Selected Title s i n Thi s Serie s

11 Kunihik o Kodaira , Editor , Basi c analysis : Japanes e grad e 11 , 199 6

10 Kunihik o Kodaira , Editor , Algebr a an d geometry : Japanes e grad e 11 , 199 6

9 Kunihik o Kodaira , Editor , Mathematic s 2 : Japanes e grad e 11 , 199 6

8 Kunihik o Kodaira , Editor , Mathematic s 1 : Japanes e grad e 10 , 199 6

7 Dmitr y Fomin , Serge y Genkin , an d Ili a Itenberg , Mathematica l circle s (Russia n experience), 199 6

6 Davi d W . Farme r an d Theodore B . Stanford , Knot s an d surfaces : A guid e t o discoverin g mathematics, 199 6

5 Davi d W . Farmer , Group s an d symmetry : A guid e t o discoverin g mathematics , 199 6

4 V . V . Prasolov , Intuitiv e topology , 199 5

3 L . E. Sadovski ï an d A . L . Sadovskiï , Mathematic s an d sports , 199 3

2 Yu . A . Shashkin , Fixe d points , 199 1

1 V . M . Tikhomirov , Storie s abou t maxim a an d minima , 199 0

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Mathematical World • Volume 8

Mathematics 1 Japanes e Grad e 10

Kunihiko Kodaira, Editor Hiromi Nagata,Translator

George Fowler, Translation Editor

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American Mathematical Society

The University of Chicago School Mathematics Project

http://dx.doi.org/10.1090/mawrld/008

The Universit y o f Chicag o Schoo l Mathematic s Projec t

Zalman Usiskin , Directo r Izaak Wirszup , Director , Resourc e Developmen t Componen t

The translatio n an d publicatio n o f thi s boo k were mad e possibl e b y the generou s suppor t o f

The Amoc o Foundation , Inc .

Translated b y Hiromi Nagat a Translation edite d b y George Fowler

1991 Mathematics Subject Classification. Primar y 00-01 .

Library o f Congres s Cataloging-in-Publicat io n Dat a Sügaku I . English .

Mathematics 1 : Japanes e grad e 1 0 / edite d b y Kunihik o Kodaira ; Hirom i Nagata , translator ; George Fowler , transla t ion editor .

p. cm . — (Mathematica l world , ISS N 1055-9426 ; v . 8 ) Includes index . ISBN 0-8218-0583- 5 (acid-fre e paper ) 1. Mathematics . I . Kodaira , Kunihiko , 1915 - . II . Title . III . Series .

QA39.2.S8313 199 6 512M2—dc20 96-2312 9

CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofl t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permission@ams.org .

Copyright © 199 6 b y th e Universit y o f Chicag o Schoo l Mathematic s Project . All right s reserved . Printe d i n th e Unite d State s o f America .

Original Japanes e editio n publishe d i n 198 3 b y Toky o Shosek i Co. , Ltd. , Tokyo , and approve d b y th e Japanes e Ministr y o f Education .

Copyright © 198 3 b y Toky o Shosek i Co. , Ltd . Al l right s reserved . Translated wit h thei r permission .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

10 9 8 7 6 5 4 3 2 1 0 1 0 0 9 9 9 8 9 7 9 6

Textbook Serie s Prefac e

The Universit y o f Chicag o Schoo l Mathematic s Projec t

This textboo k i s par t o f a serie s o f foreig n mathematic s text s tha t hav e bee n translated b y th e Resourc e Developmen t Componen t o f th e Universit y o f Chicag o Schoo l Mathematics Projec t (UCSMP) . Establishe d i n 198 3 with majo r fundin g fro m th e Amoc o Foundation, UCSM P has been sinc e tha t tim e the nation' s larges t curriculu m developmen t and implementation projec t i n schoo l mathematics . Th e international focu s o f it s resourc e component, together with the project's publication experience , makes UCSMP well suited to disseminate these remarkable materials.

The textbook s wer e originally translate d t o give U.S . educators an d researchers a first-hand loo k at the content of mathematics instruction in educationally advanced countries. More specifically , the y provide d inpu t fo r UCSM P a s i t develope d ne w instructiona l strategies, textbooks, and materials of its own; the resource component's translations of over 40 outstandin g foreig n schoo l mathematic s publications , includin g texts , workbooks , an d teacher aids , hav e bee n use d i n UCSMP-relate d researc h an d experimentatio n an d i n th e creation of innovative textbooks.

