Intermediate algebra and analytic geometry made simple

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Intermediale
A comprehensive
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and
Algebra William R.Gondin
and Bernard Sohmer
Analytic
Geometry
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ALGEBRA
INTERMEDIATE
and
ANALYTIC GEOMETRY
MADE
SIMPLE
BY
WILLIAM GONDIN,
R. Ph.D.
Associate Professor, The Crty College of N.Y.
with
practice exercises and answers
BY
BERNARD SOHMER,
Ph.D.
Instructor, The City College of N.Y.
MADE SIMPLE BOOKS
DOUBLEDAY & COMPANY, INC.
GARDEN CITY,NEWYORK
Copyright©1959 by Doubleday 4 Company, Inc.
All Rights Reserved
Printed in the UnitedStates ofAmerica
ABOUT BOOK
THIS
In an age when science and engineering make international headline news, he who would keep
abreast of the times must have more than an elementary knowledge of mathematics. For mathe-
matics is still "the first of the sciences." And as the key to all the other sciences, it has become
the engineer's most indispensable tool.
Today, an ever-increasing number of people who never before studied the subject beyond ele-
mentary algebra and geometry, or who remember it only as an irksome "requirement" of their
earlier "schooling," now find that they need some command of its more advanced techniques on
the jobs by which they earn their liv-ing. Countless others have long since discovered that, with-
out an understanding of what advanced mathematics is about, one cannot feel altogether com-
fortably at home in the culture our twentieth century has produced.
This book, together with its companion volume, Advanced Algebra and Calculus in the same
Made Simple series, is designed to help meet such needs on a plan explained in detail in the intro-
ductory chapter which follows. But first a few words of more general preface are in order here.
The concern of these books is not with mathematics as a set of arbitrary rules to be followed
blindly as an exercis—e in compliance with classroom "discipline." Rather, it is with mathematics
as a way of thinking as an application, in a somewhat specialized subject matter, of ordinary com-
monsense and reasoning power.
Of course, the subject does have a notorious reputation for being "abstruse." But one of the
convictions upon which the writing of these books has been based is that the seeming difficulty
of mathematics is due largely to the fact that too many students are introduced to it in such a
way that they never get to apply their native wits to best advantage.
The core of the trouble appears to be that, unlike more complex subject matters, the peculiarly
simple subject matter of mathematics lends itself to abbreviated treatment. It lends itself to abbre-
viated representation by theshorthand ofmathematicalsymbols. And itlendsitself toabbreviated
exposition by the shortcut of formal demonstration. This is both its unique marvel and its main
stumbling block for the uninitiated.
Once we understand the nature of the problems and ideas in a given branch ofmathematics, we have
the key to its systejiis of abbreviation. Then we can appreciate formal treatment for its beauty and
economy .... perhaps the trial-and-error research of generations of mathematicians over the
centuries, brilliantly concentrated in a few crisp lines of "proof."
But until we have grasped what these problems and ideas are all about, we lack the key to their
codes of abbreviation. Then formalism may compound confusion with despah-. And all too often,
we may attempt the impossible task of frantically trying to "memorize" what we do not under-
stand.
Especially for the benefit of readers who must study without the help of a teacher sensitive to
this difficulty, these books therefore introduce each new topic by a consideration of its mider-
m
lying ideas terms of the problems out of which these ideas arise. Only after such matters have
been carefully discussed is the pace later quickened by the more usual, formal methods of pres-
entation. Even then, more space than usual is devoted to such questions as: What kinds of
answers may be expected from the described formal method of solving problems? Does the method
8
4 About This Book
always work? If not, why and under what conditions does itfail? And what interpretation are we
to place upon the conditions of our problem in each case?
For valuable suggestions on the treatment of several topics in these books I am indebted to
Associate Dean Sherburne F, Barber of the City College School of Liberal Arts and Science. For
aidincheckinggalleysof thefirstvolume, IntermediateAlgebraandAnalyticGeometryMade Simple,
I am indebted to Mr. Henry Jacobowitz.
I am also so indebted to my colleague, Dr. Bernard Sohmer, for most of the practice exercises,
for preparing the answers to the exercises in the appendix, and for re-checking the entire work
from tj-pescript to page-proof, that I have insisted his name share its cover and title-page.
The original plan and execution of the text have been its author's sole responsibility, however,
and any shortcomings of either must be charged to his failure to make the most of these gentle-
men's better counsel.
—
William R. Gondin
TABLE OF CONTENTS
—
SECTION ONE INTRODUCTION
Chapter I
PRELIMIN.AJIY REVIEW ,.,.,, 11
The Purpose of These Books 11
What is Advanced Mathematics? , . , . . 11
Advanced Approaches to Mathematics 11
Recommended Sequence of Study 12
Recommended Pace of Study 12
Recommended Use of Practice Exercises 13
Assumed Background of the Reader 13
Note on Tables of Formulas 13
Review Table of Elementary Formulas 14
R-Series Formulas 14
Fundamental Operations 14
Operations Involving Parentheses 14
,
Operations With Fractions 14
Operations With Exponents 15
Special Factors 15
Operations With Logarithms 15
Formulas From Eleinentary Geometry 16
Formulas From Trigonometry 16
Axioms oj Equality 16
SECTION TWO — INTERMEDIATE ALGEBRA
Chapter II
LINEAR EQUATIONS IN TWO UNKNOWNS
18
Preliminary Definitions 18
,
Systems of Two Equations IS
Determinate and Indeterminate Systems 19
Consistent and Inconsistent Equations 20
Independent and Dependent Equations 22
,
Defective and Redundant Systems 23
,
Summary
25
Chapter III
VARE4BLES, FUNCTIONS, AND GRAPHS
27
Algebra and Geometry 27
Variables and Functions 27
Tables of Values 28
,
Explicit and Implicit Functions 29
<:
8 Table of Contents
First Quadrant Graphs 30
Rectangular Coordinates
31
Graphic Interpretations
33
Summary
3q
Note on Sequence of Study , 33
Chapter IV
QUADRATIC EQUATIONS IN ONE VARIABLE
38
A Different Type of Problem 33
Preliminary Definitions 38
...
