A Geometric Approach to Defining Multiplication

TL;DR

This paper introduces a purely geometric definition of multiplication using basic plane geometry, proving its properties and connections to area and similarity without limits, aiming to enhance understanding for students and teachers.

Contribution

It provides a novel geometric approach to defining and proving properties of multiplication, independent of algebraic or limit-based methods.

Findings

Geometric definition of multiplication is independent of area and similar triangles.

All properties of multiplication are proven using only plane geometry axioms.

The approach relates multiplication to the area of right triangles and the Pythagorean Theorem.

Abstract

In this paper we will do the following: (1) show how to geometrically define multiplication, using only basic plane geometry, independently of area and any notion of similar triangles; (2) prove all the properties of multiplication using only the axioms of plane geometry and the geometric definition of multiplication; (3) explain how the geometric definition of multiplication relates to the area of a right triangle (or rectangle); and (4) explain how by using only the geometric definition of multiplication and the Pythagorean Theorem one can prove that two triangles have the same angles if and only if the lengths of their corresponding sides are proportional. The interesting and surprising thing, from a pedagogical and/or mathematical point of view, is that all of these results can be proven using only simple geometry (no limits needed). As we shall see, parallel lines in our geometric approach will play a role similar to limits in the standard algebraic approach. We believe that both prospective K-12 mathematics teachers and STEM students could especially benefit by being exposed to the material in this paper before graduating from university.