PROPERTIES OF SUBTRACTION OF RATIONAL NUMBERS

About "Properties of subtraction of rational numbers"

Properties of subtraction of rational numbers :

Students who would like to practice problems on subtracting rational numbers must be aware of the properties of subtracting rational numbers.

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

Let us explore these properties on the binary operation subtraction.

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

(ii) 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.

After having gone through the stuff given above, we hope that the students would have understood "Properties of subtraction of rational numbers".

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