Solution by Extracting Square Roots ••........., 38
Solution by Factoring 39
Solution by Completing the Square 40
Solutions by Formula
41
"Imaginary" Roots
42
Equations Equivalent to Quadratics 43
Graphic Interpretation of Quadratic Roots 45
Physical Interpretation of Quadratic Roots 47
Summary
, 49
Chapter V
SYSTEMS INVOLVING QUADRATICS
51
Preliminary Definitions 51
Note on Sequence of Topics , 51
Defective Systems 52
,
Systems with a Linear Equation 56
Systems Quadratic in Only One Variable 60
Other Systems Treatable by Comparison 61
Systems Linear in Quadratic Terms ^ , . , ^ ^ 61
Systems with Only One Linear Term 62
Systems of Other Special Types 64
Systems Treatable by Substitution 65
Summary
^ 66
Note on Sequence of Study 66
Chapter VI
LINEAR EQUATIONS IN N VARIABLES
67
Systems in A^ Variables 67
Linear Dependen<;e and Independence 69
.
Linear Consistency and Inconsistency 71
Geometric Interpretation in Schematic Form 73
Summary
^ ^ ^ 77
Chapter VII
DETERMINANTS
78
A
Note on Sequence of Topics 78
Second Order Determinants 78
Table of Contents 7
Determinants of Higher Order 80
Expansion by Minors 81
Further Properties of Determinants 83
AppHcations of Cramer's Rule 86
Properties of Determinants, Continued . , 87
Exceptions to Cramer's Rule . 88
Summary 90
.
Table of Determinant Formulas . , , , 90
Chapter VIII
.........
TRIGONOMETRIC FUNCTIONS AND E.QU.AT.IO.NS.......*.«. 92
Trigonometric Equations 92
Prelimmary Algebraic Treatment 92
Inverse Trigonometric Functions 94
Angles of Any Magnitude 95
.
Radian Measure of Angles . , 97
Trigonometric Functions of Any Angle » 97
Table: Typical Values of Trigonometric Functions ........... 99
Applications of Generalized Trigonometric Functions 102
..,
Complete Solution of Trigonometric Equations 104
Graphs of Trigonometric Functions 106
Variations of Trigonometric Functions : 108
Summary 109
,
—
SECTION THREE PLANE ANALYTIC GEOMETRY
Chapter IX
...
POINTS, DISTANCES, AND SLOPES Ill
,
What is Analytic Geometry? , Ill
Positions of Points .......,...,,,,,,.,. 112
Distances or Lengths 114
Slopes 115
Angles Between Lines 118
Problem-Solving Technique 119
.
Summary with Formulas 123
, ,
Chapter X
STRAIGHT LINES
..,..,... 125
,
Point-Slope Line Formulas 125
Two-Point Line Formulas 127
Points on Lines .,,....,,,,. 128
Intersection Poin.ts o•f L•ine.s ...,,...., 129
Parallel Lines *,......,. 132
.
Perpendicular Lines 134
Perpendicular Distances . 135
Problem Solving Technique 138
Summary with Formulas 139
.
8 Table of Contents
Chapter XI
—
CONIC SECTIONS PARAEOLAS
141
Conic Sections as Loci 141
Parabolas: Definitions and Construction 141
Standard Equations for Parabolas 142
Applications of Parabola Formulas 144
, . .
Chapter XII
ELLIPSES AND CIRCLES 147
Ellipses: Definitions and Construction of Ellipses 147
Eccentricity of Ellipses 147
The Standard Equation for an Ellipse 148
Applications of Ellipse Formulas 149
Circles: Formula and Applications 153
Chapter XIII
HYPERBOLAS
155
Definitions and Construction 155
,
Eccentricity of Hyperbolas 156
Standard Equation for a Hyperbola 157
Applications of Hyperbola Formulas 159
Summary (Chapters XI, XII, XIII) with Formulas 160
—
SECTION FOUR SOLID ANALYTIC GEOMETRY
Chapter XIV
POINTS AND DIRECTIONS IN SPACE 162
Three-Dimensional Rectangular Coordinates 162
Point and Length Formulas 164
Direction Numbers and Cosines ......•,. 167
Parallel and Perpendicular Directions 169
Chapter XV
SURFACES AND LINES IN SPACE ,171
Normal Plane-Formulas 171
,
Parallel and Perpendicular Planes 172
, . .
Point Plane-Formulas 173
, . ,
Lines as Intersections of Planes 174
Quadric Surfaces , 176
Footnote on Hyper-Geometry 178
Summary (Chapters XIV and XV) with Formulas 178
ANSWERS
181
INDEX
191