Mathematics for Business Analysis E-Book

Paul Turner

• Herausgeber: Packt Publishing
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 411

Veröffentlichungsjahr: 2024

Beschreibung

This book is designed for students, faculty, and professionals, describing the role of mathematics in economics and business. Starting with the basics of numbers and advancing into complex topics like hyperreal numbers and set theory, the book builds a strong foundation of mathematical concepts. Python code is used throughout to illustrate problems numerically, making the material accessible and practical.
Readers will learn about linear simultaneous equations for market equilibrium analysis and the mechanics of matrices for solving larger systems. The book covers calculus, with applications in economics, including profit maximization and factor optimization, and introduces infinitesimal methods. Chapters on differential and difference equations reveal their importance in analyzing dynamic systems, all illuminated through practical examples and images from economics and business.
The course begins with fundamental mathematical concepts, advancing to complex applications in economics and business. By integrating computational understanding with Python code, the book ensures learners can tackle real-world problems. This structured approach provides the tools needed for economic analysis and business decision-making, making it an invaluable resource for anyone in these fields.

Leseprobe

MATHEMATICSFORBUSINESS ANALYSIS

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MATHEMATICSFORBUSINESS ANALYSIS

PAUL TURNER, PHDANDJUSTINE WOOD, PHD

MERCURY LEARNING AND INFORMATIONBoston, Massachusetts

Copyright ©2024 by MERCURY LEARNING AND INFORMATION, An Imprint of DeGruyter, Inc. All rights reserved.

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P. Turner and J. Wood. Mathematics for Business Analysis.

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I would like to dedicate this book to my Mum and Dad who have given me all the support I could possibly ask for throughout my academic career.

—Paul Turner

I dedicate this book to my parents, for their continuous love and support.

