Distributive properties of the rationals

TL;DR

This paper investigates which combinations of the four basic rational operations satisfy the distributive property, revealing that most cases require specific values, while some involve complex number theory analysis.

Contribution

It systematically analyzes all possible operation combinations on rationals to determine when the distributive property holds, extending understanding of algebraic structures.

Findings

Two operation combinations always satisfy distributivity.

Most cases require a zero or one for the property to hold.

Three complex cases involve advanced number theory.

Abstract

If o and * are two binary operations in a number system, then three elements a,b,c in that number system are said to satisfy the distributive property of the operation o over the operation * if, ao(b*c)= (aob)*(aoc) Now, suppose that the number system is the rationals,and the operations o and * are among the four usual operations of addition, multiplication, subtraction, and division. If we allow for o and * to be the same operation, then there are precisely 16 combinations with the operation o being one of the four usual operations in Q; and likewise for the operation *. Two of these 16 combinations are when o is the multiplication operationand * being the addition operation; and when is o is multiplication and * is subtraction. For these two combinations, the above stated distributive property is universally satisfied; that is, for ane three rational numbers a,b,and c. In this work, we examine the other fourteen combinations, to find out when the distributive property is satisfied. Of these 14 combinations or cases, eleven are easy/straightforward, in that almost always, one of the three rational numbers a, b, c; must be zero or 1. The remaining three cases or combinations are much more complicated, and number theory is involved.