permanence of form, the discussion of the irrational, the theory of fractional exponents, and complex numbers. The principle of the permanence of form arises in extending the fundamental laws and operations which are applicable to positive integers to the new numbers which arise in Algebra-zero, the negative, the fraction, irrational and complex numbers. This point of view was first suggested by Peacock in his Arithmetic and Symbolic Algebra in 1842, and fully developed for the negative, the fraction, and the imaginary by Hankel in his Complexe Zahlensysteme, in 1867, and was completed by Cantor's theory of the irrational in 1871. A careful distinction is made between an equation and an identity. It is pointed out that the solution of an equation or a system of equations depends upon one's ability to construct equivalent equations or equivalent systems of equations; that the same is true in solving inequalities or systems of inequalities. The great central problem of Algebra is the solution of the equation, and with the irrational and complex numbers the system of algebraic numbers is complete. Full discussions are given of equivalent systems of simultaneous quadratic equations, of the graphs of their solutions and of the equations themselves, and of problems in maximum and minimum values of fractions which can be solved by means of quadratic equations. Following the principle of the permanence of form and the equation, are infinite series and their properties, the tests for their convergence and divergence, the expansion of fractions into infinite series and their summation. Because of their value in discussing the properties of infinite series and other problems, emphasis is placed upon the method of mathematical induction, upon the properties of a variable and its limit, and upon the rigorous proof of the theorem of undetermined coefficients. Attention is directed to the fact that the sum of an infinite series is the limit of a variable sum; to the geometric illustrations of the derivation and meaning of each theorem; to the distinction between absolutely and conditionally convergent series; to the fact that an absolutely convergent series may be treated like any other number in algebraic calculation, a property which a conditionally convergent series does not have; and finally to the value of infinite convergent series in numerical calculation. The author herewith expresses his obligations to the many others who have preceded him in this field, some of whose works he has used in the classroom for many years. He would mention especially Dr. E. Bardey, who courteously granted permission to use exercises from his Aufgabensammlung. He also desires to express his indebtedness to Dr. F. R. Moulton for his special care in reading proofs of the book, to Dr. Henry Gale for his assistance in reading the manuscript and proofs of Book IV, and to Mr. A. W. Smith for critically reading a part of the manuscript. THE UNIVERSITY OF CHICAGO, MAY, 1901. JAMES HARRINGTON BOYD. PREFACE TABLE OF CONTENTS Book I THE FUNDAMENTAL OPERATIONS OF COMMON ALGEBRA— INTRODUCTION The Nature of Numbers and the Fundamental Postulate of Arithmetic The Equality of Two Groups Symbolic Representation of Numbers The Equation and Inequalities Counting CHAPTER I ADDITION AND MULTIPLICATION The Sum of Two Groups-Parenthesis Addition of Positive Integers and its Fundamental Laws 11 12 13 14 15 17 18 19 21 24 27 28 29 30 35 36 CHAPTER III SUBTRACTION AND THE NEGATIVE INTEGER-GENERALIZED DISCUSSION Numerical Subtraction Determinateness of Numerical Subtraction-Formal Rules of Subtraction 40 Division of Polynomials - First and Second Rules Third and Fourth Rules of Division Indeterminateness of Division by 0-Determinateness of Symbolic Division 69 The Vanishing of a Product 70 61 67 CHAPTER VI APPLICATIONS OF THE FUNDAMENTAL OPERATIONS-SIMPLE EQUATIONS An Identity-An Equation of Condition-The Unknown Quantity |