—Justine Wood

CONTENTS

Preface

CHAPTER 1: SETS, NUMBERS, AND ALGEBRA

1.1 Sets and Numbers

Review Exercises – Section 1.1

1.2 Rules of Algebra

Commutative Property

Associative Property

Distributive Property

Review Exercises – Section 1.2

1.3 Complex Numbers and Hyperreal Numbers

Complex Numbers

Hyperreal Numbers

Principle 1: The Extension Principle

Principle 2: The Transfer Principle

Principle 3: The Standard Part Principle

Rules for Infinitesimal Numbers

Rules for Infinite Numbers

Review Exercises – Section 1.3

1.4 Intervals

Review Exercises – Section 1.4

1.5 Expanding and Factorizing Mathematical Expressions

Review Exercises – Section 1.5

1.6 A Numerical Method for Finding Roots

Review Exercises Section 1.6

CHAPTER 2: LINES, CURVES, FUNCTIONS, AND EQUATIONS

2.1 The Cartesian Plane

Review Exercises – Section 2.1

2.2 Functions

Review Exercises – Section 2.2

2.3 Limits

Review Exercises – Section 2.3

2.4 Power Functions

Review Exercises – Section 2.4

2.5 Exponential and Logarithmic Functions

Review Exercises – Section 2.5

2.6 Polynomial Functions

Review Exercises – Section 2.6

2.7 Sine, Cosine, and Tangent Functions

Review Exercises – Section 2.7

CHAPTER 3: SIMULTANEOUS EQUATIONS

3.1 Linear Equations

Review Exercises – Section 3.1

3.2 Systems of Linear Simultaneous Equations

Review Exercises – Section 3.2

3.3 Some Examples from Economics

Review Exercises – Section 3.3

3.4 Nonlinear Simultaneous Equations

Review Exercises – Section 3.4

3.5 Numerical Methods

Review Exercises – Section 3.5

CHAPTER 4: DERIVATIVES AND DIFFERENTIATION

4.1 Differential Calculus

Review Exercises – Section 4.1

4.2 Differentiation from First Principles

Review Exercises – Section 4.2

4.3 Rules for Differentiation

Rule 1: Multiplication by a Constant

Rule 2: Sum–Difference Rule

Rule 3: The Product Rule

Rule 4: The Quotient Rule

Rule 5: The Power Function Rule

Rule 6: The Chain Rule

Rule 7: The Inverse Function Rule

Generalization of the Power Function Rule

Review Exercises – Section 4.3

4.4 Some Economic Examples

Review Exercises – Section 4.4

4.5 Higher-Order Derivatives

Review Exercises – Section 4.5

4.6 Numerical Methods

Review Exercises – Section 4.6

CHAPTER 5: OPTIMIZATION

5.1 Identifying Critical Points

Review Exercises – Section 5.1

5.2 Some Economic Examples

Review Exercises – Section 5.2

5.3 Convexity and Concavity

Review Exercises – Section 5.3

5.4 Numerical Methods for Finding Turning Points

Review Exercises – Section 5.4

CHAPTER 6: OPTIMIZATION OF MULTIVARIABLE FUNCTIONS

6.1 Multivariable Functions

Review Exercises – Section 6.1

6.2 Partial Derivatives

Review Exercise – Section 6.2

6.3 Differentials and the Total Derivative

Review Exercises – Section 6.3

6.4 Optimization with Multivariable Functions

Review Exercises – Section 6.4

6.5 Optimization with Constraints

Review Exercises – Section 6.5

6.6 Numerical Methods

Review Exercises – Section 6.6

CHAPTER 7: INTEGRATION

7.1 Definite Integration

Review Exercises – Section 7.1

7.2 The Fundamental Theorem of Calculus

Review Exercises – Section 7.2

7.3 Integration by Substitution and by Parts

Review Exercises – Section 7.3

7.4 Some Economic Applications

Review Exercises – Section 7.4

7.5 Numerical Methods of Integration

Review Exercises – Section 7.5

CHAPTER 8: MATRICES

8.1 Matrix Algebra

Addition or Subtraction of Matrices

Matrix Transposition

Scalar Multiplication

Vector Multiplication

Matrix Multiplication

Review Exercises – Section 8.1

8.2 Determinants

Review Exercises – Section 8.2

8.3 Matrix Inversion

Review Exercises – Section 8.3

8.4 Solving Simultaneous Equations with Matrices

Review Exercises – Section 8.4

8.5 Eigenvalues and Eigenvectors

Review Exercises – Section 8.5

CHAPTER 9: FIRST-ORDER DIFFERENTIAL EQUATIONS

9.1 Separable Differential Equations

Review Exercises – Section 9.1

9.2 First-order Linear Differential Equations with Constant Coefficients

Review Exercises – Section 9.2

9.3 Solutions Using an Integrating Factor

Review Exercises – Section 9.3

9.4 The Method of Undetermined Coefficients

Review Exercises – Section 9.4

9.5 Numerical Methods

Review Exercises – Section 9.5

9.6 Some Economic Examples

Review Exercises – Section 9.6

CHAPTER 10: SECOND-ORDER DIFFERENTIAL EQUATIONS

10.1 Homogeneous Second-Order Linear Differential Equations

Review Exercises – Section 10.1

10.2 Initial Value Problems with Second-Order Differential Equations

Review Exercises – Section 10.2

10.3 Nonhomogeneous Second-Order Linear Differential Equations

Review Exercises – Section 10.3

10.4 Numerical Solution for Second-Order Equations

Review Exercises – Section 10.4

Appendix: The Principle of Superposition

Appendix: Derivation of the Complementary Function When the Roots are Complex

CHAPTER 11: DIFFERENCE EQUATIONS

11.1 First-Order Difference Equations

Review Exercises – Section 11.1

11.2 Second-Order Difference Equations

Review Exercises – Section 11.2

11.3 Solution by Backward Substitution

Review Exercises – Section 11.3

11.4 Boundary Conditions and Expectations

Review Exercises – Section 11.4

Appendix: Solution for the Case of Complex Roots

APPENDIX A: CODING IN PYTHON

APPENDIX B: ODD NUMBERED EXERCISES ANSWERS

INDEX

PREFACE

In developing this book, we have drawn on our experiences of teaching mathematics to economics and business students over a long period of time. This is, more often than not, a challenging task because mathematics can be viewed as unpopular with students, who may regard it as a chore rather than a pleasure. Nevertheless, we believe that teaching mathematics as part of economics and business programs can be of immense value to the students concerned and can even be enjoyable for the staff involved. What is needed is a clear program of study and a willingness to explain the subject from basics rather than as just a set of unrelated techniques. This is what we attempt to do in this book.

Our approach is as follows: first, in Chapter 1, we develop the very basics of mathematics in terms of the nature of numbers, starting with the natural numbers and progressing to the integers, real numbers and finally, introducing more exotic concepts such as complex numbers and the hyperreals. This naturally allows us to develop the idea of sets which act as a basic organizing structure in mathematics. Chapter 2 then builds on this to develop the idea of mathematical functions as a ‘mapping’ from one set to another. Much of this initial material is designed to allow students to become comfortable with the language of mathematics and to enable them to express familiar concepts in a more formal manner.

The basic material of Chapters 1 and 2 is then followed by applications which make use of mathematical functions to address topics of interest for students of economics and business. In Chapter 3, we look at the solution of systems of simultaneous equations. This has obvious applications in the analysis of interactions between economic agents and the determination of market equilibrium. We consider methods for the solution of systems of equations and show how these can be applied to both linear and non-linear systems. Our initial treatment of this topic is limited to small systems containing only two or three equations, but this is later extended in Chapter 8 when we introduce the method of matrices as a way of extending our solution methods to larger systems.

