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Basic Mathematics for Economics, Business and Finance Free Course

Basic Mathematics for Economics, Business and Finance

1. Review of basics

 

1.1 Introduction

Economic activities have played an important role in the lives of humans for centuries past. We now know that they have an even greater influence on our modern lives. The economic agents in the old civilizations too possessed some perception, though not as sophisticated as we do today, of some of the economic phenomena that affected their lives.

But the difference is that they needed only the rudiments of mathematics to analyze and comprehend these phenomena. It was under these circumstances that some of the earliest writers on economics communicated their misty visions.

However, events such as the Renaissance and the Industrial Revolution resulted in radical transformations in production, consumption, trade, and economic management. These transformations are now bolstered by the advent of information technology. These events and the accompanying transformations have made modern economic life highly complex. This suggests that we can no longer be complacent about the rudimentary mathematics that was sufficient until about the beginning of the twentieth century.

One simple example can illuminate the argument we made above. Assume that a consumer wishes to purchase a good offered for sale. But, we are aware of the fact that the consumer’s demand for the good depends, ceteris paribus, on the price of the good. We know that this is a highly simplified version of reality.

In fact, the consumer’s demand for the good is also influenced by factors such as the price of related goods (determined in the markets for the related goods); the consumer’s income (determined in the factor market); events taking place in the government sector; and so on. Although we started with the simple proposition that a consumer’s demand for a good depends on the price of the good, we ended up with a complex situation involving many markets or sectors of the economy.

It would be difficult to analyze such a complex structure as the one presented above without mathematics. The reason is that mathematics can reduce the complexity to manageable limits. Mathematics can help define the elements of a theory precisely; can help generate new insights; and can help in the applicability of the theory. The following view of Fisher (1925: 119), a celebrated American economist, is a testimony to our above statements (italics added):

The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria1 disappears. We see better. We also see further.

The above presented necessity generated by the complexity of the economic world paved the way for the advent of mathematics in economic sciences. Mathematics has, in fact, become the language of modern economics, business, and finance. Students of these subjects require a wide variety of mathematical tools of varying degrees of complexity.

Since several of the mathematical tools used in these subjects are far beyond the scope of a basic book such as this, we include here only those necessary tools that are required by students for the successful completion of undergraduate programs, and to prepare them for graduate programs, in these subjects.

In this chapter we review some of the essential topics that we will use later. This review will include the basics of topics such as set theory; the number system; exponents; logarithms; equations; inequalities, intervals, and absolute values; relations and functions; limits and continuity; sequences and series; and summation and product notations.

Section 1.2 discusses the fundamental concepts in set theory. This is followed by the number system and the associated properties in Section 1.3. Exponents and their laws are covered in Section 1.4. Section 1.5 reviews logarithms and their properties. A review of the basics of equations is provided in Section 1.6. Section 1.7 presents inequalities, intervals, and absolute values. A review of the fundamental ideas of relations and functions is given in Section 1.8. Limits and continuity are dealt with in Section 1.9. Sequences and series are covered in Section 1.10. We introduce some of the sum and product notations in Section 1.11.

 

1.2 Set Theory

1.2.1 Meaning of sets

Sets play a crucial role in almost all branches of mathematics and are being increasingly used in economics, business, and finance. It is sometimes convenient to consider many items together. Such a collective entity is called a set. A set is defined as any well-defined list, collection, or class of objects. The objects in a set can be anything: students, numbers, vehicles, countries, trees, or anything else. Examples of sets include:

The people living in the city of New York.
The even numbers between 0 and 10.
The odd numbers between 0 and 10.
The numbers 1, 2, 3, 4, and 5.

 

1.2.2 Set notations

Sets are usually denoted by uppercase letters such as A, B, C, X, Y, Z, etc. The objects in a set are called the elements or members of the set. These objects are usually denoted by lowercase letters such as a, b, c, x, y, z, etc. If x is an object in the set A, then x is called an element of the set and is denoted as

x ∈ A, and is read “x belongs to A” or “x is a member of A

If x is not an object in A, then we may write it as

x ∈/ A, and is read “x does not belong to A” or “x is not a member of A

We can represent a set by listing its elements and using {} notation. Assume that the set A
consists of numbers 2, 4, 6, 8, and 10. Then we may write the set A as

A = {2,4,6,8,10}

Notice that in the set A above we separated the elements by commas and enclosed them in curly brackets. We call this form of representation of a set the tabular form. Sets can also be represented by stating properties that its elements must satisfy. Assume that we want a set B of even numbers. Then we may write it as

B = {x | x is even}

which we read as “B is the set of numbers x such that x is even.” This form of representation
of a set is called the set-builder form.

