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The Project Gutenberg EBook of First Course in the Theory of Equations, byLeonard Eugene DicksonThis eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever. You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at www.gutenberg.orgTitle: First Course in the Theory of EquationsAuthor: Leonard Eugene DicksonRelease Date: August 25, 2009 [EBook #29785]Language: EnglishCharacter set encoding: ISO-8859-1*** START OF THIS PROJECT GUTENBERG EBOOK THEORY OF EQUATIONS ***

Produced by Peter Vachuska, Andrew D. Hwang, Dave Morgan,and the Online Distributed Proofreading Team athttp://www.pgdp.netTranscriber’s NoteThis PDF file is formatted for printing, but may be easily formattedfor screen viewing. Please see the preamble of the LATEX source file forinstructions.Table of contents entries and running heads have been normalized.Archaic spellings (constructible, parallelopiped) and variants(coordinates/coördinates, two-rowed/2-rowed, etc.) have been retainedfrom the original.Minor typographical corrections, and minor changes to thepresentational style, have been made without comment. Figures mayhave been relocated slightly with respect to the surrounding text.

FIRST COURSEIN THETHEORY OF EQUATIONSBYLEONARD EUGENE DICKSON, Ph.D.CORRESPONDANT DE L’INSTITUT DE FRANCEPROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGONEW YORKJOHNWILEY&SONS,Inc.London: CHAPMAN & HALL, Limited

Copyright, 1922,byLEONARD EUGENE DICKSONAll Rights ReservedThis book or any part thereof must notbe reproduced in any form withoutthe written permission of the publisher.Printed in U. S. A.PRESS OFBRAUNWORTH & CO., INC.BOOK MANUFACTURERSBROOKLYN, NEW YORK

PREFACEThe theory of equations is not only a necessity in the subsequent mathematical courses and their applications, but furnishes an illuminating sequel togeometry, algebra and analytic geometry. Moreover, it develops anew and ingreater detail various fundamental ideas of calculus for the simple, but important, case of polynomials. The theory of equations therefore affords a usefulsupplement to differential calculus whether taken subsequently or simultaneously.It was to meet the numerous needs of the student in regard to his earlier andfuture mathematical courses that the present book was planned with great careand after wide consultation. It differs essentially from the author’s ElementaryTheory of Equations, both in regard to omissions and additions, and since itis addressed to younger students and may be used parallel with a course indifferential calculus. Simpler and more detailed proofs are now employed.The exercises are simpler, more numerous, of greater variety, and involve morepractical applications.This book throws important light on various elementary topics. For example, an alert student of geometry who has learned how to bisect any angleis apt to ask if every angle can be trisected with ruler and compasses and ifnot, 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 whatis involved in the simplest discussion to date of these questions, as given inChapter III. Other chapters throw needed light on various topics of algebra. Inparticular, the theory of graphs is presented in Chapter V in a more scientificand practical manner than was possible in algebra and analytic geometry.There is developed a method of computing a real root of an equation withminimum labor and with certainty as to the accuracy of all the decimals obtained. We first find by Horner’s method successive transformed equationswhose 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

ivPREFACEhigher degrees, and the work is completed by abridged division. The methodcombines speed with control of accuracy.Newton’s method, which is presented from both the graphical and thenumerical standpoints, has the advantage of being applicable also to equationswhich 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 agraph, 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 fromthe standpoint of matrices.For valuable suggestions made after reading the preliminary manuscript ofthis book, the author is greatly indebted to Professor Bussey of the Universityof Minnesota, Professor Roever of Washington University, Professor Kempnerof the University of Illinois, and Professor Young of the University of Chicago.The revised manuscript was much improved after it was read critically byProfessor Curtiss of Northwestern University. The author’s thanks are duealso to Professor Dresden of the University of Wisconsin for various usefulsuggestions on the proof-sheets.Chicago, 1921.