Chapters 4 to 7 comprise a largely self-contained section which can be used as the basis for a course in elementary calculus. Chapter 4 introduces both the idea of the derivative of a function and covers the standard methods of differentiation. We then use this in Chapter 5, to develop methods for finding maximum and minimum points of functions. In particular, we apply these methods to standard problems in economics and business such as finding profit maximizing levels of output or cost minimizing combinations of factors of production. In Chapter 5, we limit this discussion to the case of functions with a single input variable. In Chapter 6 however, we extend this to deal with multivariable functions which allow for multiple inputs. We also introduce the idea of constraints to optimization problems which require the use of Lagrangian methods. At all stages, we develop the mathematical discussion using examples drawn from economics and business to illustrate the relevance of these methods to problems of interest for students. The calculus section is completed in Chapter 7, with an introduction to integral calculus and the process of integration. Again, we take care to develop the methods we introduce using examples of interest drawn from the relevant literature.

A novel feature of our treatment of calculus is the use of infinitesimal methods. This differs from the standard treatment in many textbooks which typically use the method of limits to develop both derivative and integral calculus. The use of infinitesimal methods requires some initial investment in technique in that it requires the use of hyperreal numbers, which we introduce in Chapter 1. These are numbers which are either infinitesimally small, that is smaller than any non-zero real number, or infinitely large, that is greater than any real number. However, we believe that this framework offers significant advantages over the conventional limits approach in terms of increased intuition and ease of development of methods for the processes of differentiation and integration.

Chapters 1 to 7 cover most of the essential material for an introductory undergraduate module in calculus for economics and business studies. Most such programs will, however, find it useful to introduce more advanced mathematical methods at a later stage. In Chapters 8 to 11, we therefore cover a number of topics which feature in the later stage of undergraduate programs and in master’s programs. Chapter 8 introduces the use of matrix methods to solve systems of equations which generalizes the introductory material of Chapter 2 to permit the solution of simultaneous systems consisting of many equations. Finally, in Chapters 9, 10, and 11, we introduce the idea of differential and difference equations. These are systems of equations which allow for the analysis of dynamic systems, that is variables which change through time in response to external stimulus. As with earlier chapters, we illustrate the utility of these methods using economics or business examples at every stage.

A novel feature of our approach is the integration of numerical methods throughout the book. We do this using computer code written in the PYTHON computing language. This allows many of our examples to be illustrated numerically, which we believe helps students both understand the material more clearly and appreciate how it can be applied in practical situations. The code for our applications is provided in all cases and is available for teachers to both use and adapt as they wish. Companion files from the book are available by writing to the publisher at [email protected].

We would like to acknowledge the input of Jim Walsh, Shane Stanton, and Jennifer Blaney for help in turning the manuscript into a finished product with the usual proviso that any remaining errors are the responsibility of the authors.

Our book has been developed based on our experience in teaching mathematics to students on a wide range of different programs. It reflects what we have found to be useful and interesting for students. We hope very much that users of this book, whether teachers or students, find our approach to be of use.

Paul TurnerJustine WoodOctober 2023

CHAPTER 1

SETS, NUMBERS, AND ALGEBRA

Numbers are the raw material of mathematics. In this chapter, we define the types of numbers that you will encounter as part of your studies. To do this, we make use of the concept of a set or collection of objects—which is fundamental in mathematics. We also discuss the rules of arithmetic and algebra, allowing us to manipulate mathematical objects consistently.

1.1 SETS AND NUMBERS

The idea of a set in mathematics is a very general concept that includes any collection of objects. In mathematics, we are particularly interested in sets consisting of numbers, where a number is a mathematical object which we use to count, label, or measure other objects. The simplest numbers are the counting or whole numbers, 1, 2, 3, etc. We can define a set as a collection of objects with a rule for determining which objects belong to the set and which do not. For example, suppose we define set A to be the set of positive whole numbers less than four. This can be written in mathematical notation as , where the elements of the set are listed between curly parentheses, also known as curly brackets or braces. For small sets, we can simply list all the elements. However, this becomes cumbersome when sets become larger, and impossible when there is an infinite number of elements. A set is described as finite when the number of elements is limited and infinite when the number of elements is unlimited. The set A is finite because it contains only three elements, but it is easy to define sets which contain an infinite number of elements. For example, let B be the set of all positive whole numbers greater than 3, i.e., . The ellipsis, or dots, in this expression indicates that there are further elements in this set that increase according to the rule established by the elements shown. That is each new element increases by one relative to the preceding element.

A set is said to be well-defined if there is a clear rule for deciding whether a particular object is an element of the set. For example, in the case of A, it is clear that the number 2 is an element, but the number 5 is not. Similarly, in the case of B, it is clear by the definition that the number 2 does not belong in the set whereas the number 100 does. Defining a set in terms of a rule is often easier than simply listing its elements. Set theory allows the elements of a set to consist of any type of object, providing we can define rules for their inclusion or exclusion. For example, the set of additive primary colors consists of three colors, red, green, and blue, which can be mixed to produce almost any other color. We can define this as the set . Again, there is a clear rule for determining which colors belong in the set and which do not.

