Fundamental Concepts of Mathematics

Table of Contents

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pp. v-vi

PREFACE

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pp. vii-viii

Fundamental Concepts of Mathematics is the first of several volumes of a proposed series on fundamental mathematics which serve primarily as textbooks for students in preparation for the A-Level or other public examinations. However they go into more depth than what is required by these examinations, and contain topics ...

1. SET NOTATION

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pp. 1-36

Every environment breeds its own language and common words tend to acquire different meanings for different groups of people. For example, a Mediterranean farmer, a chairman of a committee and a young lady may have quite different ideas about the word 'date'. A peasant, a physicist, a mathematician ...

2. MATHEMATICAL INDUCTION

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pp. 37-60

In this chapter we shall study in some detail a very important and useful property of natural numbers - the principle of mathematical induction, upon which is based the well-known technique of the proof by induction. Instead of just recapitulating the standard procedure of this particular method of proof which ...

3. COMBINATORICS

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pp. 61-93

In this chapter we are interested in counting the different ways in which a given event can occur. There is a great variety of such counting problems that we encounter in every day life and in mathematics. Here we shall discuss a few useful methods that can be applied to such problems. ...

4. ARITHMETIC

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pp. 94-124

In this chapter we shall put together in a systematic and rigorous manner some results of elementary arithmetic that are familiar to our readers. ...

5. REAL NUMBERS

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pp. 125-156

For the purpose and the scope of this course, it will be sufficient for us to think of the real numbers as the points along a straight line which extends indefinitely in both directions. (Fig. 5.1) This straight line is referred to as the number line. Any finite portion of the number Iine can serve as a ruler with which we may measure ...

6. LIMIT AND CONVERGENCE

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pp. 157-192

Traditionally mathematics is divided into three large categories geometry, algebra and analysis - though at times it is very difficult to put some modern topics of mathematics in one of these categories. A very rough outline of the classification can be given as follows. The study of space inaugurated by the ancient ...

7. COMPLEX NUMBERS

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pp. 193-242

Instead of starting with a formal definition of complex number and then proceeding to study the properties of the new number system, we shall begin with a brief review of our old number systems N, Z, Q and R in order to discover a certain inadequacy in each concerning the provision of solutions of equations. ...

ANSWERS TO EXERCISES

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pp. 243-252

INDEX

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pp. 253-255