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Unknown Quantity
Prime Obsession taught us not to be afraid to put the math in a math book. Unknown Quantity heeds the lesson well. So grab your graphing calculators, slip out the slide rules, and buckle up! John Derbyshire is introducing us to algebra through the ages-and it promises to be just what his die-hard fans have been waiting for. \"Here is the story of algebra.\" With this deceptively simple introduction, we begin our journey. Flanked by formulae, shadowed by roots and radicals, escorted by an expert who navigates unerringly on our behalf, we are guaranteed safe passage through even the most treacherous mathematical terrain. Our first encounter with algebraic arithmetic takes us back 38 centuries to the time of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah. Moving deftly from Abel's proof to the higher levels of abstraction developed by Galois, we are eventually introduced to what algebraists have been focusing on during the last century. As we travel through the ages, it becomes apparent that the invention of algebra was more than the start of a specific discipline of mathematics-it was also the birth of a new way of thinking that clarified both basic numeric concepts as well as our perception of the world around us. Algebraists broke new ground when they discarded the simple search for solutions to equations and concentrated instead on abstract groups. This dramatic shift in thinking revolutionized mathematics. Written for those among us who are unencumbered by a fear of formulae, Unknown Quantity delivers on its promise to present a history of algebra. Astonishing in its bold presentation of the math and graced with narrative authority, our journey through the world of algebra is at once intellectually satisfying and pleasantly challenging.
eBook
Episodes in the History of Modern Algebra (1800–1950)
Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia. The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call \"modern algebra\" is even shorter still. The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a \"rising sea\" in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century. The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.
eBook
Al-Karaji : tenth century mathematician and engineer
Tenth-century mathematician al-Karaji is best known for his writings on algebra and for freeing algebra from geometry. The scholar spent most of his life in Baghdad, where he established a school for algebra and served as a vizier for the Abbasid government. Al-Karaji also was an accomplished engineer who wrote extensively on water extraction. Many of his hydrological ideas are still used in the Middle East today. While some modern scholars question his originality, others maintain he was an important transition between ancient mathematics and modern algebra. Bibliography, Detailed Table of Contents, Full-Color Photographs, Further Information Section, Glossary, Illustrations, Index, Primary Sources, Sidebars, Websites.
Book
SYNTAX AND MEANING AS SENSUOUS, VISUAL, HISTORICAL FORMS OF ALGEBRAIC THINKING
Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and other symbols (e.g. \"+\", \"=\") made it possible to achieve what is considered one of the greatest cultural accomplishments in human history, namely, the constitution of a symbolic algebraic language and the concomitant rise of symbolic thinking. Because of their profound historical ties with natural language, the emerging syntax and meanings of symbolic algebraic language were marked in a definite way by the syntax and meanings of the former. However, at a certain point, these ties were loosened and algebraic symbolism became a language in its own right. Without alluding to the theory of recapitulation, in this paper, we travel back and forth from history to the present to explore key passages in the constitution of the syntax and meanings of symbolic algebraic language. A contextual semiotic analysis of the use of algebraic terms in 9th century Arabic as well as in contemporary students' mathematical activity, sheds some light on the conceptual challenges posed by the learning of algebra.
Journal Article
Una introducció a l’àlgebra al segle XVI: «La regla de la cosa» de Juan Pérez de Moya / Une introduction à l’algèbre au XVIe siècle : « La règle de la chose » de Juan Pérez de Moya
El propòsit del present treball és el de mostrar, a través d'un dels textos més destacats de l'època, com s'introduïa l'àlgebra en l'ensenyament pràctic de la matemàtica. Entre els objectius que es pretenen aconseguir, a més dels derivats del pur coneixement històric dels orígens d'aquest saber, estan els d’apreciar alguns aspectes de caràcter metodològic d'interès per a la didàctica de la matemàtica: el paper de les notacions i la utilització dels símbols en el desenvolupament de la matemàtica, les limitacions que imposava el desconeixement de certs recursos (com és el cas dels nombres decimals), o la importància determinant que els aspectes aplicats tenen en la introducció i posterior desenvolupament de l'àlgebra.___________________________________________________________________Le propos du présent travail est de montrer, au travers de l’un des textes les plus remarquables de l’époque, comment fut introduit l’algèbre dans l’enseignement pratique de la mathé-matique. Parmi les objectifs que nous prétendons atteindre, en plus de ceux découlant de la pure connaissance historique des origines de ce savoir, se trouvent ceux d’apprécier certains aspects de caractère méthodologique d’intérêt pour la didactique de la mathématique : le rôle des notations et l’utilisation des symboles dans le développement de la mathématique, les limitations qu’imposait l’ignorance de certaines ressources —comme c’est le cas des nombres décimaux—, ou encore l’importance déterminante que les aspects appliqués purent avoir dans l’introduction et le développement postérieur de l’algèbre.
This article examines the most notable sixteenth-century treatises on algebra to show how this subject was introduced in mathematics education. The objectives are to explain the particular origins of algebra as a subject in mathematics in this period and to study certain aspects of the methodology used in mathematics education: the role of notation and symbols in the de-velopment of mathematics, the limitations caused by the absence of certain resources (e.g., decimal numbers) and the role played by different factors in the introduction and subsequent development of algebra.
El propósito del presente trabajo es el de mostrar, a través de uno de los textos más destacados de la época, como se introducía el álgebra en la enseñanza práctica de la matemática. Entre los objetivos que se pretenden alcanzar, además de los derivados del puro conocimiento histórico de los orígenes de este saber, están los de apreciar algunos aspectos de carácter metodológico de interés para la didáctica de la matemática: el papel de las notaciones y la utilización de los símbolos en el desarrollo de la matemática, las limitaciones que imponía el desconocimiento de ciertos recursos (como es el caso de los números decimales), o la importancia determinante que los aspectos aplicados tienen en la introducción y posterior desarrollo del álgebra.
Journal Article
Berkeley's philosophy of mathematics
In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of the Scientific Revolution. Jesseph begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a \"science of abstractions.\" Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique treatments of geometry and arithmetic and his famous critique of the calculus in The Analyst. By putting Berkeley's mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science.
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