CONTENTSNumbers refer to pages.CHAPTER IComplex NumbersSquare Roots, 1. Complex Numbers, 1. Cube Roots of Unity, 3. GeometricalRepresentation, 3. Product, 4. Quotient, 5. De Moivre’s Theorem, 5. CubeRoots, 6. Roots of Complex Numbers, 7. Roots of Unity, 8. Primitive Roots ofUnity, 9.CHAPTER IITheorems on Roots of EquationsQuadratic Equation, 13. Polynomial, 14. Remainder Theorem, 14. SyntheticDivision, 16. Factored Form of a Polynomial, 18. Multiple Roots, 18. IdenticalPolynomials, 19. Fundamental Theorem of Algebra, 20. Relations between Rootsand Coefficients, 20. Imaginary Roots occur in Pairs, 22. Upper Limit to the RealRoots, 23. Another Upper Limit to the Roots, 24. Integral Roots, 27. Newton’sMethod for Integral Roots, 28. Another Method for Integral Roots, 30. RationalRoots, 31.CHAPTER IIIConstructions with Ruler and CompassesImpossible Constructions, 33. Graphical Solution of a Quadratic Equation, 33.Analytic Criterion for Constructibility, 34. Cubic Equations with a ConstructibleRoot, 36. Trisection of an Angle, 38. Duplication of a Cube, 39. Regular Polygonof 7 Sides, 39. Regular Polygon of 7 Sides and Roots of Unity, 40. ReciprocalEquations, 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

viCONTENTSCHAPTER IVCubic and Quartic EquationsReduced 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 VThe Graph of an EquationUse 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 VIIsolation of Real RootsPurpose and Methods of Isolating the Real Roots, 81.Descartes’ Rule ofSigns, 81. Sturm’s Method, 85. Sturm’s Theorem, 86. Simplifications of Sturm’sFunctions, 88. Sturm’s Functions for a Quartic Equation, 90. Sturm’s Theoremfor Multiple Roots, 92. Budan’s Theorem, 93.CHAPTER VIISolution of Numerical EquationsHorner’s Method, 97. Newton’s Method, 102. Algebraic and Graphical Discussion, 103. Systematic Computation, 106. For Functions not Polynomials, 108.Imaginary Roots, 110.CHAPTER VIIIDeterminants; Systems of Linear EquationsSolution of 2 Linear Equations by Determinants, 115. Solution of 3 Linear Equations by Determinants, 116. Signs of the Terms of a Determinant, 117. Even andOdd Arrangements, 118. Definition of a Determinant of Order n, 119. Interchangeof Rows and Columns, 120. Interchange of Two Columns, 121. Interchange of TwoRows, 122. Two Rows or Two Columns Alike, 122. Minors, 123. Expansion, 123.Removal of Factors, 125. Sum of Determinants, 126. Addition of Columns orRows, 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.

viiCONTENTSLaplace’s Development by Columns, 137.Product of Determinants, 139.Laplace’s Development by Rows, 138.CHAPTER IXSymmetric FunctionsSigma Functions, Elementary Symmetric Functions, 143. Fundamental Theorem, 144. Functions Symmetric in all but One Root, 147. Sums of Like Powersof the Roots, 150. Waring’s Formula, 152. Computation of Sigma Functions, 156.Computation of Symmetric Functions, 157.CHAPTER XElimination, Resultants And DiscriminantsElimination, 159. Resultant of Two Polynomials, 159.Elimination, 161. Bézout’s Method of Elimination, 164.Elimination, 166. Discriminants, 167.Sylvester’s Method ofGeneral Theorem onAPPENDIXFundamental Theorem of AlgebraAnswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