The first set of numbers of interest to us is the set of natural numbers. This is an infinite set which consists of the numbers we use for counting purposes. We write this set as:

(1.1)

Note that we can form the set of natural numbers by merging the sets A and B, which we defined earlier. This defines the union of the two sets and is written as . If a number x is an element of either of the sets A or B, then it is, by definition, an element of the set . Since the set B is an infinite set, it follows that the set is also infinite. This set is sometimes referred to as the counting numbers since it comprises the basic numbers used to count other objects.

Set theory has an associated notation; it is important to become familiar with its conventions. We have already made use of the symbol , which means the union of two sets, that is, a set that contains all the elements of two other sets. Similarly, the symbol is used to mean the intersection of two sets, that is, the elements which are present in both sets. A Venndiagram provides a useful way of illustrating and understanding this distinction. In Figure 1.1, we have two sets of numbers and which are shown as being contained with circles. The union of these sets consists of all numbers which are contained in either of the two sets, that is, , while the intersection of the sets consists of the single number 4, which is the only number that is an element of both sets, so

FIGURE 1.1  Venn diagram representation of sets.

In some cases, there may be no intersection between sets. For example, let and These sets have no elements in common. In situations like this, the intersection of the two sets defines the null or empty set. This is a set that contains no elements and is written as Another way to describe this situation is to say that sets A and B are distinct sets or mutually exclusive, in the sense that they have no common elements. Note that the empty set does not contain the number zero. If zero is a common element of two sets, then their intersection cannot be said to be empty.

Another useful item of notation is the symbol which is used to indicate that one set is a subset of another set. For example, if and then all the elements of A are present in B and, therefore, A is a subset of B. This is written as and, by definition, this makes B a superset of A, which we write as . Note that this definition of a subset allows the case in which the sets are simply identical, i.e., . If we modify the symbol to exclude the horizontal line, then a statement of the form indicates that A is a proper subset of B, i.e., all the elements of A are present in B, and there is at least one element of B that is not present in A. For example, if and , then A is a proper subset of B because all the elements of B are also present in A, and the number 3 is present in the set B but not in set A. A line through this symbol indicates the opposite interpretation. For example, means that A is not a proper subset of B. This would be the case, for example, if and because the number 1 is an element of set A but not of set B.

Another symbol that you will see frequently is , which is used to indicate that an element is present in a set. That is, the statement indicates that the object x is an element of the set A. For example, the number 100 is a natural number, and we can therefore write . On the other hand, the fraction ½ is not a natural number, and we would therefore use the symbol to indicate that it does not belong in this set, i.e., In general, can be read as “x is not an element of the set A.”

At this stage, we have introduced quite a lot of new concepts and associated notation. It is, therefore, useful to consolidate this new information and provide some examples. Table 1.1 provides a summary of the set definitions we have introduced so far and gives examples of the standard notation, which should help to clarify these definitions.

TABLE 1.1  Set theory notation.

A set is said to be closed for a mathematical operation if the application of that operation to two or more of its elements creates a third element which is also an element of the original set. For example, the set of natural numbers is closed under both addition and multiplication. This means that if x and y are natural numbers () then it is always the case that and . However, the set of natural number is not closed under subtraction or division. This is easily demonstrated by providing contradictory examples. For instance, which demonstrates that the set of natural numbers is not closed under subtraction because negative numbers are not contained within the set . Similarly, we have which is not a natural number, and therefore establishes that is not closed under division.

Although the set of natural numbers is not closed under the operation of subtraction, we can define a new set that has this property. This set is referred to as the set of integers and is generally described using the symbol The set of integers includes all the natural numbers, as well as the number zero and the negative counterparts of the natural numbers. It can therefore be written as

(1.2)

Note that the set of natural numbers is a proper subset of the set of integers because every element of the set of natural numbers is also an element of the set of integers, but there are integers that are not natural numbers. In mathematical notation, this relationship is written as . The set of integers is closed under subtraction because if x and y are integers, then will also be an integer. However, the set of integers is not closed under division, as we have already demonstrated using the example.

A useful way to think of the integers is as evenly spaced points lying along an infinitely long line, as illustrated in Figure 1.2.

FIGURE 1.2  The number line showing integers.

This line extends infinitely in both directions from point 0, which we refer to as the origin. The number line is useful because it gives us a visual representation of some of the basic operations of arithmetic. We can think of the operation of addition as a rightward movement along this line. Adding the number two to the number one means starting at point 1 and moving two spaces to the right to position 3. Similarly, subtraction involves a leftward movement; subtracting the number two from the number one indicates the operation of starting at position 1 and moving two spaces to the left to position −1. Finally, we can think of multiplication as repeated movements along the number line, which are rightward, in the case of multiplication by a positive number and leftward, in the case of multiplication by a negative number. For example, can be thought of as two successive rightward “jumps” of three units starting from the origin to reach the value 6. Similarly, can be thought of as three successive leftwards “jumps” from the origin, to give a value of −3.

