Elements of mathematics for economics and finance

Vassilis C. Mavron and Timothy N. Phillips

Elements of
Mathematics for
Economics and Finance
With 77 Figures


Vassilis C. Mavron, MA, MSc, PhD
Institute of Mathematical and Physical
Sciences
University of Wales Aberystwyth
Aberystwyth SY23 3BZ
Wales, UK

Timothy N. Phillips, MA, MSc, DPhil,
DSc
Cardiff School of Mathematics
Cardiff University
Senghennydd Road
Cardiff CF24 4AG
Wales, UK

Mathematics Subject Classification (2000): 91-01; 91B02
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2006928729
ISBN-10: 1-84628-560-7
ISBN-13: 978-1-84628-560-8

e-ISBN 1-84628-561-5

Printed on acid-free paper

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(HAM)


Preface

The mathematics contained in this book for students of economics and finance
has, for many years, been given by the authors in two single-semester courses
at the University of Wales Aberystwyth. These were mathematics courses in
an economics setting, given by mathematicians based in the Department of

Mathematics for students in the Faculty of Social Sciences or School of Management. The choice of subject matter and arrangement of material reflect this
collaboration and are a result of the experience thus obtained.
The majority of students to whom these courses were given were studying for degrees in economics or business administration and had not acquired
any mathematical knowledge beyond pre-calculus mathematics, i.e., elementary
algebra. Therefore, the first-semester course assumed little more than basic precalculus mathematics and was based on Chapters 1–7. This course led on to
the more advanced second-semester course, which was also suitable for students
who had already covered basic calculus. The second course contained at most
one of the three Chapters 10, 12, and 13. In any particular year, their inclusion
or exclusion would depend on the requirements of the economics or business
studies degree syllabuses. An appendix on differentials has been included as an
optional addition to an advanced course.
The students taking these courses were chiefly interested in learning the
mathematics that had applications to economics and were not primarily interested in theoretical aspects of the subject per se. The authors have not attempted to write an undergraduate text in economics but instead have written
a text in mathematics to complement those in economics.
The simplicity of a mathematical theory is sometimes lost or obfuscated
by a dense covering of applications at too early a stage. For this reason, the
aim of the authors has been to present the mathematics in its simplest form,
highlighting threads of common mathematical theory in the various topics of
v


vi

Elements of Mathematics for Economics and Finance

economics.
Some knowledge of theory is necessary if correct use is to be made of the
techniques; therefore, the authors have endeavoured to introduce some basic
theory in the expectation and hope that this will improve understanding and
incite a desire for a more thorough knowledge.

Students who master the simpler cases of a theory will find it easier to go on
to the more difficult cases when required. They will also be in a better position
to understand and be in control of calculations done by hand or calculator
and also to be able to visualise problems graphically or geometrically. It is
still true that the best way to understand a technique thoroughly is through
practice. Mathematical techniques are no exception, and for this reason the
book illustrates theory through many examples and exercises.
We are grateful to Noreen Davies and Joe Hill for invaluable help in preparing the manuscript of this book for publication.
Above all, we are grateful to our wives, Nesta and Gill, and to our children, Nicholas and Christiana, and Rebecca, Christopher, and Emily, for their
patience, support, and understanding: this book is dedicated to them.

Vassilis C. Mavron
Aberystwyth
United Kingdom

Timothy N. Phillips
Cardiff
United Kingdom
March 2006


Contents

1.

Essential Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.3 Evaluation of Arithmetical Expressions . . . . . . . . . . . . . . . .
1.3 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Decimal Representation of Numbers . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Percentages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Powers and Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Simplifying Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Multiplying Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
3
3
4
5
7
8
10
10
12
16
16
18

2.

Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Solution of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Solution of Simultaneous Linear Equations . . . . . . . . . . . . . . . . . . .
2.4 Graphs of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Slope of a Straight Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Budget Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Supply and Demand Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Multicommodity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
24
27
30
34
37
40
44

vii


viii

Contents

3.

Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Graphs of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Applications to Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
50
56
61

4.

Functions of a Single Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Reciprocal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
69
72
72
75
81

5.

The Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Returns to Scale of Production Functions . . . . . . . . . . . . . . . . . . . . 95
5.4.1 Cobb-Douglas Production Functions . . . . . . . . . . . . . . . . . . 97
5.5 Compounding of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.6 Applications of the Exponential Function in
Economic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.

Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.1 Constant Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.2 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.3 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.4 Sums and Differences of Functions . . . . . . . . . . . . . . . . . . . . 114
6.2.5 Product of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.6 Quotient of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 119
6.4 Marginal Functions in Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4.1 Marginal Revenue and Marginal Cost . . . . . . . . . . . . . . . . . 121
6.4.2 Marginal Propensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 Approximation to Marginal Functions . . . . . . . . . . . . . . . . . . . . . . . 125
6.6 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.7 Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


Contents

ix

7.

Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Local Properties of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.1 Increasing and Decreasing Functions . . . . . . . . . . . . . . . . . . 138
7.2.2 Concave and Convex Functions . . . . . . . . . . . . . . . . . . . . . . 138
7.3 Local or Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4 Global or Absolute Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.5 Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.6 Optimization of Production Functions . . . . . . . . . . . . . . . . . . . . . . . 146
7.7 Optimization of Profit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.8 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.

Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 Functions of Two or More Variables . . . . . . . . . . . . . . . . . . . . . . . . 160
8.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.4 Higher Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.5 Partial Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.6 The Chain Rule and Total Derivatives . . . . . . . . . . . . . . . . . . . . . . 168
8.7 Some Applications of Partial Derivatives . . . . . . . . . . . . . . . . . . . . 171
8.7.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.7.2 Elasticity of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.7.3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8.7.4 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.7.5 Graphical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.2 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
9.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.3.1 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.3.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.3.3 The Lagrange Multiplier λ: An Interpretation . . . . . . . . . . 201
9.4 Iso Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

10. Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2.1 Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
10.2.2 Matrix Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.2.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
10.3 Solutions of Linear Systems of Equations . . . . . . . . . . . . . . . . . . . . 220


x

Contents

10.4 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
10.5 More Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10.6 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

11. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
11.2 Rules of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
11.3 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
11.4 Definite Integration: Area and Summation . . . . . . . . . . . . . . . . . . . 243
11.5 Producer’s Surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
11.6 Consumer’s Surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
12. Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.2 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.3 First Order Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . 264
12.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
12.5 The Cobweb Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
12.6 Second Order Linear Difference Equations . . . . . . . . . . . . . . . . . . . 273
12.6.1 Complementary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
12.6.2 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.6.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
13. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
13.2 First Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 288
13.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
13.3 Nonlinear First Order Differential Equations . . . . . . . . . . . . . . . . . 292
13.3.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
13.4 Second Order Linear Differential Equations . . . . . . . . . . . . . . . . . . 296
13.4.1 The Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
13.4.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
13.4.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
A. Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

1
Essential Skills

1.1 Introduction
Many models and problems in modern economics and finance can be expressed
using the language of mathematics and analysed using mathematical techniques. This book introduces, explains, and applies the basic quantitative methods that form an essential foundation for many undergraduate courses in economics and finance. The aim throughout this book is to show how a range of
important mathematical techniques work and how they can be used to explore
and understand the structure of economic models.
In this introductory chapter, the reader is reacquainted with some of the
basic principles of arithmetic and algebra that formed part of their previous
mathematical education. Since economics and finance are quantitative subjects
it is vitally important that students gain a familiarity with these principles
and are confident in applying them. Mathematics is a subject that can only be
learnt by doing examples, and therefore students are urged to work through
the examples in this chapter to ensure that these key skills are understood and
mastered.

1


2

Elements of Mathematics for Economics and Finance

1.2 Numbers
For most, if not all, of us, our earliest encounter with numbers was when we were
taught to count as children using the so-called counting numbers 1, 2, 3, 4, . . ..
The counting numbers are collectively known as the natural numbers. The

natural numbers can be represented by equally spaced points on a line as shown
in Fig. 1.1. The direction in which the arrow is pointing in Fig. 1.1 indicates the
direction in which the numbers are getting larger, i.e., the natural numbers are
ordered in the sense that if you move along the line to the right, the numbers
progressively increase in magnitude.

1

2

3

Figure 1.1

4

X

5

6

7

The natural numbers.

