Negative numbers in the 18th and 19th centuries: phenomenology and representations Negative numbers in the 18th and 19th centuries: phenomenology and representations History of Mathematics and Mathematics Education (original) (raw)

Negative numbers in the 18th and 19th centuries: phenomenology and representations

2009

This article presents a categorization of the phenomena and representations used to introduce negative numbers in mathematics books published in Spain during the 18th and 19th centuries. Through a content analysis of fourteen texts which were selected for the study, we distinguished four phenomena typologies: physical, accounting, temporal and mathematical. Four types of representations are also identified: verbal, numeric, graphic and algebraic. These results reflect the same level of understanding and treatment of negative numbers in Spain as is found in other European countries during this time period.

Negative numbers as an epistemic difficult concept: Some lessons from history

Proceedings of the History and Pedagogy of …, 2008

Historical studies on the development of mathematical concepts will serve mathematics teachers to relate their students' difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians as Arnauld, Leibniz, Wallis, Euler and d'Alembert. Not only the division by negative numbers poses problems for the number line, but also the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics we argue for the introduction of negative numbers in education within the context of symbolic operations.

Historical objections against the number line

Science and Education, 2011

Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations.

Old Arithmetic Books: Mathematics in Spain in the First Half of the Sixteenth Century

International Electronic Journal of Mathematics Education

During the sixteenth century, a relevant number of books related to practical arithmetic were published in Spain. This paper presents a comparative study that aims to identify the object and target of five old mathematics books and the main contents of these books. In order to do so, all the contents and all the examples found in five arithmetic books written during the first half of the 16th century were identified and categorised. A historical-mathematical analysis using a content analysis technique was then applied. The results show the authors' concerns about mathematics at the time and the different contents that were included in some arithmetic books of this century.

Mathematics in the Spanish press: a case study of the 18th century journal Semanario de Salamanca

Humanities & social sciences communications, 2023

Old mathematics books and textbooks have always been sources of relevant information for researchers working on the history of mathematics and mathematics education. However, helpful information about this field can also be found in journals, ministerial decrees, or notes. Consequently, this article analyzes the contents related to mathematics or mathematics education in an 18th-century weekly journal published in the Spanish city of Salamanca, Semanario de Salamanca. To this end, we conducted a qualitative and descriptive investigation using a content analysis technique, which is widely used in research in this related area. Our results show that despite the fact that the weekly publication under investigation was specialized neither in mathematics nor in science, it included mathematical problems and solutions, mathematical texts, job advertisements for mathematics teachers, mathematics book reviews, and opinions about mathematics. Therefore, the present work expands knowledge of the presence and dissemination of mathematics and mathematics education in 18th-century Spanish society.

On the reception and spread of metaphysical explanations of imaginary numbers in Spain

Revista de la …, 2006

The introduction in Spain orthe ideas on complex quamities is doc:umented lhrough the worit of José Maria Rey Heredia, a Professor of Logic and st:lf•taught malhematician who wrole during his last life yean thc book Teoría Transcendental tk las Canlidotks/moginor;lU. Rey W15 a follower of Kant and Kraust. and lried lo inscribe lhe dcvdopment of Ihe lheory into thc ¡ntelleetua! (nuncwon: ofTransccndental Logic. His .nempt is alengthy lOO sometimes cnoncous commmt, on the M~mo;re on imaginary quantities published by lhe Abbé 8u« in 1he Philosophical TntnSactions the year 1106. Rey had a number of rollowers \liho introduced his ideas in teadUng through thc sccond half of (he 19* Century as ao a¡de to !he introduetion of varioos geometricaJ concepts. Resumen La introducción en Espaila de las ideas filosóficas sobre los números complejos se encuentran documentadas en la obra de Jose Maria Rey Heredia, Profesor de Lógica y matemático, quien escribió una especie de testamento matemático filosófico durante los últimos 81\0S de su vida, la Teoria Tran.fcendental de las Cantidades Imaginarias. Rey fue seguidor de Kant y de Krause e intentó situar los números complejos en el marco de la Lógiea Trascendental kantiana. El texto de Reyes un largo y desigual comentario, a la luz del esquema de las categorlas, de la Memoria publicada por el Abate Buée en las PhilosophicaJ TrallSoctiolls el ailo 1806. Rey tuvo varios seguidores que introdujeron sus ideas en la ensdlanz.a media durante la segunda mitad del XIX como apoyo al estudio de la Geometria.

The conception of negative numbers

Indian Journal of History of Science, 2024

The conceptualisation of negative numbers represents a fascinating aspect of mathematical development, characterised by its evolution over time and its multifaceted applications across various disciplines. This paper embarks on a scholarly exploration of this intriguing subject, focusing on a salient segment from Kṛṣṇa Daivajña’s Bījapallava, a detailed commentary of Bhāskarācārya’s Bījagaṇita. It situates the discussion within its historical and scholarly context by examining the complexities inherent in understanding ‘negative’ quantities. While briefly referencing the evolution of negative numbers by notable mathematicians such as Brahmagupta and Bhāskara, the primary focus remains on dissecting a section of the “dhanarṇaṣaḍvidha” excerpt from Bījapallava, thereby shedding light on significant aspects of the discussion on negative numbers. Keywords: dhana · ṛṇa · dhanarṇaṣaḍvidha · Bījapallava · Kṛṣṇa Daivajña · Negative numbers · Indian mathematics

Introduction - The algebrization of mathematics during the 17th and 18th centuries

The algebrization of mathematics during the 17th and 18th centuries , 2023

This book explores the major historical phenomenon of the algebraization of mathematics in the second half of the 17th and 18th centuries, offering a broader understanding of the consolidation of analytic geometry and infinitesimal calculus as disciplines. The authors examine the external (intellectual, geographical, and political) factors that influenced these transformations and shed light on the process of acquisition and integration of analytical mathematics into traditional curricula. Drawing on new trends in historiography of science, this book emphasizes the importance of "dwarfs", that is mathematicians but also technicians, artisans, military personnel, engineers, and architects, often ignored or marginalized in traditional histories, in the circulation of original mathematical knowledge, and of peripheral countries such as Italy and Spain as important sites for the appropriation and production of such knowledge.