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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