two protagonists Alice and Bob.
Alice claims that a statement is true and Bob challenges her to
prove it. Each time Alice makes a statement, Bob challenges
her to prove it. It is probably best to write each statement of
a proof on a separate line followed by the justification for that
statement.
2.8. Terminology involving proofs. Every subject generates its own
jargon. Here are some words that you will find when reading proofs.
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Don’t worry about this section now, simply use it as a reference as you
are working through the course.
In mathematics our goal is to distinguish between what is true and
what is false.
Sentences which are capable of being either true or false are said to
be statements.
A proof is a way of showing that a statement is true. The conclusion
of a proof is marked using the symbol 2. This replaces the older use
of QED.
A counterexample is an example that tells us that something we
thought might be true, isn’t. For example, suppose I make claim that
every prime number is odd. Then a counterexample to this claim is
the prime 2. To show that a statement is false we need only find one
counterexample.
If we believe something might be true but there isn’t yet a proof we
say that it is a conjecture.
The things we can prove fall, roughly, into the following categories:
a theorem is a major result, worthy of note; a proposition is a result,
and a lemma is an auxiliary result, a tool, useful in many different
places; a corollary is a result we can deduce with little or no effort from
a proposition or theorem.
An algorithm is a recipe for doing a calculation: programs are algorithms.
A check is a way of detecting gross errors in calculations.
Let A and B be two statements.
We say that A implies B or if A then B if I can prove B using A
as an assumption. You will sometimes see this abbreviated to A ⇒ B.
We saw an example of this above if n is odd then n2 is even. It is
certainly not true that n2 is always even but it is if we assume that n
is even.
The statement B implies A is called the converse of A implies B. It
says something quite different and whether the statement A implies B
is true or false is independent of whether the statement B implies A is
true or false.
The statement A if and only if B or A iff B or A ⇔ B is in fact two
statements in one. It means (1) A implies B and (2) B implies A.
So, to prove the statement A if and only if B we have to prove
TWO statements: we have to prove A implies B and we have to prove
B implies A.
The use of the word iff is peculiar to mathematical English.
24
DR MARK. V. LAWSON
2.9. The origin of proofs. Mathematics as a subject originated in
Mesopotamia about 5,000 years ago. There is evidence from clay
tablets of quite sophisticated mathematics from that time. But the
idea of proving things seems to have been invented only once in human
history and that was in Ancient Greece sometime around 500 BCE.
The best documentary evidence we have is from a collection of thirteen books known as the Elements written by Euclid about 300 BCE.
The Elements is mainly an account of plane gemetry starting with the
axioms and progressing step-by-step to the proof of more sophisticated
results. Book I, for example, leads up to a proof of Pythagoras’ theorem. However, in addition to geometry, there are also important results
about prime numbers. Some of Euclid’s theorem will be discussed in
Chapter 1 of this course.
3. High-school algebra revisted
In this section, I am going to review some of the important results
from algebra that you learnt at school but I shall present them in the
way that they are needed at university. This is partly revision and
partly to emphasize that mathematics is based on rules or axioms.
3.1. The axioms. Algebra deals with the manipulation of symbols.
In high-school, the algebra you studied was based on the properties
of the real numbers, even if no one told you so. This means that
when you write x you mean an unknown or yet-to-be-determined real
number. I shall now describe the rules, or axioms, that you use for doing
algebra with real numbers. As usual, I shall abbreviate the operation
of multiplication x × y by writing simply xy.
(F1): Addition is associative. Let x, y and z be any real numbers.
Then (x + y) + z = x + (y + z). The word associative means
that how you bracket the numbers does not effect the result.
The brackets in the sum (x + y) + z tell you that you have to
calculate first x + y and then add z to the result.
(F2): There is an additive identity. The number 0 (zero) is the
additive identity. This means that for an real number x we have
that x + 0 = x = 0 + x. Thus adding zero to a number leaves
it unchanged.
(F3): Each element has a unique additive inverse. This means
that for each number x there is another number, written −x,
with the property that x + (−x) = 0 = (−x) + x. The number
−x is called the additive inverse of the number x.
ALGEBRA A: CHAPTER ZERO
THE NATURE OF MATHEMATICS
25
(F4): Addition is commutative. Let x and y be any real numbers.
Then x + y = y + x. The word commutative means that the
order in which you add the numbers does not matter.
(F4): Multiplication is associative. Let x, y and z be any real
numbers. Then (xy)z = x(yz). Once again associativity means
that how we bracket a calculation does not affect the result.
The difference is that this time the operation is multiplication
rather than addition.
