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PREFACE
The theory of equations is not only a necessity in the subsequent mathematical
courses and their applications, but furnishes an illuminating sequel to
geometry, algebra and analytic geometry. Moreover, it develops anew and in
greater detail various fundamental ideas of calculus for the simple, but important,
case of polynomials. The theory of equations therefore affords a useful
supplement to differential calculus whether taken subsequently or simultaneously.
It was to meet the numerous needs of the student in regard to his earlier and
future mathematical courses that the present book was planned with great care
and after wide consultation. It differs essentially from the author’s Elementary
Theory of Equations, both in regard to omissions and additions, and since it
is addressed to younger students and may be used parallel with a course in
differential calculus. Simpler and more detailed proofs are now employed.
The exercises are simpler, more numerous, of greater variety, and involve more
practical applications.
This book throws important light on various elementary topics. For example,
an alert student of geometry who has learned how to bisect any angle
is apt to ask if every angle can be trisected with ruler and compasses and if
not, why not. After learning how to construct regular polygons of 3, 4, 5, 6,
8 and 10 sides, he will be inquisitive about the missing ones of 7 and 9 sides.
The teacher will be in a comfortable position if he knows the facts and what
is involved in the simplest discussion to date of these questions, as given in
Chapter III. Other chapters throw needed light on various topics of algebra. In
particular, the theory of graphs is presented in Chapter V in a more scientific
and practical manner than was possible in algebra and analytic geometry.
There is developed a method of computing a real root of an equation with
minimum labor and with certainty as to the accuracy of all the decimals obtained.
We first find by Horner’s method successive transformed equations
whose number is half of the desired number of significant figures of the root.
The final equation is reduced to a linear equation by applying to the constant
term the correction computed from the omitted terms of the second and
iv PREFACE
higher degrees, and the work is completed by abridged division. The method
combines speed with control of accuracy.
Newton’s method, which is presented from both the graphical and the
numerical standpoints, has the advantage of being applicable also to equations
which are not algebraic; it is applied in detail to various such equations.
In order to locate or isolate the real roots of an equation we may employ a
graph, provided it be constructed scientifically, or the theorems of Descartes,
Sturm, and Budan, which are usually neither stated, nor proved, correctly.
The long chapter on determinants is independent of the earlier chapters.
The theory of a general system of linear equations is here presented also from
the standpoint of matrices.
For valuable suggestions made after reading the preliminary manuscript of
this book, the author is greatly indebted to Professor Bussey of the University
of Minnesota, Professor Roever of Washington University, Professor Kempner
of the University of Illinois, and Professor Young of the University of Chicago.
The revised manuscript was much improved after it was read critically by
Professor Curtiss of Northwestern University. The author’s thanks are due
also to Professor Dresden of the University of Wisconsin for various useful
suggestions on the proof-sheets.
The theory of equations is not only a necessity in the subsequent mathematical
courses and their applications, but furnishes an illuminating sequel to
geometry, algebra and analytic geometry. Moreover, it develops anew and in
greater detail various fundamental ideas of calculus for the simple, but important,
case of polynomials. The theory of equations therefore affords a useful
supplement to differential calculus whether taken subsequently or simultaneously.
It was to meet the numerous needs of the student in regard to his earlier and
future mathematical courses that the present book was planned with great care
and after wide consultation. It differs essentially from the author’s Elementary
Theory of Equations, both in regard to omissions and additions, and since it
is addressed to younger students and may be used parallel with a course in
differential calculus. Simpler and more detailed proofs are now employed.
The exercises are simpler, more numerous, of greater variety, and involve more
practical applications.
This book throws important light on various elementary topics. For example,
an alert student of geometry who has learned how to bisect any angle
is apt to ask if every angle can be trisected with ruler and compasses and if
not, why not. After learning how to construct regular polygons of 3, 4, 5, 6,
8 and 10 sides, he will be inquisitive about the missing ones of 7 and 9 sides.
