# Properties of Rational Numbers

(i) Closure property:

The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q is closed under addition

If p/q and r/s are any two rational numbers, then (p/q) + (r/s) is also a rational number.

Example:

4/9 + 8/9  =  12/9  =  4/3 is a rational number.

(ii) Commutative property:

Addition of two rational numbers is commutative.

If p/q and r/s are any two rational numbers,

then (p/q) + (r/s)  =  (r/s) + (p/q)

Example:

2/9 + 4/9  =  6/9  =  2/3

4/9 + 2/9  =  6/9  =  2/3

Hence, 2/9 + 4/9  =  4/9 + 2/9

(iii) Associative property:

Addition of rational numbers is 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 + 5/9  =  7/9

(2/9 + 4/9) + 1/9  =  6/9 + 1/9  =  7/9

Hence, 2/9 + (4/9 + 1/9)  =  (2/9 + 4/9) + 1/9

The sum of any rational number and zero is the rational number itself.

If a/b is any rational number,

then a/b + 0 = 0 + a/b  =  a/b

Zero is the additive identity for rational numbers.

Example:

2/7 + 0 = 0 + 2/7 = 27

(- a/b) is the negative or additive inverse of (a/b)

If a/b is a rational number,then there exists a rational number (-a/b) such that a/b + (-a/b) = (-a/b) + a/b = 0

Example:

Additive inverse of 3/5 is (-3/5)

Additive inverse of (-3/5) is 3/5

Additive inverse of 0 is 0 itself.

Let us look at the next stuff on "Properties of rational numbers"

## Subtraction

(i) 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.

(ii) 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.

(iii) 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.

Let us look at the next stuff on "Properties of rational numbers"

## Multiplication

(i) Closure property:

The product of two rational numbers is always a rational number. Hence Q is closed under multiplication.

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

then (a/b)x (c/d) = ac/bd is also a rational number.

Example:

5/9 x 2/9  =  10/81 is a rational number.

(ii) Commutative property:

Multiplication of rational numbers is commutative.

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

then (a/b)x (c/d) = (c/d)x(a/b).

5/9 x 2/9  =  10/81

2/9 x 5/9  =  10/81

Hence, 5/9 x 2/9  =  2/9 x 5/9

Therefore, Commutative property is true for multiplication.

(iii) Associative property:

Multiplication of rational numbers is associative.

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

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

Example:

2/9 x (4/9 x 1/9)  =  2/9 x 4/81  =  8/729

(2/9 x 4/9) x 1/9  =  8/81 x 1/9  =  8/729

Hence, 2/9 x (4/9 x 1/9)  =  (2/9 x 4/9) x 1/9

Therefore, Associative property is true for multiplication.

(iv) Multiplicative identity:

The product of any rational number and 1 is the rational number itself. ‘One’ is the multiplicative identity for rational numbers.

If a/b is any rational number,

then a/b x 1 = 1 x a/b  =  a/b

Example:

5/7 x 1 = 1x 5/7  =  5/7

(v) Multiplication by 0:

Every rational number multiplied with 0 gives 0.

If a/b is any rational number,

then a/b x 0 = 0 x a/b  =  0

Example:

5/7 x 0 = 0x 5/7  =  0

(vi) Multiplicative Inverse or Reciprocal:

For every rational number a/b, a≠0, there exists a rational number c/d such that a/b x c/d = 1. Then c/d is the multiplicative inverse of a/b.

If b/a is a rational number,

then a/b is the multiplicative inverse or reciprocal of it.

Example:

The reciprocal of 2/3 is 3/2

The reciprocal of 1/3 is 3

The reciprocal of 3 is 1/3

The reciprocal of 1 is 1

The reciprocal of 0 is undefined

Let us look at the next stuff on "Properties of rational numbers"

## Division

(i) Closure property:

The collection of non-zero rational numbers is closed under division.

If a/b and c/d are two rational numbers, such that c/d ≠ 0,

then a/b ÷ c/d is always a rational number.

Example:

2/3 ÷ 1/3  =  2/3 x 3/1  =  2 is a rational number.

(ii) Commutative property:

Division of rational numbers is not commutative.

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

then a/b ÷ c/d  ≠  c/d ÷ a/b

Example:

2/3 ÷ 1/3  =  2/3 x 3/1  =  2

1/3 ÷ 2/3  =  1/3 x 3/2  =  1/2

Hence, 2/3 ÷ 1/3  ≠  1/3 ÷ 2/3

Therefore, Commutative property is not true for division.

(iii) Associative property:

Division 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 ÷ 4  =  1/18

(2/9 ÷ 4/9) ÷ 1/9  =  1/2 - 1/9  =  7/18

Hence, 2/9 ÷ (4/9 ÷ 1/9)  ≠  (2/9 ÷ 4/9) ÷ 1/9

Therefore, Associative property is not true for division.

Let us look at the next stuff on "Properties of rational numbers"

## Distributive Property

(i) Distributive property of multiplication over addition:

Multiplication of rational numbers is distributive over addition.

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 3/5  =  1/5

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

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

Therefore, Multiplication is distributive over addition.

(ii) Distributive property of multiplication over subtration:

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.