The number line provides an important visual tool for understanding the relationship between integers and the arithmetic operations of addition, multiplication, and subtraction. However, it is also important because it allows us to define and understand more general definitions of numbers. We have defined the integers as evenly spaced points on the line, but is there a meaningful interpretation of the points which lie between the integers? One possibility is to interpret these points as fractions or, more formally, as rational numbers. Fractions can be thought of as points on the line, which can be expressed as the ratio of two integer numbers. For example, the point lying halfway between zero and one can be defined as 1/2, and the point lying halfway between zero and −1 can be defined as −1/2. We define the set of rational numbers as all numbers which can be written in the form where a and b are integers with no common factors.1 Alternatively, we can define rational numbers as those numbers which are solutions to equations of the form , where b and a are integers with no common factors. Note this definition is only meaningful when . The set of rational numbers is written as and both the set of natural numbers and the set of integers are proper subsets of the set of rational numbers.

The rational numbers can be written in the form of fractions or as decimal numbers. Decimal numbers are written as a sequence of digits with a single separator referred to as the decimal point. For example, we can write or . Not all decimal representations of rational numbers will have a finite number of digits or “decimal places.” An obvious example here is the rational number . If we divide one by three using the standard methods of division, then there will always be a remainder. We can write the results of this calculation as , the ellipsis here indicates that this sequence will continue forever. An alternative notation for this is , which indicates a sequence of threes which continues infinitely. If a number is rational and has an infinite decimal representation, then it can be shown that the pattern of numbers in the expansion eventually repeats, for example, .

The number of possible rational numbers between any two points on the number line is infinite. For example, consider the two points, 0 and 1. Averaging these two values gives the rational number which lies halfway between these points. Now consider the interval defined by the numbers 0 and , the rational number which lies halfway between these is . We can now divide by two again to get the rational number which lies halfway between zero to get and there is no limit to the number of times we can do this. We can continue to define rational numbers using smaller and smaller intervals on the number line, and however small we make this interval, it can always be subdivided further by dividing it into two smaller subintervals.

Since all the rational numbers can be represented as points on the number line, and an infinite number of rational numbers lie between any two points on the line, it is tempting to think that any point on the line can be represented as a rational number. However, this is not true. To illustrate this, we will make use of a counter-example. Consider the equation . This states that x is equal to the square root of two. To find x, we look for a number which, when multiplied by itself, gives the integer value 2. However, it is not possible to find a rational number with this property. This can be demonstrated by the method of proof by contradiction. That is, we assume that the statement is true and then show that it implies a logical contradiction. If is a rational number, then we should be able to find integers a and b (with no common factors) such that . If this statement is true, then we have

(1.3)

It, therefore, follows that a is even. Let us write where k is an integer. From our definition of b, we have

(1.4)

It, therefore, follows that b must also be even. The number 2 is, therefore, a common factor for both integers a and b, which contradicts the original assumption they have no common factors. Therefore, it is not possible to write as a rational number. However, it is possible to write down an approximation to using decimal notation as 1.41421..., but this decimal representation has an infinite number of terms and never settles down into a repeating pattern. Numbers without repeating patterns are referred to as irrational numbers.

Note that all numbers that can be written as a finite decimal expression are, by definition, rational. This should be immediately obvious. For example, suppose we have , then we can equivalently write this as . As we have already noted, however, not all numbers with infinite decimal expressions are irrational, for example, where the number 6 repeats indefinitely. For a number to be irrational, it must have an infinite decimal expression that never repeats. Some of the most important numbers in mathematics fall into this category. Two examples are , the ratio of the circumference of a circle to its diameter, and Euler’s number e, which is defined as These are both irrational numbers that have infinite, nonrepeating decimal representations. However, both can be represented by approximations. We have and to an accuracy of four decimal places.

We define the set of real numbers as the set of all numbers which can be written as infinite decimal expressions. This includes all the natural numbers, integers, rational numbers, and all those numbers that can be written as infinite, nonrepeating decimal expressions. The symbol for this set is and, since all elements of this set can be thought of as points on the number line, it is usual to refer to this line as the real line. The set of real numbers is closed under addition, subtraction, and multiplication. That is, if a and b are real numbers, then , and will also be real numbers. Moreover, it is closed under the operation of division, if we exclude the special case of division by zero. That is, if a and b are real numbers, then is also real number except for the case .