The resourc e component' s translation s includ e th e entire mathematic s curriculu m (grades 1-10 ) use d in the former Sovie t Union, Standard Japanese texts for grades 7-11, and innovative elementary textbooks from Hungary and Bulgaria.

The conten t o f Japan' s compulsor y nationa l curriculu m fo r grade s 7-1 1 i s mad e available fo r th e firs t tim e i n English , thank s i n par t t o th e generosit y o f th e Japanes e publisher, Tokyo Shoseki Company, Ltd., which provided the copyright permissions .

Japanese Secondar y Schoo l Mathematic s Textbook s

The achievemen t o f Japanes e elementar y an d secondar y student s gaine d worl d prominence largely a s a result of their superb performance i n the International Mathematic s Studies conducte d b y th e Internationa l Associatio n fo r th e Evaluatio n o f Educationa l Achievement. Th e Secon d Internationa l Mathematic s Stud y surveye d mathematic s achievement i n 24 countries i n 1981-8 2 an d released it s flndings i n 1984 . Th e result s ar e recapitulated i n a 198 7 national report entitled "The Underachieving Curriculum: Assessin g U.S. Schoo l Mathematic s fro m a n Internationa l Perspective " ( A Nationa l Repor t o n th e Second International Mathematics Study, 1987) .

Let us take a brief loo k at the schooling behind muc h of Japan's economie success . The Japanes e schoo l syste m consist s o f a six-yea r primar y school , a three-yea r lowe r secondary school , an d a three-yea r uppe r secondar y school . Th e firs t nin e grade s ar e

v

VI

compulsory, an d enrollmen t i s no w 99.99% . According t o 199 0 statistics , 95.1 % of age -group children are enrolled in upper secondary school , and the dropout rate is 2.2%. I n terms of achievement, a typical Japanese student graduates from secondary school with roughly fou r more year s o f educatio n tha n a n averag e America n hig h schoo l graduate . Th e leve l o f mathematics training achieved by Japanese students can be inferred from the following data :

Japanese Grade 7 Mathematics (New Mathematics 1 ) explores integers, positive and negative numbers , letter s an d expressions , equations , function s an d proportions , plan e figures, an d figures i n space . Chapte r headings i n Japanese Grade 8 Mathematics includ e calculating expressions, inequalities, systems of equations, linear functions, paralle l lines and congruent figures , parallelograms , simila r figures , an d organizing data . Japanese Grade 9 Mathematics cover s square roots, polynomials, quadratic equations, functions, circles , figures and measurement , an d probability an d statistics . Th e material i n these three grade s (lowe r secondary school ) is compulsory for all students.

The textbook Japanese Grade 10 Mathematics 1 covers material that is compulsory. This course , which i s completed b y over 97% of al l Japanese students , is taught fou r hour s per week and comprises algebra (including quadratic functions, equations , and inequalities) , trigonometrie functions, and coordinate geometry.

Japanese Grade 11 General Mathematics 2 is intended for the easier of the electives offered i n tha t grad e an d i s taken b y about 40% of th e students . I t covers probabilit y an d statistics; vectors; exponential, logarithmic, and trigonometrie functions ; an d differentiatio n and integration in an informal presentation .

The other 60% of students in grade 11 concurrently take two more extensive courses using th e text s Japanese Grade 11 Algebra and Geometry an d Japanese Grade 11 Basic Analysis. Th e firs t consist s o f fulle r treatment s o f plan e an d soli d coordinat e geometry , vectors, and matrices. Th e second includes a more thorough treatment of trigonometry an d an informal bu t quite extensive introduction to differential an d integral calculus.

Some 25 % of Japanes e student s continu e wit h mathematic s i n grad e 12 . Thes e students tak e a n advance d cours e usin g th e tex t Probability and Statistics an d a secon d rigorous course with the text Differential and Integral Calculus.

One o f th e author s o f thes e textbook s i s Professo r Hirosh i Fujita , wh o spok e a t UCSMP's Internationa l Conferences o n Mathematic s Educatio n i n 1985 , 1988 , and 1991 . Professor Fujita' s pape r o n Japanese mathematic s educatio n appeare d i n Developments in School Mathematics Education Around the World, volume 1 (NCTM, 1987) . Th e curren t school mathematics reform i n Japan is described in the article "The Reform o f Mathematic s Education a t th e Uppe r Secondar y Schoo l (USS ) Leve l i n Japan " b y Professor s Fujita , Tatsuro Miwa, and Jerry Becker in the proceedings o f the Second Internationa l Conference , volume 2 of Developments.

vu

Acknowledgments

It goe s withou t sayin g tha t a publicatio n projec t o f thi s scop e require s th e commitment an d cooperatio n o f a broa d networ k o f institution s an d individuals . I n acknowledging thei r contributions, we would like first of all to express our deep appreciation to the Amoco Foundation. Withou t the Amoco Foundation's generou s long-term suppor t of the University o f Chicago Schoo l Mathematics Projec t thes e books migh t neve r hav e been translated for use by the mathematics education community .

