F17CC1 ALGEBRA A Algebra, geometry and combinatorics

214 Pages·2013·0.83 MB·English

F17CC1 ALGEBRA A
Algebra, geometry and combinatorics
Dr Mark V Lawson
July 12, 2013
ii
Contents
1 The fundamental theorem of arithmetic 1
1.1 Basic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Writing numbers down . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 From tallies to the Hindu-Arabic number system . . . . 5
1.2.2 Number bases . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 The fundamental theorem of arithmetic . . . . . . . . . . . . . 13
1.3.1 Greatest common divisors . . . . . . . . . . . . . . . . 13
1.3.2 Primes: the atoms of number . . . . . . . . . . . . . . 18
1.4 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Irrational numbers . . . . . . . . . . . . . . . . . . . . 24
1.4.2 Decimal fractions . . . . . . . . . . . . . . . . . . . . . 25
1.5 *The prime number theorem* . . . . . . . . . . . . . . . . . . 28
1.6 *Proofs by induction* . . . . . . . . . . . . . . . . . . . . . . 31
1.7 Learning outcomes for Chapter 1 . . . . . . . . . . . . . . . . 37
1.8 Further reading and exercises . . . . . . . . . . . . . . . . . . 37
2 The fundamental theorem of algebra 39
2.1 Complex number arithmetic . . . . . . . . . . . . . . . . . . . 39
2.1.1 Solving quadratic equations . . . . . . . . . . . . . . . 40
2.1.2 Introducing complex numbers . . . . . . . . . . . . . . 42
2.2 The fundamental theorem of algebra . . . . . . . . . . . . . . 50
2.2.1 The arithmetic of polynomials . . . . . . . . . . . . . . 51
2.2.2 Roots of polynomials . . . . . . . . . . . . . . . . . . . 52
2.3 Complex number geometry . . . . . . . . . . . . . . . . . . . . 59
2.3.1 sin and cos . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3.2 The complex plane . . . . . . . . . . . . . . . . . . . . 60
2.3.3 Arbitrary roots of complex numbers . . . . . . . . . . . 63
2.3.4 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . 66
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ii CONTENTS
2.4 *Making sense of complex numbers* . . . . . . . . . . . . . . 68
2.5 *Morning duel: cubics, quartics, quintics and beyond* . . . . . 69
2.6 *Analogies* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.7 *Rational functions* . . . . . . . . . . . . . . . . . . . . . . . 71
2.7.1 Numerical partial fractions . . . . . . . . . . . . . . . . 71
2.7.2 Partial fractions . . . . . . . . . . . . . . . . . . . . . . 74
2.7.3 Integrating rational functions . . . . . . . . . . . . . . 77
2.8 Learning outcomes for Chapter 2 . . . . . . . . . . . . . . . . 80
2.9 Further reading and exercises . . . . . . . . . . . . . . . . . . 80
3 Matrices 81
3.1 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.1 Basic matrix definitions . . . . . . . . . . . . . . . . . 81
3.1.2 Addition, subtraction, scalar multiplication and the
transpose . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.3 Matrix multiplication . . . . . . . . . . . . . . . . . . . 85
3.1.4 Summary of matrix mutiplication . . . . . . . . . . . . 89
3.1.5 Special matrices . . . . . . . . . . . . . . . . . . . . . . 90
3.1.6 Linear equations . . . . . . . . . . . . . . . . . . . . . 91
3.2 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2.1 Properties of matrix addition . . . . . . . . . . . . . . 94
3.2.2 Properties of matrix multiplication . . . . . . . . . . . 95
3.2.3 Properties of scalar multiplication . . . . . . . . . . . . 97
3.2.4 Properties of the transpose . . . . . . . . . . . . . . . . 97
3.2.5 Polynomials of matrices . . . . . . . . . . . . . . . . . 98
3.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4 Solving systems of linear equations . . . . . . . . . . . . . . . 110
3.4.1 Some theory . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.2 Gaussian elimination . . . . . . . . . . . . . . . . . . . 112
3.5 Blankinship’s algorithm . . . . . . . . . . . . . . . . . . . . . 120
3.6 *Some proofs* . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.7 *Matrix inverses* . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.7.1 The key idea . . . . . . . . . . . . . . . . . . . . . . . 125
3.7.2 Invertible and noninvertible matrices . . . . . . . . . . 126
3.7.3 The matrix inverse method for solving linear equations 129
3.8 *Complex numbers via matrices* . . . . . . . . . . . . . . . . 135
3.9 Learning outcomes for Chapter 3 . . . . . . . . . . . . . . . . 136
3.10 Further reading and exercises . . . . . . . . . . . . . . . . . . 136