It is sometimes useful and necessary to talk in terms of numbers less than
zero. For example, a person with an overdraft on their bank account essentially
has a negative balance or debt, which needs to be cancelled before the account
is in credit again. In the physical world, negative numbers are used to report

temperatures below 00 Centigrade, which is the temperature at which water
freezes. So, for example, −50 C is 50 C below freezing.
If the line in Fig. 1.1 is extended to the left, we can mark equally spaced
points that represent zero and the negatives of the natural numbers. The natural numbers, their negatives, and the number zero are collectively known as
the integers. All these numbers can be represented by equally spaced points
on a number line as shown in Fig. 1.2. If we move along the line to the right,
the numbers become progressively larger, while if we move along the line to the
left, the numbers become smaller. So, for example, −4 is smaller than −1 and
we write −4 < −1 where the symbol ‘<’ means ‘is less than’ or, equivalently,
−1 is greater than −4 and we write −1 > −4 where the symbol ‘>’ means ‘is
greater than’. Note that these symbols should not be confused with the symbols
‘≤’ and ‘≥’, which mean ‘less than or equal to’ and ‘greater than or equal to’,
respectively.


1. Essential Skills

-4

3

-3

-2

Figure 1.2

-1

0

X

1

2

3

4

5

Integers on the number line.

1.2.1 Addition and Subtraction
Initially, numerical operations involving negative numbers may seem rather
confusing. We give the rules for adding and subtracting numbers and then
appeal to the number line for some justification. If a and b are any two numbers,
then we have the following rules
a + (−b)
a − (+b)

a − (−b)

= a − b,

(1.1)

= a + b.

(1.3)

= a − b,

(1.2)

Thus we can regard −(−b) as equal to +b.
We consider a few examples:
4 + (−1) = 4 − 1 = 3,
and
3 − (−2) = 3 + 2 = 5.
The last example makes sense if we regard 3 − (−2) as the difference between
3 and −2 on the number line. Note that a − b will be negative if and only if
a < b. For example,
−2 − (−1) = −2 + 1 = −1 < 0.

1.2.2 Multiplication and Division
If a and b are any two positive numbers, then we have the following rules for
multiplying positive and negative numbers:
a × (−b)

(−a) × b

(−a) × (−b)

= −(a × b),

= −(a × b),
= a × b.

(1.4)
(1.5)
(1.6)

So multiplication of two numbers of the same sign gives a positive number,
while multiplication of two numbers of different signs gives a negative number.


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Elements of Mathematics for Economics and Finance

For example, to calculate 2 × (−5), we multiply 2 by 5 and then place a minus
sign before the answer. Thus,
2 × (−5) = −10.
It is usual in mathematics to write ab rather than a × b to express the multiplication of two numbers a and b. We say that ab is the product of a and b.
Thus, we can write (1.6) in the form
(−a)(−b) = ab.
These multiplication rules give, for example,
(−2) × (−3) = 6, (−4) × 5 = −20, 7 × (−5) = −35.
The same rules hold for division because it is the same sort of operation as
multiplication, since
a
1
=a× .
b
b
So the division of a number by another of the same sign gives a positive number,
while division of a number by another of the opposite sign gives a negative

number. For example, we have
(−15) ÷ (−3) = 5, (−16) ÷ 2 = −8, 2 ÷ (−4) = −1/2.

1.2.3 Evaluation of Arithmetical Expressions
The order in which operations in an arithmetical expression are performed is
important. Consider the calculation
12 + 8 ÷ 4.
Different answers are obtained depending on the order in which the operations
are executed. If we first add together 12 and 8 and then divide by 4, the result
is 5. However, if we first divide 8 by 4 to give 2 and then add this to 12, the
result is 14. Therefore, the order in which the mathematical operations are
performed is important and the convention is as follows: brackets, exponents,
division, multiplication, addition, and subtraction. So that the evaluation of
expressions within brackets takes precedence over addition and the evaluation
of any number or expressions raised to a power (an exponential) takes precedence over division, for example. This convention has the acronym BEDMAS.
However, the main point to remember is that if you want a calculation to be
done in a particular order, you should use brackets to avoid any ambiguity.


1. Essential Skills

5

Example 1.1
Evaluate the expression 23 × 3 + (5 − 1).
Solution. Following the BEDMAS convention, we evaluate the contents of
the bracket first and then evaluate the exponential. Therefore,
23 × 3 + (5 − 1) = 23 × 3 + 4
=

8 × 3 + 4.

Finally, since multiplication takes precedence over addition, we have
23 × 3 + (5 − 1) = 24 + 4 = 28.