The teacher will be in a comfortable position if he knows the facts and what
is involved in the simplest discussion to date of these questions, as given in
Chapter III. Other chapters throw needed light on various topics of algebra. In
particular, the theory of graphs is presented in Chapter V in a more scientific
and practical manner than was possible in algebra and analytic geometry.
There is developed a method of computing a real root of an equation with
minimum labor and with certainty as to the accuracy of all the decimals obtained.
We first find by Horner’s method successive transformed equations
whose number is half of the desired number of significant figures of the root.
The final equation is reduced to a linear equation by applying to the constant
term the correction computed from the omitted terms of the second and
iv PREFACE
higher degrees, and the work is completed by abridged division. The method
combines speed with control of accuracy.
Newton’s method, which is presented from both the graphical and the
numerical standpoints, has the advantage of being applicable also to equations
which are not algebraic; it is applied in detail to various such equations.
In order to locate or isolate the real roots of an equation we may employ a
graph, provided it be constructed scientifically, or the theorems of Descartes,
Sturm, and Budan, which are usually neither stated, nor proved, correctly.
The long chapter on determinants is independent of the earlier chapters.
The theory of a general system of linear equations is here presented also from
the standpoint of matrices.
For valuable suggestions made after reading the preliminary manuscript of
this book, the author is greatly indebted to Professor Bussey of the University
of Minnesota, Professor Roever of Washington University, Professor Kempner
of the University of Illinois, and Professor Young of the University of Chicago.
The revised manuscript was much improved after it was read critically by
Professor Curtiss of Northwestern University. The author’s thanks are due
also to Professor Dresden of the University of Wisconsin for various useful
suggestions on the proof-sheets.
CONTENTS
Numbers refer to pages.
CHAPTER I
Complex Numbers
Square Roots, 1. Complex Numbers, 1. Cube Roots of Unity, 3. Geometrical
Representation, 3. Product, 4. Quotient, 5. De Moivre’s Theorem, 5. Cube
Roots, 6. Roots of Complex Numbers, 7. Roots of Unity, 8. Primitive Roots of
Unity, 9.
CHAPTER II
Theorems on Roots of Equations
Quadratic Equation, 13. Polynomial, 14. Remainder Theorem, 14. Synthetic
Division, 16. Factored Form of a Polynomial, 18. Multiple Roots, 18. Identical
Polynomials, 19. Fundamental Theorem of Algebra, 20. Relations between Roots
and Coefficients, 20. Imaginary Roots occur in Pairs, 22. Upper Limit to the Real
Roots, 23. Another Upper Limit to the Roots, 24. Integral Roots, 27. Newton’s
Method for Integral Roots, 28. Another Method for Integral Roots, 30. Rational
Roots, 31.
CHAPTER III
Constructions with Ruler and Compasses
Impossible Constructions, 33. Graphical Solution of a Quadratic Equation, 33.
Analytic Criterion for Constructibility, 34. Cubic Equations with a Constructible
Root, 36. Trisection of an Angle, 38. Duplication of a Cube, 39. Regular Polygon
of 7 Sides, 39. Regular Polygon of 7 Sides and Roots of Unity, 40. Reciprocal
Equations, 41. Regular Polygon of 9 Sides, 43. The Periods of Roots of Unity, 44.
Regular Polygon of 17 Sides, 45. Construction of a Regular Polygon of 17 Sides, 47.
Regular Polygon of n Sides, 48.
v
vi CONTENTS
CHAPTER IV
Cubic and Quartic Equations
Reduced Cubic Equation, 51. Algebraic Solution of a Cubic, 51. Discriminant,
53. Number of Real Roots of a Cubic, 54. Irreducible Case, 54. Trigonometric
Solution of a Cubic, 55. Ferrari’s Solution of the Quartic Equation, 56.
Resolvent Cubic, 57. Discriminant, 58. Descartes’ Solution of the Quartic Equation,
59. Symmetrical Form of Descartes’ Solution, 60.
CHAPTER V
The Graph of an Equation
Use of Graphs, 63. Caution in Plotting, 64. Bend Points, 64. Derivatives, 66.
