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MODERN ALGEBRA
Structure and Method: Book 1
COVER
These thousands of tiny, doughnut-shaped, ferromagnetic rings, threaded on
wires, are memory units in a computer. They are a symbol of modern man’s de¬
pendence on mathematics. An electrical current passing along the wire sets up a
magnetic field which magnetizes the rings. Since current in the opposite direction
reverses the magnetic field, the direction of the magnetic field may represent a
0 or 1, a + or —, a yes or no condition. This is the electronic mechanism for
storing information in the binary number system used in most modern computers.
Knowledge of algebra has made possible these computers which initially were
built to handle scientific and engineering problems. Today, electronic data process¬
ing systems are invaluable in business, industry, and research, where they help to
untangle and to simplify, in a matter of seconds, calculations and paper work that
formerly took days to do.
TITLE PAGE
The illustration on the title pages indicates that future vocational plans are
dependent upon your high school preparation. Are you interested in business,
chemistry, medicine? Do you look forward to being an engineer, an architect, a
home economist, a machinist, a housewife, or a psychologist? Regardless of your
vocational plans, algebra is essential to the modern educated person. More im¬
portant is the fact that algebra is essential to many future vocational opportunities
that do not even exist today. Algebra is equipment you will need if you are to take
your place as an educated person in the modern world of today and tomorrow.
MARY P. DOLCIANI
SIMON L. BERMAN
JULIUS FREILICH
EDITORIAL ADVISER
ALBERT E. M E D E R, Jr.
STRUCTURE
AND
METHOD
•BOOK ONE*
Houghton Mifflin Company Boston
•
NEW YORK . ATLANTA • GENEVA, ILL • DALLAS • PALO ALTO
ABOUT THE AUTHORS
Mary P. Dolciani,
Professor of Mathematics, Hunter College, New York.
Dr. Dolciani has been a member of the School Mathematics Study Group and a
director and teacher in numerous National Science Foundation and New York
State Education Department institutes for mathematics teachers, and visiting
secondary school lecturer for the Mathematical Association of America.
L. Berman,
SiaiOll Chairman, Department of Mathematics, Stuyvesant High
School, New York, and formerly instructor in mathematics at Brooklyn Poly¬
technic Institute.
JullUS Freilich,
Principal, Floyd Bennett School, formerly chairman of the
mathematics department of Brooklyn Technical High School and instructor at
Brooklyn Polytechnic Institute.
EDITORIAL ADVISER
Albert E. Meder, Jr.,
Dean and Vice Provost, Rutgers University. Dr. Meder
was executive director of the Commission on Mathematics, College Entrance
Examination Board, and is an advisory committee member of the SMSG.
COPYRIGHT © 1962 BY HOUGHTON MIFFLIN COMPANY
ALL RIGHTS RESERVED INCLUDING THE RIGHT TO REPRODUCE THIS BOOK
OR PARTS THEREOF IN ANY FORM. PRINTED IN THE U.S.A.