The set of real numbers is the most general set we have defined so far because all previously defined number sets are subsets of this set. To clarify, we define a hierarchy of sets as shown in (1.5). This indicates that the set of natural numbers is a proper subset of the set of integers, which is a proper subset of the set of rational numbers, and which, in turn, is a proper subset of the set of real numbers. It follows that, if we can demonstrate a particular mathematical result is true for all real numbers, it will also be true when applied to natural numbers, integers, and rational numbers.

.(1.5)

REVIEW EXERCISES – SECTION 1.1

1. To which sets do the following numbers belong

(a) 0.25

(b)

(c) –4

(d) 0.666….

(e) 5,489,127

2. Show that is a rational number.

3. Show that is irrational.

1.2 RULES OF ALGEBRA

The rules of algebra provide a consistent method for the manipulation of symbols representing numbers. It is important to familiarize yourself with these rules because you will frequently need to use them.

Algebra is the mathematics of symbols. The use of symbols to replace numbers allows us to derive general rules which apply to all numbers within a given set. In this section, we apply the method of algebra to the four basic mathematical operations: addition, multiplication, subtraction, and division. Since the real numbers are the most general set of numbers we have defined so far, we will consider operations involving these numbers.

Commutative Property

The property of commutativity is concerned with the ordering of the variables in algebraic expressions. It states that, when performing addition or multiplication, the order of the variables is not important. Commutativity holds for the addition and multiplication of real numbers but not for subtraction and division. Let a and b be real numbers, and we can define the commutative properties as follows:

Note that the commutative property does not hold for either subtraction or division. This can easily be demonstrated using counterexamples.

Associative Property

The property of associativity concerns the grouping of operations. Parentheses are used to indicate the order of operations by grouping together those operations which are to be performed first. For addition and multiplication, the associativity property states that the order in which operations are carried out does not affect the result. We can show that the following rules apply for all real numbers a, b, and c:

Again, this property does not hold for subtraction and division.

Distributive Property

Distributivity is a property that applies when addition and multiplication form part of the same expression. It can be written as follows:

Distributive law of multiplication  

The distributive law states that, when evaluating a multiple of the sum of elements, we can either perform the summation first and then multiply by the common factor, or we multiply each of the elements by the common factor and then take the sum. Note that, unlike the commutative and associative laws, the distributive law does apply to the combination of multiplication and subtraction. In general, it is true that . It also applies to the combination of division with either addition or subtraction, i.e., and assuming that

The properties of commutativity, associativity, and distributivity are fundamental to algebraic manipulation. If we carefully apply these rules, we can manipulate general expressions involving algebraic symbols to present them in more convenient forms. Although algebraic manipulation involves using only a few simple rules, it nevertheless requires practice to do this accurately and fluently.

Finally, we note that algebra also makes use of the existence of additive and multiplicative identities in the set of real numbers. The additive identity is the number 0, which has the property that . Related to this idea, there exists an additive inverse such that . The multiplicative identity is the number 1 which has the property that . A related property is the existence of a multiplicative inverse such that . Note that the multiplicative inverse is only defined if .

Mathematical expressions can often involve multiple operations. For example, we might have an expression of the form . The value of this expression is sensitive to the order in which these operations are carried out. It is, therefore, important to establish rules for the precedence of different operations. The convention is to give priority to operations in parentheses, followed by exponents (or power) operations, followed by division and multiplication, and finally, addition and subtraction. In the United States, this is associated with the mnemonic PEDMAS or parentheses, exponents, division/multiplication, addition/subtraction. In the UK, the equivalent mnemonic is BIDMAS or brackets, indices, division/multiplication, and addition/subtraction. These mnemonics are not, however, completely unambiguous. The rules define “levels” for different operations, with parentheses being the top level, followed by exponents, then division/multiplication, and finally, addition/subtraction. However, when operations of the same level are written as part of the expression, then the application of alternative orderings may give different results. For example, involves two subtraction operations. The ordering does not tell us which of these we should perform first, and we have already seen that . In cases like this, the convention is to work from the left to right of the expression so that is evaluated by first calculating b from a, and then subtracting c from the result. The use of parentheses, however, provides an unambiguous ordering for the operations and is recommended whenever the possibility of misinterpretation arises.

Some other notation conventions which are often assumed without being formally stated are:

1. The multiplication operator is often implicit. For example, is often written as

2. Division is most often indicated by a horizontal line or slash rather than the division operator. That is, we write a/b rather than .

Table 1.2 gives a few illustrative examples of how the rules of algebra should be applied in practice.

TABLE 1.2  Evaluation order for algebraic expressions.

REVIEW EXERCISES – SECTION 1.2

1. Evaluate the following expressions.

(a)

(b)

(c)

(d)

(e)

2. Remove the parentheses from the following expressions, where a, b, and c are nonzero real numbers.

(a)

(b)

(c)

(d)

(e)

1.3 COMPLEX NUMBERS AND HYPERREAL NUMBERS

It is sometimes useful to extend the set of numbers we consider to include complex numbers (which include the square roots of negative numbers) and hyperreal numbers (which can be either smaller or larger than any of the set of real numbers.) In this section, we discuss both types of numbers and show how they are related to real numbers. However, we do not make immediate use of either set, so you can safely pass over this section and return to it later if you prefer.

