College Algebra, With Applications (Classic Reprint)

Nothing causes more trouble to young inatliematicians than the traditional method of introducing complex numbers into algebra. The student knows that the square root of minus one is neither a positive nora negative number, and that it is not equal to zero. Nobody has ever told him that there are any numbers which are neither positive, nor negative, nor zero. In spite of this, square roots of negative numbers are introduced for him to reckon with. Of course he does not know what they mean and he becomes suspicious. Even later, if a concrete representation of these socalled imaginary numbers is ever presented to him, he never quite gets over his first suspicions ;the uncomfortable feeling remains within him that, somehow, he has been cheated, that imaginary numbers are impossible and without meaning. It is historically correct to introduce imaginaries in this way. But the students doubts are also justified historically. His suspicious attitude toward imaginary numbers is merely a repetition of the position taken by the whole mathematical world from the fifteenth to the nineteenth century. The historical order of presentation is not ahvays the best pedagogic order, and the subject of complex numbers is one of the most striking instances ichere the historical order should be avoided. In Chapter I of the present book, the imniber system of algebra, up to and including the system of complex numbers, is developed in a concrete and convincing way by means of the geometry of directed line-segments or vectors. No student will feel any doubt concerning the legitimacy of complex numbers, after mastering this chapter, unless his suspicions have been previously aroused in his first course in algebra. In a sense, Chapter I may be regarded as devoted to the foundations of algebra. It is not, however, a chapter on foundations in the formal technical sense in which that word
(Typographical errors above are due to OCR software and don't occur in the book.)