📝 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(bc)= (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.

💡 Analysis

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(bc)= (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.

📄 Content

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Distributive properties of the rationals

Konstantine Zelator Department of Mathematics and Computer Science Rhode Island College 600 Mount Pleasant Avenue Providence, R.I. 02908-1991, U.S.A. e-mail address: 1) Kzelator@ric.edu

2) Konstantine_zelator@yahoo.com 

Effective August 1, 2009, and for the academic year 2009-10; Konstantine Zelator Department of Mathematics 301 Thackeray Hall 139 University Place University of Pittsburgh Pittsburgh, PA 15260 U.S.A e-mail address : kzet159@pitt.edu

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1. Introduction

Well before the intricacies of irrational numbers come into play in high school math education, a student learns about the fundamental properties of the four basic operations in the number system of the rational numbers. These basic binary operations are addition +, multiplication ⋅ , subtraction − , and division ÷. Two of the fundamental properties are the distributive properties of multiplication over addition and subtraction:

For any rational numbers ; , , 3 2 1 r r r

( ) ( ) ) 2 ( )1( 3 1 2 1 3 2 1 3 1 2 1 3 2 1 r r r r r r r and r r r r r r r ⋅ − ⋅

− ⋅ ⋅ + ⋅

⋅

In a more general setting, if ∗and o are two binary operations in a number system; then the number system is said to have the distributive property of the operation o over the operation ∗if for any three elements c b a , , in the number system,

( ) ( ) ( ). c a b a c b a o o o ∗

∗

(3)

In the case of the rationals, addition, multiplication and subtraction are defined for any ordered pair of rational numbers; while division will produce a rational number outcome, only when the second number ( in the ordered pair) is nonzero.

Now, consider the following question. In the rational number system, if ∗is one of the four operations ; , , , ÷ − ⋅ + and likewise, if o is also one of these four operations. When is (3) satisfied, that is, for which rational triples is (3) satisfied? If we allow o and ∗to be the same operation, then there are preuisely 16 combinations with ∗and o being among the four basic operations. The two cases in which o is the multiplication operation and ∗is either addition or subtraction , are already answered by (1) and (2). Of the remaining 14 cases; 11 are easy/straight forward in that, almost always, one of the three rationals involved, must equal zero or 1. The last three cases are really interesting, and number theory is involved.

It appears that the determination of all such rational triples in each of these three cases is a very complicated, involved process. Instead, what we do in these three cases, is finding infinite families of rational triples with the said distributive property.

This is then the aim of this work.

To determine, fully or partly, the triples of rational numbers which satisfy the distributive property, for each of the fourteen combinations or cases.

2. Eleven easy/straight forward cases Page 3 of 11

3. Distributive property of addition over itself In this case, ( ) ( ) ( ) 0 1 3 1 2 1 3 2 1 = ⇔

= + + r r r r r r r r

Rational triples: ( ) 3 2 , ,0 r r

2. Distributive property of addition over subtraction We have, ( ) ( ) ( ) 0 1 3 1 2 1 3 2 1 = ⇔

− +

− + r r r r r r r r

Rational triples : ( ) 3 2 , ,0 r r

3. Distributive property of multiplication over itself This is , ( ) ( ) ( ) ( ) ⇔ = − ⇔ ⋅ ⋅ ⋅ = ⋅ ⋅ 0 1 1 3 2 1 3 1 2 1 3 2 1 r r r r r r r r r r r

( )1 0 1 3 2 1

= ⇔ r or r r r . We have two nondisjoint families: Family 1: rational triples ( ) 3 2 1 , , r r r with 0 3 2 1

r r r

Family 2: rational triples ( ) 3 2, ,1 r r

4. Distributive property of multiplication over division

We have , ( ) ( ) ( ) 3 1 2 1 3 2 1 r r r r r r r ⋅ ÷ ⋅

÷ ⋅ , with
.0 3 1 ≠ r r

Equivalently , ( ) ⇔       ≠

− ⇔       ≠

0 0 1 0 , 3 1 1 2 3 1 3 1 2 1 3 2 1 r r r r r r r r r r r r r

( ) ( ) 0 1 0 3 1 1 2 ≠

= ⇔ r r and r or r .

Rational triples : Two nondisjoint families:

Family 1: Rational triples ( ) 3 1 , 0 , r r with 0 3 1 ≠ r r

Family 2: Rational triples ( ) 3 2, ,1 r r with 0 3 ≠ r

5. Distributive property of subtraction over itself

This is the case of ( ) ( ) ( ) 0 1 3 1 2 1 3 2 1

⇔ − − −

− − r r r r r r r r

Rational triples :
( ) 3 2 , , 0 r r

6. Distributive property of subtraction over addition

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In this case, ( ) ( ) ( ) 0 1 3 1 2 1 3 2 1

⇔ − + −

− r r r r r r r r

Rational triples : ( ) 3 2 , , 0 r r

7. Distributive property of division over itself

This property says that ( ) ( ) ( ), 3 1 2 1 3 2 1 r r r r r r r ÷ ÷ ÷

÷ ÷

with 0 3 2 ≠ r r . This statement is equivalent to, ( ). 0 1 0 3 2 1 3 2 1 2 3 2 3 1 ≠

⇔       ≠

r r and r r r r and r r r r r

Rational triples: ( ) 3 2, ,1 r r with .0 3 2 ≠ r r

8. Distributive property of division over multiplication.

This proper

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