CONTENTS iii
4 Vectors 137
4.1 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.1.1 Addition and scalar multiplication of vectors . . . . . . 139
4.1.2 Inner, scalar or dot products . . . . . . . . . . . . . . . 145
4.1.3 Vector or cross products . . . . . . . . . . . . . . . . . 147
4.1.4 Scalar triple products . . . . . . . . . . . . . . . . . . . 150
4.2 Vector arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.2.1 i’s, j’s and k’s . . . . . . . . . . . . . . . . . . . . . . . 152
4.3 Geometry with vectors . . . . . . . . . . . . . . . . . . . . . . 156
4.3.1 Position vectors . . . . . . . . . . . . . . . . . . . . . . 156
4.3.2 Linear combinations . . . . . . . . . . . . . . . . . . . 157
4.3.3 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.3.4 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . 165
4.4 Summary of vectors . . . . . . . . . . . . . . . . . . . . . . . . 170
4.5 *Two vector proofs* . . . . . . . . . . . . . . . . . . . . . . . 173
4.6 *Quaternions* . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.7 Learning outcomes for Chapter 4 . . . . . . . . . . . . . . . . 177
4.8 Further reading and exercises . . . . . . . . . . . . . . . . . . 177
5 Counting 179
5.1 More set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.1.1 Operations on sets . . . . . . . . . . . . . . . . . . . . 179
5.1.2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.1.3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.2 Ways of counting . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2.1 Counting principles . . . . . . . . . . . . . . . . . . . . 186
5.2.2 Counting sequences . . . . . . . . . . . . . . . . . . . . 187
5.2.3 The power set . . . . . . . . . . . . . . . . . . . . . . . 187
5.2.4 Counting arrangements: permutations . . . . . . . . . 188
5.2.5 Counting choices: combinations . . . . . . . . . . . . . 190
5.2.6 Examples of counting . . . . . . . . . . . . . . . . . . . 192
5.3 The binomial theorem . . . . . . . . . . . . . . . . . . . . . . 194
5.4 *An introduction to infinite numbers* . . . . . . . . . . . . . . 200
5.5 *Proving things about sets* . . . . . . . . . . . . . . . . . . . 202
5.6 Learning outcomes for Chapter 5 . . . . . . . . . . . . . . . . 203
5.7 Further reading and exercises . . . . . . . . . . . . . . . . . . 204
iv CONTENTS
Afterword 205
Chapter 1
The fundamental theorem of
arithmetic
In everday life a number is a number, but in mathematics we distinguish
between different kinds of numbers according to their properties and uses.
The goal of this chapter is to describe those numbers that should be familiar
to you whereas in the next chapter, we shall introduce numbers that might
be unfamiliar to you: the complex numbers that are so important in the
later study of mathematics. There are two essential results in this chapter:
the fact that every natural number greater than or equal to 2 can be written
uniquely as a product of powers of primes — this is the fundamental theorem
of arithmetic — and the proof that certain numbers are irrational.
1.1 Basic sets
Set theory, invented by Georg Cantor (1845–1918) in the last quarter of the
nineteenth century, provides a precise language for doing mathematics. This
section is mainly a phrasebook of the most important terms we shall need
for the first four chapters, whereas in Chapter 5 we shall study this language
in slightly more detail. The starting point of set theory is the following two
deceptively simple definitions:
• A set is a collection of objects which we wish to regard as a whole. The
members of a set are called its elements.
• Two sets are equal precisely when they have the same elements.
1
2 CHAPTER 1. THE FUNDAMENTAL THEOREM OF ARITHMETIC
We often use capital letters to name sets: such as A, B, or C or fancy capital
letters such as N and Z. The elements of a set are usually denoted by lower
case letters. If x is an element of the set A then we write
x ∈ A
and if x is not an element of the set A then we write
x ∈/ A.
A set should be regarded as a bag of elements, and so the order of the
elements within the set is not important. In addition, repetition of elements
is ignored.1
Examples 1.1.1.
(i) Thefollowingsetsareallequal: {a,b},{b,a},{a,a,b},{a,a,a,a,b,b,b,a}
because the order of the elements within a set is not important and any
repetitions are ignored. Despite this it is usual to write sets without
repetitions to avoid confusion. We have that a ∈ {a,b} and b ∈ {a,b}
but α ∈/ {a,b}.
