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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 consumerās 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 consumerās 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 consumerās 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 consumerās 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 ...