Horizontal Tangents, 68. Multiple Roots, 68. Ordinary and Inflexion Tangents, 70.
Real Roots of a Cubic Equation, 73. Continuity, 74. Continuity of Polynomials, 75.
Condition for a Root Between a and b, 75. Sign of a Polynomial at Infinity, 77.
Rolle’s Theorem, 77.
CHAPTER VI
Isolation of Real Roots
Purpose and Methods of Isolating the Real Roots, 81. Descartes’ Rule of
Signs, 81. Sturm’s Method, 85. Sturm’s Theorem, 86. Simplifications of Sturm’s
Functions, 88. Sturm’s Functions for a Quartic Equation, 90. Sturm’s Theorem
for Multiple Roots, 92. Budan’s Theorem, 93.
CHAPTER VII
Solution of Numerical Equations
Horner’s Method, 97. Newton’s Method, 102. Algebraic and Graphical Discussion,
103. Systematic Computation, 106. For Functions not Polynomials, 108.
Imaginary Roots, 110.
CHAPTER VIII
Determinants; Systems of Linear Equations
Solution of 2 Linear Equations by Determinants, 115. Solution of 3 Linear Equations
by Determinants, 116. Signs of the Terms of a Determinant, 117. Even and
Odd Arrangements, 118. Definition of a Determinant of Order n, 119. Interchange
of Rows and Columns, 120. Interchange of Two Columns, 121. Interchange of Two
Rows, 122. Two Rows or Two Columns Alike, 122. Minors, 123. Expansion, 123.
Removal of Factors, 125. Sum of Determinants, 126. Addition of Columns or
Rows, 127. System of n Linear Equations in n Unknowns, 128. Rank, 130. System
of n Linear Equations in n Unknowns, 130. Homogeneous Equations, 134.
System of m Linear Equations in n Unknowns, 135. Complementary Minors, 137.
CONTENTS vii
Laplace’s Development by Columns, 137. Laplace’s Development by Rows, 138.
Product of Determinants, 139.
CHAPTER IX
Symmetric Functions
Sigma Functions, Elementary Symmetric Functions, 143. Fundamental Theorem,
144. Functions Symmetric in all but One Root, 147. Sums of Like Powers
of the Roots, 150. Waring’s Formula, 152. Computation of Sigma Functions, 156.
Computation of Symmetric Functions, 157.
CHAPTER X
Elimination, Resultants And Discriminants
Elimination, 159. Resultant of Two Polynomials, 159. Sylvester’s Method of
Elimination, 161. Bézout’s Method of Elimination, 164. General Theorem on
Elimination, 166. Discriminants, 167.
APPENDIX
Fundamental Theorem of Algebra
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Numbers refer to pages.
CHAPTER I
Complex Numbers
Square Roots, 1. Complex Numbers, 1. Cube Roots of Unity, 3. Geometrical
Representation, 3. Product, 4. Quotient, 5. De Moivre’s Theorem, 5. Cube
Roots, 6. Roots of Complex Numbers, 7. Roots of Unity, 8. Primitive Roots of
Unity, 9.
CHAPTER II
Theorems on Roots of Equations
Quadratic Equation, 13. Polynomial, 14. Remainder Theorem, 14. Synthetic
Division, 16. Factored Form of a Polynomial, 18. Multiple Roots, 18. Identical
Polynomials, 19. Fundamental Theorem of Algebra, 20. Relations between Roots
and Coefficients, 20. Imaginary Roots occur in Pairs, 22. Upper Limit to the Real
Roots, 23. Another Upper Limit to the Roots, 24. Integral Roots, 27. Newton’s
Method for Integral Roots, 28. Another Method for Integral Roots, 30. Rational
Roots, 31.
CHAPTER III
Constructions with Ruler and Compasses
Impossible Constructions, 33. Graphical Solution of a Quadratic Equation, 33.
