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If o and * are two binary operations in a number system, then three elementsa,b,c in that number system are said to satisfy the distributive property ofthe operation o over the operation * if, ao(b*c)= (aob)*(aoc) Now, suppose thatthe number system is the rationals,and the operations o and * are among thefour usual operations of addition, multiplication, subtraction, and division.If we allow for o and * to be the same operation, then there are precisely 16combinations 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 isthe multiplication operationand * being the addition operation; and when is ois multiplication and * is subtraction. For these two combinations, the abovestated distributive property is universally satisfied; that is, for ane threerational numbers a,b,and c. In this work, we examine the other fourteencombinations, to find out when the distributive property is satisfied. Of these14 combinations or cases, eleven are easy/straightforward, in that almostalways, one of the three rational numbers a, b, c; must be zero or 1. Theremaining three cases or combinations are much more complicated, and numbertheory is involved.

Distributive properties of the rationals