(F5): There is a multiplicative identity. The number 1 is the multiplicative identity. This means that for any real number x we
have that 1x = x = x1.
(F6): 0 6= 1.
(F7): The additive identity is a multiplicative zero. This means
that 0x = 0 = x0. If you multiply any real number by 0 then
you get 0.
(F8): Multiplication is commutative. Let x and y be any real
numbers. Then xy = yx. Once again the word commutative
means that the order in which you carry out the operations
doesn’t matter. In this case, the operation is multiplication.
(F9): Multiplication distributes over addition on the left and the
right. This is the first axiom that connects the two operations of
addition and multiplication. There are actually two distributive
laws: the left distributive law
x(y + z) = xy + xz
and the right distributive law
(y + z)x = yx + zx.
(F10): Each non-zero number has a unique multiplicative inverse.
Let x 6= 0. Then there is a unique real number written x−1
with the property that x−1 x = 1 = xx−1 . The number x−1
is called the multiplicative inverse of x. It is, of course, the
number x1 . It is very important to observe that zero does not
have a multiplicative inverse.
I won’t use this terminology in this course, but for the record any
collection of numbers that satisfies these ten axioms is called a field.
On a point of notation, we define a − b to mean a + (−b).
I also assume the usual properties of = such as
a = b ⇒ a + c = b + c and a = b ⇒ ac = bc.
Example 3.1. When I talked about algebra above, I mentioned that
the usual way of solving a linear equation in one unknown depended
26
DR MARK. V. LAWSON
on the properties of real numbers. Let me now show you how we use
the above axioms to solve ax + b = 0 where a 6= 0.
ax + b
(ax + b) + (−b)
ax + (b + (−b))
ax + 0
ax
ax
−1
a (ax)
(a−1 a)x
1x
x
=
=
=
=
=
=
=
=
=
=
0
0 + (−b) by (F3)
0 + (−b) by (F1)
0 + (−b) by (F3)
0 + (−b) by (F2)
−b by (F2)
a−1 (−b) by (F10)
a−1 (−b) by (F5)
a−1 (−b) by (F10)
a−1 (−b) by (F5)
I don’t propose that you go into quite such gory detail when solving
equations, but I wanted to show you what actually lay behind the rules
that you might have been taught at school.
Example 3.2. We can now prove that −1×−1 = 1 something which is
hard to understand in any other way. By definition, −1 is the additive
inverse of 1. This means that 1+(−1) = 0. Let us calculate (−1)(−1)−
1. We have that
(−1)(−1) − 1 =
=
=
=
=
(−1)(−1) + (−1) by definition of subtraction
(−1)(−1) + (−1)1 since 1 is the multiplicative identity
(−1)[(−1) + 1] by the left distributivity law
(−1)0 by properties of additive inverses
0 by properties of zero
Hence (−1)(−1) = 1. In other words, the result follows from the usual
rules of algebra.
Example 3.3. Why is division by zero not defined in the rationals or
reals? Let’s pretend that zero is an ordinary number and manipulate
it like an ordinary number. By definition ab = c if and only if a = bc.
Thus 0 = 0c for all numbers c. Thus 0 = 01 and 0 = 02. If it were
legitimate to divide by zero we would then deduce that 1 = 2.
The associative, commutative and distributive laws can also be extended. I will not prove the following results. Instead, you can treat
them as axioms:
ALGEBRA A: CHAPTER ZERO
THE NATURE OF MATHEMATICS
27
• The associative law for addition applies to any finite sum of
numbers. So, how you bracket such a sum will not affect the
answer you get.
• The associative law for multiplication applies to any finite product of numbers. So, how you bracket such a product will not
affect the answer you get.
• The order in which you add up a finite number of numbers will
not affect the answer.
• The order in which you multiply a finite number of numbers
will not affect the answer.
• The above results mean that a straight sum of numbers or a
straight product of numbers do not have to be bracketed.
• The left distributive law extends as follows
a(b1 + b2 + b3 + . . . + bn ) = ab1 + ab2 + ab3 + . . . + abn
and the right distributive law likewise.
The above results mean that the following make sense. We abbreviate 2x = x + x and nx = x + . . . + x where the x occurs n times. We
have 1x = x and 0x = 0. We abbreviate x2 = xx and more generally
xn = x . . . x where the x occurs n times. We define x1 = x. We define
x0 = 1 as long as x 6= 0.
An extreme case! What about 00 ? This is a can of worms. For this
course, it is probably best to define 00 = 1.