Analytic Criterion for Constructibility, 34. Cubic Equations with a Constructible
Root, 36. Trisection of an Angle, 38. Duplication of a Cube, 39. Regular Polygon
of 7 Sides, 39. Regular Polygon of 7 Sides and Roots of Unity, 40. Reciprocal
Equations, 41. Regular Polygon of 9 Sides, 43. The Periods of Roots of Unity, 44.
Regular Polygon of 17 Sides, 45. Construction of a Regular Polygon of 17 Sides, 47.
Regular Polygon of n Sides, 48.
v
vi CONTENTS
CHAPTER IV
Cubic and Quartic Equations
Reduced Cubic Equation, 51. Algebraic Solution of a Cubic, 51. Discriminant,
53. Number of Real Roots of a Cubic, 54. Irreducible Case, 54. Trigonometric
Solution of a Cubic, 55. Ferrari’s Solution of the Quartic Equation, 56.
Resolvent Cubic, 57. Discriminant, 58. Descartes’ Solution of the Quartic Equation,
59. Symmetrical Form of Descartes’ Solution, 60.
CHAPTER V
The Graph of an Equation
Use of Graphs, 63. Caution in Plotting, 64. Bend Points, 64. Derivatives, 66.
Horizontal Tangents, 68. Multiple Roots, 68. Ordinary and Inflexion Tangents, 70.
Real Roots of a Cubic Equation, 73. Continuity, 74. Continuity of Polynomials, 75.
Condition for a Root Between a and b, 75. Sign of a Polynomial at Infinity, 77.
Rolle’s Theorem, 77.
CHAPTER VI
Isolation of Real Roots
Purpose and Methods of Isolating the Real Roots, 81. Descartes’ Rule of
Signs, 81. Sturm’s Method, 85. Sturm’s Theorem, 86. Simplifications of Sturm’s
Functions, 88. Sturm’s Functions for a Quartic Equation, 90. Sturm’s Theorem
for Multiple Roots, 92. Budan’s Theorem, 93.
CHAPTER VII
Solution of Numerical Equations
Horner’s Method, 97. Newton’s Method, 102. Algebraic and Graphical Discussion,
103. Systematic Computation, 106. For Functions not Polynomials, 108.
Imaginary Roots, 110.
CHAPTER VIII
Determinants; Systems of Linear Equations
Solution of 2 Linear Equations by Determinants, 115. Solution of 3 Linear Equations
by Determinants, 116. Signs of the Terms of a Determinant, 117. Even and
Odd Arrangements, 118. Definition of a Determinant of Order n, 119. Interchange
of Rows and Columns, 120. Interchange of Two Columns, 121. Interchange of Two
Rows, 122. Two Rows or Two Columns Alike, 122. Minors, 123. Expansion, 123.
Removal of Factors, 125. Sum of Determinants, 126. Addition of Columns or
Rows, 127. System of n Linear Equations in n Unknowns, 128. Rank, 130. System
of n Linear Equations in n Unknowns, 130. Homogeneous Equations, 134.
System of m Linear Equations in n Unknowns, 135. Complementary Minors, 137.
CONTENTS vii
Laplace’s Development by Columns, 137. Laplace’s Development by Rows, 138.
Product of Determinants, 139.
CHAPTER IX
Symmetric Functions
Sigma Functions, Elementary Symmetric Functions, 143. Fundamental Theorem,
144. Functions Symmetric in all but One Root, 147. Sums of Like Powers
of the Roots, 150. Waring’s Formula, 152. Computation of Sigma Functions, 156.
Computation of Symmetric Functions, 157.
CHAPTER X
Elimination, Resultants And Discriminants
Elimination, 159. Resultant of Two Polynomials, 159. Sylvester’s Method of
Elimination, 161. Bézout’s Method of Elimination, 164. General Theorem on
Elimination, 166. Discriminants, 167.
APPENDIX
Fundamental Theorem of Algebra
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
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