Complex Numbers

The algebraic real numbers consist of the set of numbers that can be written as infinite decimal expansions and which are solutions to algebraic equations with integer coefficients. For example, the equation has solutions and which are both algebraic real numbers. However, not all algebraic equations have real solutions. Consider, for example, the equation . We need to find a solution such that , but since a squared real number is always positive, it follows that this equation has no real solution. This is unfortunate because equations like this occur naturally in all sorts of problems. To get around this problem, we define a new class of number known as complex or imaginary numbers. Let us define the symbol i to mean the square root of minus one, that is, and let a and b be real numbers. We can define the set of complex numbers as all numbers which can be written in the form .

A complex number of the form consists of two parts, a real part a, and an imaginary or “complex” part . The set of complex numbers is written and consists of all numbers that can be written in this form. Note that the set of complex numbers includes the set of real numbers as a proper subset, , because any real number can be written as a complex number with .

We can add, subtract, multiply, and divide complex numbers in the same way as we perform these operations for real numbers. To add two complex numbers together, we simply add the coefficients for the real parts and the complex parts, as shown in equation (1.6)

(1.6)

EXAMPLE

Let and , adding these numbers gives us

Subtraction of complex numbers operates by subtracting the corresponding coefficients, as shown in equation (1.7)

(1.7)

EXAMPLE

Let and , subtracting y from x gives

Multiplication of complex numbers is a little bit more complicated and requires the use of the distributive property of algebra. Using the property that gives us

(1.8)

EXAMPLE

Let and , multiplying x by y gives

Note that, if x and y are complex conjugates, that is, if and , then their product is a real number. We can show this, in general, using the distributive property of multiplication since

(1.9)

EXAMPLE

Let and The product of these two numbers is

Finally, we can divide one complex number by another using the following procedure:

(1.10)

The proof of this statement is left as Exercise 1.3.3 for the interested reader.

EXAMPLE

Let and , using equation (1.10) we can show that

Earlier, we found that the real line provides a useful visual tool for understanding the nature of real numbers. In the case of complex numbers, a similar visualization is provided by thinking of them in terms of points in a two-dimensional plane. This is illustrated in Figure 1.3. The distance along the horizontal axis represents the real part of the complex number, and the distance along the vertical axis represents the imaginary or complex part.

FIGURE 1.3  Diagrammatic representation of the complex numbers.

Points in a two-dimensional space can be represented in terms of their coordinates, as shown in Figure 1.3, with the coefficient for the real part represented on the horizontal axis and that for the complex part represented on the vertical axis. This representation leads naturally to an alternative interpretation of complex numbers in terms of polar coordinates. Polar coordinates consist of the magnitude, i.e., the distance of the point from the origin and the angle of the point relative to the horizontal axis. These are represented by the symbols r and in Figure 1.3. The relationship between the two representations can be defined by the following pair of equations:

(1.11)

Polar coordinates prove useful when we use complex numbers to capture periodic motion, that is motion repeated in equal intervals of time. Let us consider how the location of a point in the plane changes as the angle parameter changes while keeping the magnitude constant. The constant magnitude means that the length of the line from the origin to the point remains constant. Therefore, changing the angle over the range 0 to has the effect of tracing out a circle in the plane, as shown in Figure 1.4. This means that complex numbers can be used to describe cyclical or periodic motion. In economic analysis, this proves useful when modeling phenomena such as business cycles.

FIGURE 1.4  Effects of varying the parameter.

Hyperreal Numbers

Next, we turn to the set of hyperreal numbers. This extends the set of real numbers in two ways. First, to include extremely small numbers, or infinitesimals, and second, to include extremely large numbers, or infinite quantities. We introduce a discussion of these numbers here because we make use of them later in developing a treatment of calculus which is somewhat easier than the standard approach.

For many years, the use of infinitesimals in mathematics was regarded as lacking rigor. Many argued that they could not be defined clearly in the way that the real numbers are defined. In the 1960s, however, Abraham Robinson showed that infinitesimals and infinite numbers could be given rigorous mathematical definitions. This meant that the intuitive approach to the development of calculus used by Leibniz and Newton was retrospectively justified by modern mathematics. The number system that allows us to do this is referred to as the set of hyperreal numbers, and the approach to mathematical analysis which uses these numbers is referred to as nonstandard analysis. This distinguishes nonstandard analysis from standard analysis, which derives from the work of Weierstrass, which builds calculus using the method of limits.

There are three main principles of nonstandard analysis, which we set out below. Note that this is not meant to be a rigorous definition of the approach, but rather an intuitive introduction to the system that will allow us to make use of the concept of infinitesimals for the development of calculus in later chapters.

