On the Negative Numbers from the Historical and Educational Perspective

Piotr Błaszczyk; Mirosława Sajka

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia (2017)

  • Volume: 9, page 5-36
  • ISSN: 2080-9751

Abstract

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We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, ≺ ) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes to form the theory of polynomials, as developed by Descartes in his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second model into a tablet game.

How to cite

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Piotr Błaszczyk, and Mirosława Sajka. "O liczbach ujemnych z perspektywy historycznej i dydaktycznej." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 9 (2017): 5-36. <http://eudml.org/doc/296318>.

@article{PiotrBłaszczyk2017,
abstract = {We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $\prec $) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes form the theory of polynomials, as developed by Descartesin his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second modelin to a tablet game.},
author = {Piotr Błaszczyk, Mirosława Sajka},
journal = {Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia},
keywords = {Negative numbers; History of mathematics; Mathematics education},
language = {pol},
pages = {5-36},
title = {O liczbach ujemnych z perspektywy historycznej i dydaktycznej},
url = {http://eudml.org/doc/296318},
volume = {9},
year = {2017},
}

TY - JOUR
AU - Piotr Błaszczyk
AU - Mirosława Sajka
TI - O liczbach ujemnych z perspektywy historycznej i dydaktycznej
JO - Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
PY - 2017
VL - 9
SP - 5
EP - 36
AB - We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $\prec $) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes form the theory of polynomials, as developed by Descartesin his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second modelin to a tablet game.
LA - pol
KW - Negative numbers; History of mathematics; Mathematics education
UR - http://eudml.org/doc/296318
ER -

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