1.3 Fractions
A fraction is a number that expresses part of a whole. It takes the form a/b
where a and b are any integers except that b = 0. The integers a and b are known
as the numerator and denominator of the fraction, respectively. Note that
a can be greater than b. The formal name for a fraction is a rational number
since they are formed from the ratio of two numbers. Examples of statements
that use fractions are 3/5 of students in a lecture may be female or 1/3 of a
person’s income may be taxed by the government.
Fractions may be simplified to obtain what is known as a reduced fraction
or a fraction in its lowest terms. This is achieved by identifying any common
factors in the numerator and denominator and then cancelling those factors by
dividing both numerator and denominator by them. For example, consider the
simplification of the fraction 27/45. Both the numerator and denominator have
9 as a common factor since 27 = 9 × 3 and 45 = 9 × 5 and therefore it can be
cancelled:
3×9
3
27
=
= .
45
5×9
5

We say that 27/45 and 3/5 are equivalent fractions and that 3/5 is a reduced

fraction.
To compare the relative sizes of two fractions and also to add or subtract
two fractions, we express them in terms of a common denominator. The common denominator is a number that each of the denominators of the respective
fractions divides, i.e., each is a factor of the common denominator. Suppose
we wish to determine which is the greater of the two fractions 4/9 and 5/11.
The common denominator is 9 × 11 = 99. Each of the denominators (9 and


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Elements of Mathematics for Economics and Finance

11) of the two fractions divides 99. The simplest way to compare the relative
sizes is to multiply the numerator and denominator of each fraction by the
denominator of the other, i.e.,
4 × 11
44
5
5×9
45
4
=
=
, and
=
=
.
9
9 × 11
99

11
11 × 9
99

So 5/11 > 4/9 since 45/99 > 44/99.
We follow a similar procedure when we want to add two fractions. Consider
the general case first of all in which we add the fractions a/b and c/d with b = 0
and d = 0:
c
a
+
b
d

=
=

Therefore, we have

a×d
c×b
+
b×d
d×b
a×d+b×c
.
b×d

a
c

ad + bc
+ =
.
b
d
bd

(1.7)

For example,

2 3
2×5+3×7
10 + 21
31
+ =
=
=
.
7 5
7×5
35
35
The result for the subtraction of two fractions is similar, i.e.,
c
ad − bc
a
− =
.
b

d
bd

(1.8)

Example 1.2
Simplify
13
5
− .
24 16
Solution. The idea is to express each of these fractions as equivalent fractions
having a common denominator. Therefore, we have
13
5
−
24 16

=
=
=
=
=

13 × 16
5 × 24
−
24 × 16 16 × 24
208 − 120
384

88
384
11 × 8
48 × 8
11
.
48


1. Essential Skills

7

Note that a smaller common denominator, namely 48, could have been used in
this example since the two denominators, viz. 16 and 24, are both factors of
48. Thus
2 × 13
26
13
=
=
24
2 × 24
48
and
3×5
13
5
=
=

.
16
3 × 16
48
Therefore,
13
5
26 − 15
11
−
=
=
.
24 16
48
48

1.3.1 Multiplication and Division
To multiply together two fractions, we simply multiply the numerators together
and multiply the denominators together:
c
a×c
ac
a
× =
= .
b
d
b×d
bd

(1.9)

To divide one fraction by another, we multiply by the reciprocal of the divisor
where the reciprocal of the fraction a/b is defined to be b/a provided a, b = 0.
That is
c
a d
a×d
ad
a
÷ = × =
=
.
(1.10)
b
d
b
c
b×c
bc

Example 1.3
Simplify the following fractions
5 16
× ,
8 27
9
12
2.

÷ .
13 25

1.

