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First Course in the Theory of Equations
The theory of equations is not onlj'^ a necessity in the subsequent
mathematical courses and their apphcations, 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 apphcations.
This book throws important hght 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 transiv PREFACE
formed 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 higher degrees, and the work is completed
b3^ 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.
mathematical courses and their apphcations, 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 apphcations.
This book throws important hght 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 transiv PREFACE
formed 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 higher degrees, and the work is completed
b3^ 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.
166 pages, Hardcover
First published September 1, 2009
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About the author
Leonard Eugene Dickson
71 books4 followersLeonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, History of the Theory of Numbers.
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Displaying 1 of 1 review
December 21, 2020
Even though the book is labeled "First Course..", I found it too basic. Neither is the prose good, questions good nor the content is rigorous enough. Had to go through, because it was the recommended reading in my college.
Displaying 1 of 1 review