First Course inThe Theory of EquationsCHAPTER IComplex Numbers 1. Square Roots. If p is a positive real number, the symbol p is used todenote the positive square root of p. It is most easily computed by logarithms.We shall express the square roots of negative numbers in terms of thesymbol i such that the relation i2 1 holds. Consequently we denote theroots of x2 1 by i and i. The roots of x2 4 are written in the form 2i in preference to 4. In general, if p is positive, the roots of x2 p are written in the form pi in preference to p. The square of either root is thus ( p)2 i2 p. Had we used the less desirable notation p for the roots of x2 p, we might be tempted to find the square ofeither root by multiplying together the values under the radical sign and concludeerroneously thatp p p p2 p. To prevent such errors we use p i and not p.2. Complex Numbers. If a and b are any two real numbers and i2 1,a bi is called a complex number 1 and a bi its conjugate. Either is said tobe zero if a b 0. Two complex numbers a bi and c di are said to beequal if and only if a c and b d. In particular, a bi 0 if and only ifa b 0. If b 6 0, a bi is said to be imaginary. In particular, bi is called apure imaginary.1Complex numbers are essentially couples of real numbers. For a treatment from thisstandpoint and a treatment based upon vectors, see the author’s Elementary Theory ofEquations, p. 21, p. 18.

2[Ch. ICOMPLEX NUMBERSAddition of complex numbers is defined by(a bi) (c di) (a c) (b d)i.The inverse operation to addition is called subtraction, and consists in findinga complex number z such that(c di) z a bi.In notation and value, z is(a bi) (c di) (a c) (b d)i.Multiplication is defined by(a bi)(c di) ac bd (ad bc)i,and hence is performed as in formal algebra with a subsequent reduction bymeans of i2 1. For example,(a bi)(a bi) a2 b2 i2 a2 b2 .Division is defined as the operation which is inverse to multiplication, andconsists in finding a complex number q such that (a bi)q e f i. Multiplyingeach member by a bi, we find that q is, in notation and value,e fi(e f i)(a bi)ae bfaf be 2 2i.a bia2 b2a b2a b2Since a2 b2 0 implies a b 0 when a and b are real, we conclude thatdivision except by zero is possible and unique.EXERCISESExpress as complex numbers1. 9. 3. ( 25 25) 16. 5. 8 2 3. 3 5 .6.2 12. 4.4. 23 .7.3 5i.2 3i8.a bi.a bi9. Prove that the sum of two conjugate complex numbers is real and that theirdifference is a pure imaginary.

§4.]GEOMETRICAL REPRESENTATION310. Prove that the conjugate of the sum of two complex numbers is equal to thesum of their conjugates. Does the result hold true if each word sum is replaced bythe word difference?11. Prove that the conjugate of the product (or quotient) of two complex numbersis equal to the product (or quotient) of their conjugates.12. Prove that, if the product of two complex numbers is zero, at least one ofthem is zero.13. Find two pairs of real numbers x, y for which(x yi)2 7 24i.As in Ex. 13, express as complex numbers the square roots of14. 11 60i.16. 4cd (2c2 2d2 )i.15. 5 12i.3. Cube Roots of Unity. Any complex number x whose cube is equalto unity is called a cube root of unity. Sincex3 1 (x 1)(x2 x 1),the roots of x3 1 are 1 and the two numbers x for whichx2 x 1 0,(x 21 )2 34 , x 12 12 3i.Hence there are three cube roots of unity, viz.,1, ω 12 12 3i, ω 0 21 12 3i.In view of the origin of ω , we have the important relationsω 2 ω 1 0,ω 3 1.Since ωω 0 1 and ω 3 1, it follows that ω 0 ω 2 , ω ω 02 .4. Geometrical Representation of Complex Numbers. Using rectangular axes of coördinates, OX and OY , we represent the complex numbera bi by the point A having the coördinates a, b (Fig. 1).

4[Ch. ICOMPLEX NUMBERS YThe positive number r a2 b2 givingthe length of OA is called the modulus (orabsolute value) of a bi. The angle θ XOA,measured counter-clockwise from OX to OA,is called the amplitude (or argument) of a bi. Thus cos θ a/r, sin θ b/r, whence(1)ArObθaa bi r(cos θ i sin θ).XFig. 1The second member is called the trigonometric form of a bi.For the amplitude we may select, instead of θ, any of the angles θ 360 ,θ 7