Sign rule for multiplication: the notion of semantic congruence and the extension principle in mathematics

In the 3rd century B.C., the sign rule for multiplication was presented by Diophantus of Alexandria without any demonstration. Only in 1867 was the rule demonstrated by Hankel as being the only one with the advantage of satisfying both left and right distributivity. Thus, Hankel solves the problem from the mathematical point of view when he uses the extension principle to apply the distributivity property, which has been used for many years with positive numbers, to negative numbers. This paper shows that this approach is still present in mathematics teaching today. The paper also presents studies from a teaching perspective of this rule based on the idea of semantic congruence and the extension principle in mathematics.

Sign Rule; Semantic Congruence; Diophantus; Hankel's Theorem; Extension Principle