Solution.
1. The product is the fraction

5 × 16
.
8 × 27
To simplify this fraction, we note that 8 is a factor of the numerator and
denominator (since 16 = 8 × 2) and can be cancelled. Therefore, we have
5 × 16
5×8×2
10
5 16
×
=
=
=
.
8 27
8 × 27
8 × 27
27


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Elements of Mathematics for Economics and Finance

2. Using the rule (1.10) for the division of two fractions, we have
9
12
9
25
9 × 25
÷
=
×
=
.
13 25
13 12
13 × 12
Then noting that 3 is a common factor of the numerator and denominator,
we have
5 16
3 × 3 × 25
3 × 25
75
×
=
=
=
.
8 27
13 × 4 × 3
13 × 4

52

1.4 Decimal Representation of Numbers
A fraction or rational number may be converted to its equivalent decimal representation by dividing the numerator by the denominator. For example, the
decimal representation of 3/4 is found by dividing 3 by 4 to give 0.75. This is
an example of a terminating decimal since it ends after a finite number of
digits. The following are examples of rational numbers that have a terminating
decimal representation:
1
= 0.125,
8
and
3
= 0.12.
25
Some fractions do not possess a finite decimal representation – they go on
forever. The fraction 1/3 is one such example. Its decimal representation is
0.3333... where the dots denote that the 3s are repeated and we write
1
˙
= 0.3,
3
where the dot over the number indicates that it is repeated indefinitely. This
is an example of a recurring decimal. All rational numbers have a decimal
representation that either terminates or contains an infinitely repeated finite
sequence of numbers. Another example of a recurring decimal is the decimal
representation of 1/13:
1
˙
˙

= 0.0769230769230 . . . = 0.076923
0,
13
where the dots indicate the first and last digits in the repeated sequence.
All numbers that do not have a terminating or recurring decimal representation
are known as irrational numbers. Examples of irrational numbers are
√
2 and π. All the irrational numbers together with all the rational numbers


1. Essential Skills

9

form the real numbers. Every point on the number line in Fig. 1.2 corresponds
to a real number, and the line is known as the real line.
To convert a decimal to a fraction, you simply have to remember that the
first digit after the decimal point is a tenth, the second a hundredth, and so
on. For example,
1
2
= ,
0.2 =
10
5
and
375
3
0.375 =
= .

1,000
8
Sometimes we are asked to express a number correct to a certain number
of decimal places or a certain number of significant figures. Suppose that
we wish to write the number 23.541638 correct to two decimal places. To do
this, we truncate the part of the number following the second digit after the
decimal point:
23.54 | 1638.

Then, since the first neglected digit, 1 in this case, lies between 0 and 4, then
the truncated number, 23.54, is the required answer. If we wish to write the
same number correct to three decimal places, the truncated number is
23.541 | 638,

and since the first neglected digit, 6 in this case, lies between 5 and 9, then
the last digit in the truncated number is rounded up by 1. Therefore, the
number 23.541638 is 23.542 correct to three decimal places or, for short, ‘to
three decimal places’.
To express a number to a certain number of significant figures, we employ
the same rounding strategy used to express numbers to a certain number of
decimal places but we start counting from the first non-zero digit rather than
from the first digit after the decimal point. For example,
72,648

= 70,000 (correct to 1 significant figure)
=

73,000 (correct to 2 significant figures)

=

72,600 (correct to 3 significant figures)

=

72,650 (correct to 4 significant figures),

and
0.004286

= 0.004 (correct to 1 significant figure)
=

0.0043 (correct to 2 significant figures)

=

0.00429 (correct to 3 significant figures).

Note that 497 = 500 correct to 1 significant figure and also correct to 2 significant figures.


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Elements of Mathematics for Economics and Finance

1.4.1 Standard Form
The distance of the Earth from the Sun is approximately 149,500,000 km. The
mass of an electron is 0.000000000000000000000000000911 g. Numbers such as
these are displayed on a calculator in standard or scientific form. This is a

shorthand means of expressing very large or very small numbers. The standard
form of a number expresses it in terms of a number lying between 1 and 10
multiplied by 10 raised to some power or exponent. More precisely, the standard
form of a number is
a × 10b ,
where 1 ≤ a < 10, and b is an integer. A practical reason for the use of
the standard form is that it allows calculators and computers to display more
significant figures than would otherwise be possible.
For example, the standard form of 0.000713 is 7.13 × 10−4 and the standard
form of 459.32 is 4.5932 × 102 . The power gives the number of decimal places
the decimal point needs to be moved to the right in the case of a positive
power or the number of decimal places the decimal point needs to be moved to
the left in the case of a negative power. For example, 5.914 × 103 = 5914 and
6.23 × 10−4 = 0.000623. Returning to the above examples, the Earth is about
1.495 × 108 km from the Sun and the mass of an electron is 9.11 × 10−28 g.
Similarly, a budget deficit of 257,000,000,000 is 2.57 × 1011 in standard form.