UNIVERSITY
■* 'N* RT.BF.RTA LIBRARY
Symbols and Sets
NUMBERS AND THEIR RELATIONSHIPS • 1-1 Representing Num¬
bers on a Line: Order Relations, 1 1-2 Comparing Numbers:
*
The Sign of Equality, 5 1-3 Comparing Numbers: The Signs of
*
Inequality, 7 • GROUPING NUMBERS IN SETS AND SUBSETS • 1-4
Meaning of Membership in a Set, 10 • 1-5 Kinds of Sets, 13 •
1-6 The Graph of a Set, 16 • 1-7 How Subsets Relate to Sets, 18 •
USING NUMBERS IN ONE OR MORE OPERATIONS • 1-8 Punc¬
tuation Marks in Algebra, 19 • 1-9 Order of Operations, 23 •
THE HUMAN EQUATION, 25 • CHAPTER SUMMARY, 26 •
CHAPTER TEST, 27 • CHAPTER REVIEW, 28 • EXTRA FOR
EXPERTS, 30 • SURVEYORS AND MATHEMATICS, 32 •
JUST FOR FUN, 33 •
Variables and Open Sentences
ANALYZING ALGEBRAIC STATEMENTS • 2-1 Evaluating Alge¬
braic Expressions Containing Variables, 35 • 2-2 Identifying
Factors, Coefficients, and Exponents, 40 • 2-3 Solving Open
Sentences, 44 • PROBLEMS SOLVED WITH VARIABLES • 2-4 Think¬
ing with Variables: From Symbols to Words, 49 • 2-5 Thinking
with Variables: From Words to Symbols, 51 • 2-6 Solving Prob¬
lems with Open Sentences, 56 •
EXTRA FOR EXPERTS, 60 • THE HUMAN EQUATION, 62 •
CHAPTER SUMMARY, 63 • CHAPTER TEST, 64 • CHAPTER
REVIEW, 65 • JUST FOR FUN, 67 •
Axioms, Equations, and Problem Solving
*
IDENTIFYING AND USING NUMBER AXIOMS • 3-1 Axioms of
Equality, 69 • 3-2 The Closure Properties, 70 • 3-3 Commutative
and Associative Properties of Arithmetic Numbers, 73 • 3-4 The
CONTENTS
Distributive Property; Special Properties of 1 and 0, 75 • TRANS¬
FORMING EQUATIONS WITH EQUALITY PROPERTIES • 3-5 Addi¬
tion and Subtraction Properties of Equality, 80 • 3-6 Division
and Multiplication Properties of Equality, 83 • 3-7 Combining
Terms and Using Transformation Principles, 86 • 3-8 Equations
Having the Variable in Both Members, 91 •
MACHINISTS AND MATHEMATICS, 96 • CHAPTER SUMMARY,
97 • CHAPTER TEST, 98 • CHAPTER REVIEW, 100 • CUMU¬
LATIVE REVIEW: CHAPTERS 1-3, 102 • EXTRA FOR EXPERTS,
105 • JUST FOR FUN, 107 • THE HUMAN EQUATION, 109 •
The Negative Numbers 111
EXTENDING THE NUMBER LINE • 4-1 Directed Numbers, 111 •
4-2 Comparing Numbers, 114 • OPERATING WITH DIRECTED
NUMBERS • 4-3 Addition on the Number Line, 116 • 4-4 The
Opposite of a Directed Number, 120 • 4-5 Absolute Value, 123 •
4 6 Adding Directed Numbers, 124 • 4-7 Subtracting Directed
-
Numbers, 128 • 4-8 Multiplying Directed Numbers, 133 • 4-9 Di¬
viding Directed Numbers, 138 • 4-10 Averages and Directed
Numbers (Optional), 142 •
CHAPTER SUMMARY, 144 • CHAPTER TEST, 145 • CHAPTER
REVIEW, 146 • ELECTRICAL ENGINEERS AND MATHEMATICS,
151 • EXTRA FOR EXPERTS, 152 • JUST FOR FUN, 153 •
THE HUMAN EQUATION, 155 •
Equations, Inequalities, and Problem Solving i/ 157
OPEN SENTENCES IN THE SET OF DIRECTED NUMBERS • 5-1 Trans¬
forming Equations, 157 • 5-2 The Properties of Inequality, 159 •
5 3 Pairs of Inequalities (Optional), 164 • THE ANALYSIS OF
-
PROBLEMS • 5-4 A Plan for Solving Problems, 166 • 5-5 Prob¬
lems about Consecutive Integers, 170 • 5-6 Problems about
Angles, 172 • 5-7 Uniform Motion Problems, 178 • 5-8 Mixture
Problems, 182 •
THE HUMAN EQUATION, 1 85 • CHAPTER SUMMARY, 1 86 •
CHAPTER TEST, 187 • CHAPTER REVIEW, 188 • PSYCHOME-
TRISTS AND MATHEMATICS, 192 • JUST FOR FUN, 193 •
EXTRA FOR EXPERTS, 194 •