3.2. Sigma notation. Let a1 , a2 , . . . , an be n numbers. Their sum is
a1 + a2 + . . . + an which can be written more succinctly as
n
X
ai .
i=1
P
Where
is Greek ‘S’ and stands for Sum. This notation is very useful
and can be manipulated using the rules above. If 1 < s < n then we
can write
n
s
n
X
X
X
ai =
ai +
ai .
i=1
s+1
i=1
If b is any number then
b
n
X
i=1
!
ai
=
n
X
bai .
i=1
Although I have started the sum at 1, I could, in other circumstances,
have started at 0, or any other appropriate number.
28
DR MARK. V. LAWSON
P
The most complicated use of
-notation arises when we have to
sum up what is called an array of numbers aij where 1 ≤ i ≤ m and
1 ≤ j ≤ n. I shall give the example where m = 3 and n = 4. We can
therefore think of the numbers aij as being arranged in a 3 × 4 array
as follows:
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
Observe that the first subscript tells you the row and the second subscript tells you the column. Thus a23 is the number in the second row
and the third column. Now we can add these numbers up in two different ways getting the same answer in both cases. The first way is
to add the numbers up along the rows. So, we calculate the following
sums
n
n
n
X
X
X
a1j ,
a2j ,
a3j .
j=4
j=4
j=4
We then add up these three numbers
4
X
a1j +
j=1
4
X
a2j +
j=1
4
X
a3j =
j=1
3
4
X
X
i=1
!
aij
.
j=1
The second way is to add the numbers up along the columns. So, we
calculate the following sums
3
X
ai1 ,
3
X
i=1
ai2 ,
i=1
3
X
ai3 ,
3
X
ai4 .
i=1
i=1
n
X
4
3
X
X
We then add up these four numbers
n
X
i=1
ai1 +
n
X
ai2 +
i=1
n
X
ai3 +
i=1
ai4 =
i=1
j=1
!
aij
.
i=1
We therefore have in general that
!
!
m
n
n
m
X
X
X
X
aij =
aij .
i=1
j=1
j=1
i=1
What I have defined so far are finite sums and form part of algebra.
There are also infinite sums
∞
X
ai
i=1
which form part of analysis, the subject that provides the foundations
for calculus. There is one place where we use infinite sums in everyday
ALGEBRA A: CHAPTER ZERO
THE NATURE OF MATHEMATICS
29
life, and that is in the decimal representations of numbers. Thus the
fraction 13 can be written as 0 · 3333 . . . and this is in fact an infinite
sum: it means the infinite sum
∞
X
3
.
10i
i=1
Warning! ∞ is not a number. It simply tells us to keep adding on
terms for increasing values of i without end so we never write
3
.
10∞
4. Exercises zero
(1) Raymond Smullyan is both a mathematician and a magician.
Here are two of his puzzles. On an island there are two kinds
of people: knights who always tell the truth and knaves who
always lie. They are indistinguishable.
(i): You meet three such inhabitants A, B and C. You ask A
whether he is a knight or knave. He replies so softly that
you cannot make out what he said. You ask B what A
said and they say ‘he said he is a knave’. At which point
C interjects and says ‘that’s a lie!’. Was C a knight or a
knave?
(ii): You encounter three inhabitants: A, B and C.
A says ‘exactly one of us is a knave’.
B says ‘exactly two of us are knaves’.
C says: ‘all of us are knaves’.
What type is each?
(2) There are five houses, from left to right, each of which is painted
a different colour, their inhabitants are from different parts of
the British Isles, own different pets, drink different drinks and
drive different cars.
(a) There are five houses.
(b) The Welshman lives in the red house.
(c) The Cornishman owns the dog.
(d) Coffee is drunk in the green house.
(e) The Orkadian drinks tea.
(f) The green house is immediately to the right (that is: your
right) of the ivory house.
(g) The Oldsmobile driver owns snails.
(h) The Ka owner lives in the yellow house.
(i) Milk is drunk in the middle house.
30
DR MARK. V. LAWSON
(j) The Shetlander lives in the first house.
(k) The man who drives the Chevy lives in the house next to
the man with the fox.
(l) The Ka owner lives in a house next to the house where the
horse is kept.
(m) The Lotus owner drinks orange juice.
(n) The Manxman drives the Porsche.
(o) The Shetlander lives next to the blue house.
There are two questions: who drinks water and who owns the
aardvark? q
√
(3) Show that 12 and √12 and 22 all have the same value.
(4) Criticize sinx
= xy .
siny
(5) Criticize (a + b)2 = a2 + b2 .
(6) Suppose that a and b are natural numbers and that a exactly
divides b. Is it true that a + 1 exactly divides b + 1?
(7) A rectangular box has side of length 2, 3 and 7 units. What is
the length of the longest diagonal?