1.5 Percentages
To convert a fraction to a percentage, we multiply the fraction by 100%. For
example,
3
3
= × 100% = 75%,
4
4
and
3
3
=
× 100% = 23.077% (to three decimal places).

13
13
To perform the reverse operation and convert a percentage to a fraction,
we divide the number by 100. The resulting fraction may then be simplified to
obtain a reduced fraction. For example,
45% =

9
45
=
,
100
20

where the fraction has been simplified by dividing the numerator and denominator by 5 since this is a common factor of 45 and 100.


1. Essential Skills

11

To find the percentage of a quantity, we multiply the quantity by the number
and divide by 100. For example,
25% of 140 is

25
× 140 = 35,
100

and

4
× 5, 200 = 208.
100
If a quantity is increased by a percentage, then that percentage of the
quantity is added to the original. Suppose that an investment of £300 increases
in value by 20%. In monetary terms, the investment increases by
4% of 5, 200 is

20
× 300 = £60,
100
and the new value of the investment is
£300 + £60 = £360.
In general, if the percentage increase is r%, then the new value of the investment comprises the original and the increase. The new value can be found by
multiplying the original value by the factor
1+

r
.
100

It is easy to work in the reverse direction and determine the original value if the
new value and percentage increase is known. In this case, one simply divides
by the factor
r
.
1+
100

Example 1.4
The cost of a refrigerator is £350.15 including sales tax at 17.5%. What is the
price of the refrigerator without sales tax?
Solution. To determine the price of the refrigerator without sales tax, we
divide £350.15 by the factor
1+

17.5
= 1.175.
100

So the price of the refrigerator without VAT is
350.15
= £298.
1.175


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Elements of Mathematics for Economics and Finance

Similarly, if a quantity decreases by a certain percentage, then that percentage of the original quantity is subtracted from the original to obtain its new
value. The new value may be determined by multiplying the original value by
the quantity
r
.
1−
100

Example 1.5

A person’s income is e25,000 of which e20,000 is taxable. If the rate of income
tax is 22%, calculate the person’s net income.
Solution. The person’s net income comprises the part of his salary that is not
taxable (e5,000) together with the portion of his taxable income that remains
after the tax has been taken. The person’s net income is therefore
5,000 + 1 −

22
100

× 20,000

78
× 20,000
100
5,000 + 78 × 200

= 5,000 +
=
=

5,000 + 15,600

= e20,600.

1.6 Powers and Indices
Let x be a number and n be a positive integer, then xn denotes x multiplied
by itself n times. Here x is known as the base and n is the power or index
or exponent. For example,
x5 = x × x × x × x × x.

There are rules for multiplying and dividing two algebraic expressions or
numerical values involving the same base raised to a power. In the case of
multiplication, we add the indices and raise the expression or value to that new
power to obtain the product rule
xa × xb = xa xb = xa+b .
For example,
x2 × x3 = (x × x) × (x × x × x) = x5 .


1. Essential Skills

13

In the case of division, we subtract the indices and raise the expression or value
to that new power to obtain the quotient rule
xa ÷ xb =

xa
= xa−b .
xb

For example,

x×x
1
= 2,
x×x×x×x
x
and using the quotient rule we have
x2 ÷ x4 =

x2
= x2−4 = x−2 .
x4
More generally, we have

1
= x−n .
xn
Suppose now that we multiply an expression with a fractional power as
many times as the denominator of the fraction. For example, multiply x1/3 by
itself three times. We have
x1/3 × x1/3 × x1/3 = x1/3+1/3+1/3 = x1 = x.

However, the number that when multiplied by itself three times gives x is known
√
as the cube root of x, and an alternative notation for x1/3 is 3 x. The symbol
√
n
x, which sometimes appears on a calculator as x1/n , is known as the nth root
√
of x. In the case n = 2, the n is omitted in the former symbol. So we write x
√
rather than 2 x for the square root x1/2 of x.
Suppose we wish to raise an expression with a power to a power, for example
(x2 )4 . We may rewrite this as
(x2 )(x2 )(x2 )(x2 ) = x2+2+2+2 = x8 ,
using the product rule. More generally, we have
(xm )n = xmn .
These rules for simplifying expressions involving powers may be used to

evaluate arithmetic expressions without using a calculator. For example,
23

=

4

=

3
√
81
√
3
27
2−3

2 × 2 × 2 = 8,
3 × 3 × 3 × 3,

= 9,

= 3,
1
1
=
= .
23
8

Note the following two conventions related to the use of powers:


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Elements of Mathematics for Economics and Finance

1. x1 = x (An exponent of 1 is not expressed.)
2. x0 = 1 for x = 0 (Any nonzero number raised to the zero power is equal to
1.)
To summarise, we have the following rules governing indices or powers:
Rules of Indices
xa xb
xa
xb
a b
(x )
1
xa
√
a
x
√
a
xb

= xa+b

(1.11)

= xa−b

(1.12)

= xab

(1.13)

= x−a

(1.14)

= x
= x

1
a
b
a

(1.15)
(1.16)

Finally, consider the product of two numbers raised to some power. For
example, consider (xy)3 . Now
(xy)3 = (x × y) × (x × y) × (x × y) = (x × x × x) × (y × y × y) = x3 y 3 ,
since it does not matter in which order numbers are multiplied. More generally,
we have
(xy)a = xa y a .
Similarly, we have

x
y

a

=

xa
.
ya

Example 1.6
Simplify the following using the rules of indices:
1.

x2
,
x3/2

2.

x2 y 3
.
x4 y


1. Essential Skills

15

Solution.
1. Using the quotient rule (1.12), we have
√
x2
= x2−3/2 = x1/2 = x
3/2
x
2. Using the quotient and reciprocal rules, we have
x2 y 3
x4 y

=
=

x2
x4

y3
y

(x2−4 )(y 3−1 ) (using the quotient rule (1.12))

= x−2 y 2
y2
=
x2
y 2
=
.
x

(using the reciprocal rule (1.14))

Example 1.7
Write down the values of the following without using a calculator:
1. 3−3
2. 163/4 3. 16−3/4
−1/3
4. 27
5. 43/2
6. 190 .
Solution.
1. 3−3 =

1
1
=
.
3
3
27

√
2. 163/4 = (161/4 )3 = ( 4 16)3 = 23 = 8.
1
1
= .
8
163/4
1

1
1
4. 27−1/3 =
= √
= .
3
3
271/3
27
√
5. 43/2 = (41/2 )3 = ( 4)3 = 23 = 8.
3. 16−3/4 =

6. 190 = 1.
Note that we could also evaluate 43/2 as follows:
√
43/2 = (43 )1/2 = 641/2 = 64 = 8.


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Elements of Mathematics for Economics and Finance

1.7 Simplifying Algebraic Expressions
In the algebraic expression
7x3 ,
x is called the variable, and 7 is known as the coefficient of x3 . Expressions
consisting simply of a coefficient multiplying one or more variables raised to the
power of a positive integer are called monomials. Monomials can be added or
subtracted to form polynomials. Each of the monomials comprising a polynomial is called a term. For example, the terms in the polynomial 3x2 + 2x + 1

are 3x2 , 2x, and 1. The coefficient of x2 is 3, the coefficient of x is 2, and the
constant term is 1.
To add or subtract two polynomials, we collect like terms and add or subtract their coefficients. For example, if we wish to add 7x + 2 and 5 − 2x, then
we collect the terms in x and the constant terms:
(7x + 2) + (5 − 2x) = (7 + (−2))x + (2 + 5) = 5x + 7.

Example 1.8
Simplify the following:
1. (3x2 + 2x + 1) + (5x2 − x − 7),
2. (9x4 + 12x3 + 6x + 1) − (x4 + 2x2 − 4),
3. (x3 + 4x − 5) + (2x2 − x + 8).
Solution.
1. (3 + 5)x2 + (2 − 1)x + (1 − 7) = 8x2 + x − 6.
2. (9 − 1)x4 + 12x3 − 2x2 + 6x + (1 + 4) = 8x4 + 12x3 − 2x2 + 6x + 5.
3. x3 + 2x2 + (4 − 1)x + (−5 + 8) = x3 + 2x2 + 3x + 3.

1.7.1 Multiplying Brackets
There are occasions when mathematical expressions may be simplified by removing any brackets present. This process, which is also known as expanding
the brackets or multiplying out the brackets, culminates in an equivalent expression without brackets. The removal of brackets is based on the following
basic rule:
a(b + c) = ab + ac,
(1.17)