Excerpt from The Geometry of the Complex Domain
As an example of (a) we may ask how to find a geometrical representation of the complex points of a line, a circle, or a plane. Question (b) leads to mathematical considerations of a very different order. We usually assume that whatever is true in the real domain is true in the complex one also the properties of the complex portion of a curve are inferred from those of its real trace. If we are asked for our grounds for this erroneous belief, we are inclined to reply Continuity' or 'analytic continuation' or what not. But these vague generalities do not by any means exhaust the question. There are more things in Heaven and Earth than are dreamt of in our philosophy of reals. What, for instance, can be said about the totality of points in the plane such that the sum of the squares of the absolute values of their distances from two mutually perpendicular lines is equal to unity? This is a very numerous family of points indeed, depending on no less than three real parameters, so that it is not contained completely in any one curve, nor is any one curve contained completely therein; it is an absolutely different variety from any curve or system of curves in the plane.
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Paperback. Condition: New. Print on Demand. This book delves into the fascinating history and evolution of representing imaginary numbers in geometry, a challenge that has captivated mathematicians for centuries. The author traces the journey from the early intuitions of the Greeks to the groundbreaking work of mathematicians like Wallis, Kuhn, Wessel, Argand, and Gauss, who each contributed unique insights into understanding and visualizing complex numbers. The central theme revolves around the quest to find meaningful geometric interpretations for complex numbers, exploring various approaches and their implications. The author delves into the concept of chains, a system of points with specific properties, and their representation in the complex plane. The book also explores the geometry of the binary domain, examining collineations and anti-collineations, and their role in understanding transformations and invariants. Moving beyond the binary domain, the author investigates the representation of complex points in three-dimensional space, introducing concepts like the Marie representation and the Laguerre representation. The book concludes with a discussion of the von Staudt theory, which provides a foundation for understanding complex elements in pure geometry. Through its comprehensive exploration of historical developments and geometric interpretations, this book illuminates the profound impact of complex numbers on our understanding of mathematics and the world around us. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9780282529239_0
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PAP. Condition: New. New Book. Shipped from UK. Established seller since 2000. Seller Inventory # LW-9780282529239
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PAP. Condition: New. New Book. Shipped from UK. Established seller since 2000. Seller Inventory # LW-9780282529239
Quantity: 15 available