(8) An aeroplane flies around a square each side of which is 100
miles. It takes the first side at 100 miles per hour, the second
at 200 miles per hour, the third at 300 miles per hour and the
foruth at 400 miles per hour. What is its average speed?
(9) Show using only axiom (F1) of Section 3.1 that the following is
true
((u + v) + w) + x = u + ((v + w) + x).
(10) Show using only the axioms of Section 3.1 that if ab = ac and
a 6= 0 then b = c.
(11) Show using only the axioms of Section 3.1 that (−x)y = −(xy).
(12) Show using only the axioms of Section 3.1 that
(x + y)2 = x2 + 2xy + y 2 .
(13) Show using only the axioms of Section 3.1 that
x2 − y 2 = (x + y)(x − y).
(14) I draw a square. Without measuring any lengths, you now have
construct a square that has exactly twice the area.
(15) A trapezoid is a shape in the plane with 4 straight sides two of
which are parallel. If the lengths of the two parallel sides are x
and y and the perpendicular distance between them is h, find
a formula for the area of the figure.
ALGEBRA A: CHAPTER ZERO
THE NATURE OF MATHEMATICS
31
(16) A right-angled triangle has sides with lengths x, y and hy2
potenuse z. Prove that if the area of the triangle is z4 then
the triangle is isosceles.
(17) Observe the following patterns
1 + 3 = 4 = 22
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
1 + 3 + 5 = 9 = 32
1 + 3 + 5 + 7 = 16 = 42
State in words what you think the pattern is. Can you prove
that this pattern is always true? I would recommend drawing
pictures.
Prove that the last digit in the square of a positive whole number must be one of 0,1,4,5,6, or 9. Is the converse true?
Prove that a natural number is even if and only if its last digit
is even.
Prove that a natural number is exactly divisible by 9 if and only
if the sum of
√ its digits is divisible by 9.
Prove that 3√cannot be written as an exact fraction.
What does 10 2 actually mean?
Is it true that 1 = 0 · 9̇ where 0 · 9̇ = 0 · 999 . . ..
Let s = 1 − 1 + 1 − 1 + 1 − . . .. Show that by bracketing terms
appropriately you can get s = 0 or s = 1 or s = 21 . What does
this tell you?
Calculate the following. This question is just checking that you
know how to interpret sigma-notation: it is not asking you to
find formulae.
P
(i): 6i=3 i2 .
P
(ii): 5p=0 (2p + 3).
P
(iii): 4i=1 i2i .
P P
(iv): 3i=1 4j=2 ij .
The M U -puzzle. A string is just an ordered sequence of symbols. In this puzzle, you will construct strings using the letters
M, I, U . You are given the string M I which is your only axiom.
You can make new strings only by using the following rules any
number of times in succession in any order:
(I): If you have a string that ends in I then you can add a U
on at the end.
(II): If you have a string M x where x is a string then you
may form M xx.
(III): If III occurs in a string then you may make a new
string with III replaced by U .
32
DR MARK. V. LAWSON
(IV): If U U occurs in a string then you may erase it.
I shall write x → y to mean that y is the string obtained from
the string x by applying one of the above four rules. Here are
some examples:
• By rule (I), M I → M IU .
• By rule (II), M IU → M IU IU .
• By rule (III), U M IIIM U → U M U M U .
• By rule (IV), M U U U II → M U II.
The question is: can you make M U ?
(27) Take any positive natural number n; so n = 1, 2, 3, . . . If n is
even, divide it by 2 to get n2 ; if n is odd, multiply it by 3 and
add 1 to obtain 3n+1. Now repeat this process. For example, if
n = 6 you get 6, 3, 10, 5, 16, 8, 4, 2, 1. What happens if n = 11?
What about n = 27? Prove that no matter what number you
start with, you will always eventually reach 1.
References
[1] Philip J. Davis, Reuben Hersh, Elena Anne Marchisotto, The mathematical
experience, Birkhäuser, 1995.
[2] Richard Hammack, Book of proof, VCU, 2009 (and available free online).
[3] T. L. Heath, The thirteen books of the Elements: volume 1, Dover Publications,
second edition, 1956.
[4] D. E. Joyce, Euclid’s elements,
URL: http://aleph0.clarku.edu/ djoyce/java/elements/bookI/bookI.html [23
August 2013].
[5] Jenny Olive, Maths, Second Edition, CUP, 2006.
[6] J. J. O’Connor, E. F. Robertson, The MacTutor history of mathematics
archive,
URL: http://www-history.mcs.st-and.ac.uk/ [23 August 2013].
Department of Mathematics and the Maxwell Institute for Mathematical Sciences