# PROPERTIES OF SUBTRACTION OF RATIONAL NUMBERS

There are some properties of subtracting rational numbers like closure, commutative, associative and distributive.

## Closure Property

The difference between any two rational numbers is always a rational number.

Hence Q is closed under subtraction.

If a/b and c/d are any two rational numbers, then (a/b) - (c/d) is also a rational number.

Example :

5/9 - 2/9  =  3/9  =  1/3 is a rational number.

## Commutative Property

Subtraction of two rational numbers is not commutative.

If a/b and c/d are any two rational numbers,

then (a/b) - (c/d)    (c/d) - (a/b)

Example :

5/9 - 2/9  =  3/9  =  1/3

2/9 - 5/9  =  -3/9  =  -1/3

Hence, 5/9 - 2/9    2/9 - 5/9

Therefore, Commutative property is not true for subtraction.

## Associative Property

Subtraction of rational numbers is not associative.

If a/b, c/d and e/f  are any three rational numbers,

then a/b - (c/d - e/f)    (a/b - c/d) - e/f

Example :

2/9 - (4/9 - 1/9)  =  2/9 - 3/9  =  -1/9

(2/9 - 4/9) - 1/9  =  -2/9 - 1/9  =  -3/9

Hence, 2/9 - (4/9 - 1/9)    (2/9 - 4/9) - 1/9

Therefore, Associative property is not true for subtraction.

## Distributive Property

Distributive property of multiplication over subtraction :

Multiplication of rational numbers is distributive over subtraction.

If a/b, c/d and e/f  are any three rational numbers,

then a/b x (c/d - e/f)  =  a/b x c/d  -  a/b x e/f

Example :

1/3 x (2/5 - 1/5)  =  1/3 x 1/5  =  1/15

1/3 x (2/5 - 1/5)  =  1/3 x 2/5  -  1/3 x 1/5  =  (2 - 1) / 15 = 1/15

Hence, 1/3 x (2/5 - 1/5)  =  1/3 x 2/5  -  1/3 x 1/5

Therefore, multiplication is distributive over subtraction.

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