0 SOME BASIC CONCEPTS

0．1．Introductory remarks

0．2．Sets and subsets

0．3．On the logical method

1 NUMERICAL SYSTEMS

1．1．Natural numbers

1．2．Systems of numeration．Decimal scale

1．3．Addition of natural numbers

1．4．Multiplication of natural numbers

1．5．Multiples and divisors

1．6．Prime and composite numbers

1．7．Greatest common divisor and least common multiple

1．8．Number bases

1．9．Rational fractions

1．10．Operations on fractions

1．11．Decimal representation of fractions

1．12．Concepts of irrational numbers

1．13．Summary

2 THE SYSTEM OF REAL NUMBERS

2．1．Introduction

2．2．Field properties of real numbers

2．3．Summary

3 OPERATIONS ON SIGNED NUMBERS

3．1．Some order properties of real numbers

3．2．Addition of signed numbers

3．3．Subtraction of signed numbers

3．4．Multiplication of signed numbers

3．5．Powers of signed numbers

3．6．Summary

4 ALGEBRAIC EXPRESSIONS

4．1．Constants，variables，and parameters

4．2．Algebraic expressions

4．3．Degree of a polynomial

4．4．Numerical value of an algebraic expression

4．5．Ratio and proportion

4．6．Summary

5 FUNDAMENTAL OPERATIONS ON POLYNOMIALS

5．1．Introduction

5．2．Addition of polynomials

5．3．Subtraction of polynomials

5．4．Multiplication of polynomials

5．5．Division of polynomials

5．6．Factoring polynomials

5．7．Special products

5．8．Simple types of factoring

5．9．Special devices

5．10．Factoring the general quadratic trinomial

5．11．Summary

6 RATIONAL FRACTIONS

6．1．Definitions

6．2．Reduction of fractions

6．3．Addition and subtraction of fractions

6．4．Multiplication and division of fractions

6．5．Complex fractions

6．6．Summary

7 EXPONENTS AND RADICALS

7．1．Introduction

7．2．Rational exponents

7．3．Laws for radicals

7．4．Operations with radicals

7．5．Real exponents

7．6．Irrational algebraic expressions

7．7．Summary

8 RELATIONS AND FUNCTIONS

8．1．Some properties of complete ordered fields—the real number system

8．2．Geometric representations of the real numbers

8．3．Coordinates of points on a line

8．4．On the language of sets

8．5．Cartesian coordinates of points in a plane

8．6．Relations

8．7．Identities and equations

8．8．Linear equations

8．9．Functions

8．10．Variation

8．11．Summary

9 COMPLEX NUMBERS

9．1．Introduction

9．2．Equality of complex numbers

9．3．Operations on complex numbers

9．4．Absolute value of complex numbers

9．5．Square roots of complex numbers

9．6．Geometric representation of complex numbers

9．7．Euclidean coordinate geometry

9．8．Summary

10 ALGEBRAIC EQUATIONS

10．1．Introduction

10．2．Transformation of equations

10．3．Solution of linear equations

10．4．Solution of quadratic equations

10．5．Some properties of the roots of quadratic equations

10．6．Equations of higher degree

10．7．Summary

11 MATRICES，DETERMINANTS，AND SYSTEMS OF LINEAR EQUATIONS

11．1．Matrices

11．2．Different types of matrices

11．3．Equality of matrices and operations on matrices

11．4．Systems of equations

11．5．Solution of a system of linear equations by addition and subtraction

11．6．Systems of linear equations in matrix form

11．7．The inverse of a square matrix

11．8．Determinants

11．9．A formula for the inverse of a square matrix．Cramer＇s rule

11．10．Summary

12 PROPERTIES OF POLYNOMIALS IN ONE VARIABLE

12．1．Introduction

12．2．Theorem of the remainder

12．3．Divisibility

12．4．Factored form of the polynomial

12．5．Polynomial identities

12．6．Method of undetermined coefficients

12．7．Synthetic division

12．8．Polynomials expressed as polynomials in x—k

12．9．Summary

13 SOLUTION OF REAL NUMERICAL POLYNOMIAL EQUATIONS

13．1．Introduction

13．2．Upper and lower bounds of the roots

13．3．Descartes＇ rule of signs

13．4．Graphs of polynomials

13．5．Locating real roots

13．6．Rational roots of an equation

13．7．Irrational roots．Horner＇s method

13．8．Summary

14 SYSTEMS OF HIGHER DEGREE EQUATIONS

14．1．Introduction

14．2．Method of adjunction

14．3．Graphical interpretation

14．4．Problems involving systems of equations

14．5．Summary

15 INEQUALITIES

15．1．Introduction

15．2．Absolute inequalities

15．3．Conditional inequalities

15．4．Summary

16 SEQUENCES AND SERIES

16．1．Introduction

16．2．Arithmetic progressions

16．3．Harmonic progressions

16．4．Geometric progressions

16．5．Infinite geometric series

16．6．Summary

17 APPROXIMATIONS AND LOGARITHMS

17．1．Introduction

17．2．Intervals

17．3．Absolute and relative errors in approximations

17．4．Trustworthy and significant digits

17．5．Determination of the trustworthy digits of an approximation

17．6．Precision and accuracy of approximations

17．7．On the arithmetic of approximations

17．8．General method of calculation with approximations

17．9．Logarithms

17．10．Common logarithms—base 10

17．11．Some uses of common logarithms

17．12．Change of base of logarithms

APPENDIX：ON THE FOUNDATIONS OF ARITHMETIC

A．1 Introduction

A．2 Undefined elements

A．3 Assumptions

A．4 Some fundamental definitions

A．5 Mathematical induction

A．6 Exhaustion of a set

A．7 Properties of equivalent sets

A．8 Fundamental sequence

A．9 Cardinal and ordinal numbers

A．10 Equality and inequality of natural numbers

A．11 Addition of sets

A．12 Properties of set addition

A．13 Addition of numbers

A．14 Product of two sets

A．15 Multiplication of numbers

A．16 Peano＇s postulates for natural numbers

ANSWERS

INDEX