 

1.2.3 Equality of sets and subsets

Two sets A and B are said to be equal if they have the same elements; that is, if every element in A also belongs to B and if every element in B also belongs to A. Let A = {9, 8, 7, 6} and B = {8,7,9,6}. Then A = B. Notice that a set does not change if its elements are rearranged. Notice also that the set {1, 2, 3, 3, 4} = {1, 2, 3, 4}.

Let there be two sets A and B. If every element in A is also an element of B, then A is called a subset of B. In other words, A is a subset of B if a∈A and a∈B, and is denoted as A ⊆ B. For example, let A = {1,2,3} and B = {1,2,3,4,5}. Since the elements 1, 2, and 3 appear in both sets and since B contains more elements than A does, then A ⊆ B. Notice that if A=B, A⊆B and B⊆A. Assume that A⊆B. Then, we may also write B⊇A, which we read “B is a superset of A.”

Another term widely used is the proper subset. Let there be two sets A and B. Then A is called a proper subset of B if A⊆B and A̸=B, and is denoted as A⊂B. As an example, if A = {1,2,3} and B = {1,2,3,4,5}, then A ⊂ B.

 

1.2.4 Types of sets

There are a number of different types of sets. One of the basic types of sets is the null set or empty set, which is denoted by the Greek letter Φ (phi). As an example, let A be a set of people who are neither dead nor alive. We can write this set using the set-builder for as A = {x|x is a person who is neither dead nor alive}. We know that this set is a null or empty set. Notice that Φ is considered to be a subset of all other sets.

Sets can be finite or infinite. A set is said to be a finite set if it contains a finite number of different elements. Otherwise the set is called an infinite set. The set of months in a year, the set of hours in a day, etc., are examples of finite sets. The set of stars in the sky, the set of real numbers, etc., are the examples of infinite sets.

Two other important sets widely used are universal set and complementary set. The universal set consists of all the objects that are being considered in a particular situation. It is generally denoted by U . The complementary set is the set of all elements that are not the elements of a particular set (say A) but are of U. The complementary set of, say, B is denoted by B′.

Sometimes two or more sets may not have common elements. Such sets are called disjoint sets. For example, if A = {1,2,3,4} and B = {5,6,7,8}, then A and B are called disjoint sets. Another important type of set is the power set. The power set is defined as the set of all the subsets that can be generated from a given set A. It can be shown that if A has n elements, then the power set will contain 2n elements and is usually denoted as 2n(A). For example, let A = {1, 2}. Then 2n(A) = {{1, 2}, {1}, {2}, φ}.

 

1.2.5 Set operations

There are three basic set operations: union, intersection, and difference. We shall review each of them below. The union of two sets A and B is defined as the set of all elements which belong to A, or to B, or to both A and B. We denote the union of sets A and B by A∪B, which is read “A union B.” Let A={1,2,3,4} and B={4,3,5,6}. Then A∪B= {1,2,3,4,5,6}.

The intersection of two sets A and B is defined as the set of elements that are common to A and B, and is denoted by A ∩ B, which is read “A intersection B.” In our last example, A∩B={3,4}.

The difference of two sets A and B is defined as the set of elements which belong to A but not to B and is noted by A − B, which is read “A difference B” or “A minus B.” In our last example, A − B = {1, 2}. Notice that B − A = {5, 6}.

A useful way of representing sets and their operations is the Venn diagram, named after the English logician and mathematician John Venn. In a Venn diagram, the uni- versal set U is represented by a square or a rectangle within which individual sets are shown as circles. The Venn diagram representations of union, intersection, dif- ference, and complement are illustrated by the shaded areas in Figures 1.2.1(A)–(D), respectively.
sets in mathematical finance

 

Mathematical Finance - Laws of set operations

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