An Alternative View of the Expression y/x and the Expression 0/0

Abstract This paper presents an alternative interpretation of the expression y/x, including the case 0/0, together with its geometric representation in three-dimensional (3D) space and examples of possible applications. This approach does not start from division or from infinitesimal (limit) considerations. The proposed interpretation of the expression y/x is based on the solution set of the equation y = x · z over ℝ, with z as the unknown, for different combinations of arbitrarily specified real parameter values x and y, without any a priori exclusion of the case x = 0. In the geometric interpretation, an expression y/x with a numerical meaning corresponds to a proper point (x,y,z) lying on the hyperbolic paraboloid y = x · z in 3D space. For the expression 0/0 introduced here with a numerical meaning, the corresponding points lie on the Oz-axis, which is an integral part of the hyperbolic paraboloid y = x · z in 3D space. This proposal does not redefine classical division or introduce division by zero; within the proposed alternative interpretation of y/x, the symbol “/” does not denote the division operator. The paper also presents two examples of applications of the expression y/x introduced here: a simple algebraic equation and a homogeneous differential equation.