Principle 1: The Extension Principle

The set of real numbers is a proper subset of the set of hyperreal numbers. There exists at least one hyperreal number that is greater than zero but less than every positive real number. Formally, we state that there exists a nonzero number such that where a is any real number. (epsilon) is referred to as an infinitesimal number. The inverse of an infinitesimal number is an infinite number that is greater or less than any real number depending on whether is positive or negative. A hyperreal number is said to be finite if it lies between any two real numbers. Such numbers consist of a real number plus an infinitesimal. Finally, any function f, which is defined using real numbers, has a natural extension that can be applied to hyperreal numbers. For example, if is a function that is defined for real numbers x and y, then there exists an extension of this function that applies when x is a hyperreal number.

Principle 2: The Transfer Principle

Every real statement that holds for one or more real functions holds for the hyperreal natural extensions of these functions. This states that hyperreal numbers obey the same rules of arithmetic and algebra as the real numbers. For example, is true, whether a and b are real numbers or hyperreal numbers. Conventional functions and operations such as addition, subtraction, multiplication, etc., which are applicable to real numbers, can also be applied to hyperreal numbers. Note that the transfer principle, in conjunction with the extension principle, implies that there are many infinitesimal numbers. By the extension principle is infinitesimal, and, by the transfer principle, we can define the variable , which is also infinitesimal. Thus, since any multiple of an infinitesimal is itself infinitesimal, this implies that there exists an infinite number of infinitesimals.

Principle 3: The Standard Part Principle

Every finite hyperreal number is infinitely close to exactly one real number. This means that we can define any hyperreal number as the sum of a standard (or real) part and an infinitesimal. Thus, if a is a hyperreal number, then , where is real and is infinitesimal. Using these principles, we can write down the following rules for manipulating expressions that contain hyperreal numbers.

Rules for infinitesimal numbers

Let and be positive infinitesimal numbers, and let a be a nonzero real number.

1. Sum Rule is infinitesimal. is finite but not infinitesimal.2. Product Rule and are both infinitesimal.3. Quotient Rule is infinitesimal.4. Roots Rule is infinitesimal where

In addition, we have the following rules for infinite numbers, where is a positive infinitesimal number and a is a nonzero real number.

Rules for infinite numbers

1. Reciprocals Rule is infinite.2. Product Rule is infinite.3. Quotient Rule is infinitesimal. is infinite.

One immediate implication of the reciprocals rule is that there is no unique number in the set of hyperreal numbers which we can refer to as “infinity.” Instead, there are many infinite numbers depending on the values of a and , which we use to define H.

Note that these rules do not allow us to determine the nature of the product of an infinitesimal number and an infinite number which may be finite, infinitesimal, or infinite. Similarly, there are no definitive rules for the ratio of infinitesimal numbers, the ratio of infinite numbers, or the sum/difference of infinite numbers. Despite these limitations, however, the rules we have established will prove sufficient for us to derive all of the standard results of calculus using the method of nonstandard analysis.

REVIEW EXERCISES – SECTION 1.3

1. Show that the solutions of the equation are real, but those of are complex. Plot the curves and in the plane and identify what makes them different.

2. Let be a nonzero infinitesimal number and a be a nonzero real number. For the following expressions, state the type of number

(a)

(b)

(c)

(d)

(f)

3. Let and , show that

1.4 INTERVALS

An interval is a subset of one of the general sets of numbers we have defined. In this section, we introduce the idea of open intervals and closed intervals for real numbers. This will prepare the way for a discussion of functions in Chapter 2.

An interval defines a range of possible values that a number can take. We are often interested in intervals that define a range of possible values on the real line. For example, if a and b are real numbers, then the open interval can be read as “the set of all real numbers which are greater than a but less than b.” Open intervals are indicated by curved parentheses and do not include the end points. A closed interval is indicated using square parentheses, that is, . This can be read as “the set of all real numbers greater than or equal to a but less than or equal to b.” We can also define semi-open intervals, which mix these two definitions. For example, is the set of real numbers that is greater than or equal to a but less than b. All these intervals define subsets of the set of real numbers.

Intervals can be written in various ways. Table 1.3 shows some of these ways and defines some important cases. The precise definition of ranges becomes important in Chapter 2 when we consider the definition of functions. Ranges allow us to define the values of x over interval in which a function is valid, that is, its domain. For example, we may wish to restrict attention to numbers that lie in the range −1 to +1. We could indicate this by the open interval .

TABLE 1.3  Types of Interval and Methods for Defining Intervals

Intervals also provide a way of interpreting approximations of numbers that have an infinite decimal representation. We have already noted that some rational numbers have decimal representations which extend indefinitely. For example, we can state that 1/3 lies in the open interval . This is illustrated in Figure 1.5