We ar e gratefu l t o UCSM P Directo r Zalma n Usiski n fo r hi s hel p an d advic e i n making thes e valuabl e resource s availabl e t o a wid e audienc e a t lo w cost . Rober t Streit , Manager o f the Resource Development Component , did an outstanding job coordinating th e translation wor k an d collaboratin g o n th e editin g o f mos t o f th e manuscripts . Georg e Fowler, Steve n R . Young , an d Caroly n J . Ayer s mad e a meticulou s revie w o f th e translations, while Susan Chang and her technical staff a t UCSMP handled the text entry and layout wit h grea t car e an d skill . W e gratefull y acknowledg e th e dedicate d effort s o f th e translators and editors whose names appear on the title pages of these textbooks.

Izaak Wirszup, Director UCSMP Resource Development Componen t

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FOREWORD T O THE JAPANESE EDITIO N

The history of mathematics goes back thousands of years, and the discipline has undergone continuous advances. I n the last few decades, particularly, mathematics has made startling progress.

Geometry is said to have started in ancient Egypt and Babylonia with astronomical observations, measurement of land, and architecture. Thus , the advancement of mathematics was originally linked with science and technology; however, it gradually became independent of science and technology, and present-day mathematicians think freely about virtually everything possible. Therefore , mathematics is said to be a free creation of the human spirit.

This mathematics which is created freely by the human spirit can nonetheless be very useful when it is applied to various areas of study. Fo r example, complex numbers were introduced as a way to imagine non-existing entities; this fact is reflected in the tenn "imaginary number." Bu t present-day physics would be unthinkable without complex numbers. Th e realm of complex numbers is the real, existing world, and the realm of real numbers is now thought to be only a small part of that world. I n this respect, even if mathematics can be said to be a free creation of the human spirit, it is not created freely by human beings. I n other words, there exists a world of mathematical phenomena, just as physics is the study of natural phenomena, and mathematics is the science that studies those mathematical phenomena. Moreover , since the world of mathematical phenomena underlies the natural world, we must realize that mathematics is useful for the natural sciences. Thus , mathematics is a critical discipline; i t is basic for the study of various other sciences.

This book is a textbook for students who have graduated from junior high school and entered high school to study Mathematics I. I n Chapter 1 you will continue to learn basic concepts involving numbers, expressions, and fractional expressions, etc. which you also studied in junior high school. I n Chapter 2 you will learn about equations and inequalities, especially a method of solving quadratic equations, and about complex numbers in connection with this method. Yo u have already studied quadratic equations in junior high school, but in this chapter you will study quadratic equations from a broader standpoint and you will reach complex numbers. Figure s are objects with completely different properties from numbers and polynomials; however , there is a close relationship between figures and polynomials. I n Chapter 3 you will learn about the relation between figures and polynomials. Chapte r 4 presents functions and their graphs. Th e fifth and last chapter deals with trigonometry and the relation between the properties of triangles and trigonometry.

To master mathematics, it is not enough to read and memorize books. I t is important to think through the material, do calculations, and solve problems by yourselves. I n this book we try to explain the material in a simple way by using many examples and sample problems.

IX

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TABLE O F CONTENT S

Chapter 1 Number s an d Expression s 1

Section 1 Rea l Numbers 2

1 Th e Number Line and Real Numbers 2 2 Calculation s Involving Real Numbers 5 3 Classificatio n o f Real Numbers 9 4 Calculatin g Expressions Containing

Radicals 1 1 Exercises 1 5

Section 2 Number s and Sets 1 7

1 Set s 1 7 2 Th e Intersection and Union of Sets 1 9 3 Subset s 2 2 4 Complementar y Set s 2 3 5 Set s with Non-Numerical Element s 2 5

Exercises 2 6

Section 3 Integra l Expressions 2 7

1 Integra l Expression s 2 7 2 Addition , Subtraction, and Multiplication

of Integral Expressions 3 0 3 Expansio n Formulas 3 2 4 Factorin g 3 4 Exercises 4 0