(ii) The set {} is empty and is called the empty set. It is given a special
symbol ∅, which is taken from Danish and is the first letter of the
Danishwordmeaning‘empty’. Rememberthat∅meansthesamething
as {}. Take careful note that ∅ =(cid:54) {∅}. The reason is that the empty
set contains no elements whereas the set {∅} contains one element. By
the way, the symbol for the emptyset is different from the Greek letter
phi: φ or Φ.
The number of elements in a set is called its cardinality. If X is a set
then |X| denotes its cardinality. A set is finite if it only has a finite number
of elements, otherwise it is infinite. If a set has only finitely many elements
then we might be able to list them if there aren’t too many: this is done by
putting them in ‘curly brackets’ { and }. We can sometimes define infinite
sets by using curly brackets but then, because we can’t list all elements in
an infinite set, we use ‘...’ to mean ‘and so on in the obvious way’. This can
also be used to define finite sets where there is an obvious pattern. Often,
1If you want to take account of repetitions you have to use multisets.
1.1. BASIC SETS 3
we describe a set by saying what properties an element must have to belong
to the set. Thus
{x: P(x)}
means ‘the set of all things x which satisfy the condition P’. Here are some
examples of sets defined in various ways.
Examples 1.1.2.
(i) D = { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sun-
day }, the set of the days of the week. This is a small finite set and so
we can conveniently list its elements.
(ii) M = { January, February, March, ..., November, December }, the set
of the months of the year. This is a finite set but I didn’t want to write
down all the elements so I wrote ‘...’ to indicate that there were other
elements of the set which I was too lazy to write down explicitly but
which are, nevertheless, there.
(iii) A = {x: x is a prime number}. I define a set by describing the proper-
ties that the elements of the set must have.
Sets can be complicated. In particular, a set can contain elements which
are themselves sets. For example, A = {{a},{a,b}} is a set whose elements
are {a} and {a,b} which both happen to be sets. Thus {a} ∈ {{a},{a,b}}.
In this course, the following sets of numbers will play a special role. We
shall use this notation throughout and so it is worthwhile getting used to it.
Examples 1.1.3.
(i) The set N = {0,1,2,3,...} of all natural numbers.
(ii) The set Z = {...,−3,−2,−1,0,1,2,3,...} of all integers. The reason Z
is used to designate this set is because ‘Z’ is the first letter of the word
‘Zahl’, the German for number.
(iii) The set Q of all rational numbers i.e. those numbers that can be written
as fractions whether positive or negative.
(iv) The set R of all real numbers i.e. all numbers which can be represented
by decimals with potentially infinitely many digits after the decimal
point.
4 CHAPTER 1. THE FUNDAMENTAL THEOREM OF ARITHMETIC
(v) The set C of all complex numbers, which I shall introduce in Chapter 2.
Given a set A, a new set B can be formed by choosing elements from A
to put in B. We say that B is a subset of A, which is written B ⊆ A. If
A ⊆ B and A (cid:54)= B then we say that A is a proper subset of B. Observe that
two sets are equal precisely when the following two conditions hold
1. A ⊆ B.
2. B ⊆ A.
This is often the best way of showing that two sets are equal although we
won’t have a lot of use of it in this course.
Examples 1.1.4.
(i) ∅ ⊆ A for every set A, where we choose no elements from A. It is a very
common mistake to forget the empty set when listing subsets of a set.
(ii) A ⊆ A for every set A, where we choose all the elements from A.
(iii) N ⊆ Z ⊆ Q ⊆ R ⊆ C.
(iv) E, the set of even natural numbers, is a subset of N.
(v) O, the set of odd natural numbers, is a subset of N.
(vi) P = {2,3,5,7,11,13,17,19,23,...}, the set of primes, is a subset of N.
(vii) A = {x: x ∈ R and x2 = 4} which is just the set {−2,2}.
Exercises 1.1
1. LetA = {♣,♦,♥,♠},B = {♠,♦,♣,♥}andC = {♠,♦,♣,♥,♣,♦,♥,♠}.
Is it true or false that A = B and B = C? Explain.
2. Find all subsets of the set {a,b,c,d}.
3. Let X = {1,2,3,4,5,6,7,8,9,10}. Write down the following subsets of
X:
(i) The subset A of even elements of X.