On the Negative Numbers from the Historical and Educational Perspective

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