College Algebra

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Excerpt from College Algebra: With Applications Nothing causes more trouble to young mathematicians than the traditional method of introducing complex numbers into algebra. The student knows that the square root of minus one is neither a positive nor a 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 so-called "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 always the best pedagogic order, and the subject of complex numbers is one of the most striking instances where the historical order should be avoided. In Chapter I of the present book, the number 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 is being used at the present time. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.