# Fundamental Concepts of Mathematics

Publication Year: 1994

Published by: Hong Kong University Press, HKU

#### 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

E-ISBN-13: 9789882201415

Print-ISBN-13: 9789622091818

Page Count: 260

Publication Year: 1994