Teach Yourself Algebra

308 Pages·1988·11.39 MB·English

by  P. Abbott

TEACH YOURSELF
Algebra
P. Abbott

ALGEBRA
This book provides an introduction to the principles and
foundations of Algebra, with particular emphasis on its
applications to engineering and allied sciences. The progress-
ive exercises are designed both to test the reader's under-
standing of the subject and to give him practice in the
essential power of manipulation. Only a knowledge of
Arithmetic is assumed, though some reference is made to
theorems in Geometry and Trigonometry for the benefit of
those students who are acquainted with them.
TEACH YOURSELF BOOKS
Granted that it is possible to learn the subject from a book,
then this volume will serve the purpose excellently. It covers
the ground from the very beginning, along the usual paths
of equations, factors, indices and so on, to a concluding
chapter on simple progressions and an appendix which
contains notes on Permutations, the Binomial Theorem and
Quadratic Equations theory. A chapter worthy of especial
mention 5s that dealing with Determination of Graph Laws,
a topic too often neglected but one of great value and fre-
quent practical use.
Regarded as a textbook, this is probably the best value
for money on the market.
Higher Education Journal
ALGEBRA
P. Abbott
TEACH YOURSELF BOOKS
Hodder & Stoughton
in association with
Arnold Publishers
First printed 1942
Revised edition 1971
Sixteenth impression 1988
Copyright © 1971 edition
Hodder and Stoughton Ltd.
All rights reserved. No part of this publication may be reproduced
or transmitted in any form or by any means, electronic or
mechanical, including photocopy, recording, or any information
storage and retrieval system, without permission in writing from
the publisher.
ISBN 81-7031-188-8
INTRODUCTION
ALGEBRA is such a wide and comprehensive subject that this volume
cannot be regarded as anything more than an elementary intro-
duction to it. It is an endeavour to enable the private student to
learn something of the principles and foundations of the subject,
thus enabling him to proceed to the study of more detailed and
advanced treatises. It also provides, within the necessarily pre-
scribed limits of such a book, that knowledge of Algebra which is
required by a student of allied branches of Mathematics or in
applications of Mathematics to Engineering, etc. Consequently
some of those elementary sections of the subject which are of little
use for these purposes have not been included.
The exercises are progressive and designed both to enable the
student to test his knowledge of the work he has studied and also
to provide material for his training in that power of manipulation
which is so essential. They contain few of the more complicated
or academic problems which are beyond the practical requirements
of the ordinary student.
An Appendix contains, without exercises, a very brief summary
of the meaning of Permutations and Combinations, the Binomial
Theorem, and the nature of the roots of a Quadratic Equation,
together with those formulae which students may require when
beginning work on the Calculus or other branches of Mathematics.
While the fundamental laws of Algebra have not been entirely
overlooked, rigid proofs of them have been omitted, owing to
exigencies of space. It is hoped, however, that the logical basis of
the subject has not been seriously impaired by the omissions.
Some emphasis has been placed on the graphical aspects of parts
of the subject, since experience has shown that they prove stimu-
lating and provide revealing help to the student.
No previous mathematical knowledge is required for this work,
beyond that of Arithmetic. References have occasionally been
made to theorems in Geometry or Trigonometry for the benefit of
those students who have some knowledge of them. „
The Author is desirous of expressing his indebtedness to Mr.
C. E. Kerridge, B.Sc., for the use of a number of examples from
National Certificate Mathematics, Vol. I, and also to Mr. H. Marshall,
B.Sc., for the use of examples from Vol. II of the same work. He
also desires to record his gratitude to Mr. S. R. Morrell for the
valuable assistance he has given in the correction of proofs.
P. ABBOTT.

CONTENTS
CHAPTER I
THE MEANING OF ALGEBRA
PARA. PACE PARA. PAGE
1. Algebra and Arithmetic . 13 3. Transiormation of a Formula. 15
2. A Formula . . .. 13 7. Algebraic Forms 19
CHAPTER II
ELEMENTARY OPERATIONS
« Symbols of'Operation 22 17. Multiplication . 38
10. Algebraic Expression. Terms 23 18. Powers of Numbers . 39
11. Brackets . . .. 23 19. Multiplication of Powers . 30
13. Coefficient . . .. 24 21. Division of Powers 32
13. Addition and Subtraction 24 22. Easy Fractions . 33
16. Evaluation by Substitution . 26
CHAPTER III
BRACKETS AND OPERATIONS WITH THEM
as. Removal of Brackets >8. Systems of Brackets
36. Addition and Subtraction of
Expressions within Brackets 40
CHAPTER IV
POSITIVE AND NEGATIVE NUMBERS
JO. The Scale of a Thermometer . 47 37. Operations with Negative
33. Motion in Opposite Directions 48 Numbers . . .. (1
M. Positive and Negative Num- 43. Rules for Signs M
bers 4» 43. Powers . . .. to
CHAPTER V
SIMPLE EQUATIONS
Meaning of an Equation St I 47. Problems Leading to
M. Solving an Equation . M | Equations 6fi
CHAPTER VI
FORMULAE
49. Treatment of Formulae 68 S3. Literal Equations n
U. Transformation of Formulae . 71
CHAPTER VII
SIMULTANEOUS EQUATIONS
Solution of Simultaneous | 58. Problems Leading to Simul-
Equations 78 taneous Equations
CONTENTS
CHAPTER VIII
GRAPHICAL REPRESENTATION OF QUANTITIES
PAHA. PACE PARA. PAGE
<)0. The Object ot Graphical Work 88 64. Examples ol Uruphs and their
61. The Column Graph . 88 Uses . . .. 91
63. A Straight-line Graph . 89
CHAPTER IX
THE LAW OF A STRAIGHT LINE; CO-ORDINATES
68. The Law Represented by a 73. Equation of a Straight Line
Straight-line Graph . 101 Passing Through the Origin 111
Graph of an Equation of the 74. Equation of a Straight Line
First Degree .102 Not Passing Through the
Position in a Plane; Co- Origin . .. 112
ordinates . . .. 106 76. Graphic Solution of Simul-
73. A Straight Line as a Locus . 109 taneous Equations . 114
CHAPTER X
MULTIPLICATION OF ALGEBRAICAL EXPRESSIONS
Product of Binomial Expres- 83. Product of Sum and Difference
sions . . .. 118
CHAPTER XI
FACTORS
87. Binomial Factors. . 129 I 93. Factors of the Difference of
Two Squares 136
CHAPTER XII
FRACTIONS
99. Laws of Fractions 140 102. Addition and Subtraction 143
100. Reduction of Fractions 140 103. Simple Equations Involving
101. Multiplication and Division 142 Fractions . . . 146
CHAPTER XIII
GRAPHS OF QUADRATIC FUNCTIONS
104. Constants and Variables 148 109. The Curve of y = x* 162
105. Dependent and Independent 110. The Curve of y = — x* 153
Variables . . .. 149 111. The Curves of y = ax* . 164
106. Functions . . .. 149 112. The Curves of y = xB ± a 166
107. Graph of a Function ISO 113. Change of Axis lie
108. Graph of a Function of Second 116. The Graph of y = x' - 2x - 3 169
Degree . . .. 151 119. The Graph of y = 12 - x - x1 161
CHAPTER XIV
QUADRATIC EQUATIONS
120. Algebraic Solution 165 129. Problems Leading to Quadra-
124. Solution by Factorisation . 17t> tics 175
120. General Formula . . . 172 130. Simultaneous Quadratics . 178