Why do we need the real numbers? How should we construct them? These questions arose in the nineteenth century, along with the ideas and techniques needed to address them. Nowadays it is commonplace for apprentice mathematicians to hear 'we shall assume the standard properties of the real numbers' as part of their training. But exactly what are those properties? And why can we assume them? This book is clearly and entertainingly written for those students, with historical asides and exercises to foster understanding. Starting with the natural (counting) numbers and then looking at the rational numbers (fractions) and negative numbers, the author builds to a careful construction of the real numbers followed by the complex numbers, leaving the reader fully equipped with all the number systems required by modern mathematical analysis. Additional chapters on polynomials and quarternions provide further context for any reader wanting to delve deeper.
Why do we need the real numbers? How should we construct them? These questions arose in the nineteenth century, along with the ideas and techniques needed to address them. Nowadays it is commonplace for apprentice mathematicians to hear 'we shall assume the standard properties of the real numbers' as part of their training. But exactly what are those properties? And why can we assume them? This book is clearly and entertainingly written for those students, with historical asides and exercises to foster understanding. Starting with the natural (counting) numbers and then looking at the rational numbers (fractions) and negative numbers, the author builds to a careful construction of the real numbers followed by the complex numbers, leaving the reader fully equipped with all the number systems required by modern mathematical analysis. Additional chapters on polynomials and quarternions provide further context for any reader wanting to delve deeper.
"This work focuses on the number theory of quadratic irrationalities in various forms, including continued fractions, orders in quadratic number fields, and binary quadratic forms. It presents classic
Trading floors are a thing of the past. Thanks to a combination of computers, high-speed networks and algorithms, millions of financial transactions now happen in fractions of a second. This book studies the automation of stock markets in the United Kingdom and the United States of America, identifying the invisible actors, devices, and politics that were central to the creation of electronic trading. In addition to offering a detailed account of how stock exchanges wrestled with technology, the book also invites readers to rethink the nature of markets in modern societies. Markets, it argues, are sites for the creation of relations, and in studying how these relations changed through technology, the book highlights the sources, dynamics, and consequences of automation. In this respect, the book is both a history of automation in finance and a sociological analysis of the way in which automation gradually changed the lives and work of key financial actors.