Section 4 Divisio n of Integral and Fractional Expressions 4 1

1 Divisio n of Integral Expressions 4 1 2 Th e Greatest Common Divisor and Least

Common Multiple of Integral Expressions 4 4

3 Calculation s Involving Fractiona l Expressions 4 6

4 Expansion s of Exponents 5 0 Exercises 5 3

• Chapte r Exercises A and B 5 4

xi

Equations an d Inequalitie s 57

Section 1 Quadrati c Equations 5 8

1 Quadrati c Equations 5 8 2 Comple x Numbers 5 9 3 Th e Quadratic Formula 6 5 4 Relation s between Solutions and

Coefficients 7 3 Exercises 7 8

Section 2 Simultaneou s Equations and Higher Degree Equations 7 9

1 Simultaneou s Linear Equations in Three Variables 7 9

2 Simultaneou s Linear and Quadratic Equations 8 0

3 Simpl e Higher Degree Equations 8 2 4 Th e Factor Theorem 8 4 5 So l ving Equations of Higher Degree Using

the Factor Theorem 8 8 Exercises 9 1

Section 3 Inequalitie s 9 3

1 Solution s of Inequalities 9 3 2 Quadrati c Inequalities 9 4 Exercises 10 2

Section 4 Expression s and Proofs 10 3

1 Provin g Equalities 10 3 2 Provin g Inequalities 11 0 3 Th e Contrapositive an d Indirec t Proof 11 4 Exercises 11 9

• Chapte r Exercises A and B 12 0

xii i

Chapter 3 Plan e Figure s an d Expression s 12 3

Section 1 Coordinate s of Points 12 4

1 Coordinate s of Points on a Straight Line 12 4 2 Coordinate s of Points in a Plane 12 7 Exercises 13 2

Section2 Straigh t Lines 13 3

1 Equation s of Straight Lines 13 3 2 Paralle l and Perpendicular Straight Lines 13 6 3 Equatio n of a Locus 14 3 Exercises 14 7

Section 3 Circle s 14 8

1 Equation s of Circles 14 8 2 Circle s and Straight Lines 15 1 3 Translatio n 15 7 Exercises 16 0

Section 4 Region s Represented by Inequalities 16 1

1 Region s Represented by Inequalities 16 1 2 Region s Represented by Simultaneous

Inequalities 16 4 3 Applicatio n of Regions Represented by

Inequalities 16 6 Exercises 16 9

• Chapte r Exercises A and B 17 0

Chapter 4 Function s 17 3

Section 1 Quadrati c Functions 17 4

1 Function s and Graphs 17 4 2 Quadrati c Functions and Their Graphs 17 6 3 Equation s and Inequalities and the

Graphs of Functions 18 4 Exercises 19 1

Section 2 Simpl e Fractional Functions and Irrational Functions 19 3

1 Fractiona l Functions and Their Graphs 19 3 2 Irrationa l Functions and Their Graphs 19 7 3 Invers e Functions 20 3 Exercises 20 8

• Chapte r Exercises A and B 20 9

XIV

Chapter 5 Trigonometri e Ratio s 21 1

Section 1 Trigonometri e Ratios 21 2

1 Tangen t 21 2 2 Sin e and Cosine 21 4 3 Relation s among Trigonometrie Ratios 21 7 4 Extendin g Trigonometrie Ratios 21 9 Exercises 22 5

Section 2 Application s of Trigonometrie Ratios 22 7

1 Th e Sine Theorem 22 7 2 Th e Cosine Theorem 23 0 3 Th e Area of a Triangle 23 2 Exercises 23 5

• Chapte r Exercises A and B 23 6

A p p e n d i x 23 9

Answers 24 0

Index 24 3

Greek Letter s 24 5

Numerical Tables

Table of Squares, Square Roots , and Reciprocals 24 6

Table o f Trigonometri e Ratio s 24 7

To th e Studen t

(Example )

(Demonstration

(Note:

(Problem 1 )

Exercises

Chapter Exercise s

Reference

This marker designates a concrete example to help you understand the text.

J Thi s heading precedes a Standard problem for better understanding of the material. Boxe s labeled [Solution] and [Proof ] giv e mode l answers .

This marker indicates an explanation to help you understand a particular point.

Problems for rapid mastery of current material and for introducing new material appear in the text with this label.

At the end of each section problems are provided for practice with the material in that section.

At the end of each chapter problems are provided for review of the entire chapter and practical application of the material. A problems involve primarily basic material, while 3 problem s are a bit more advanced.

Refers to topics from other elective courses.

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