PROPERTIES OF RATIONAL NUMBERS

The properties of rational numbers are

∙ Closure Property

 Commutative Property

 Associative Property

 Distributive Property

Identity Property

∙ Inverse Property

Let us explore these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.

Closure Property

(i) Addition : 

The sum of any two rational numbers is always a rational number. Thus, Q is closed under addition.

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

(ᵃ⁄b + ᶜ⁄d) is also a rational number

Example : 

²⁄₉ + ⁴⁄₉ = ⁶⁄₉

= 

 is a rational number

(ii) Subtraction : 

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

= 

⅓ is a rational number.

(iii)  Multiplication :

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

ᵃᶜ⁄bd is a rational number.

Example : 

⁵⁄₉ x ²⁄₉ = ¹⁰⁄₈₁

¹⁰⁄₈₁ is a rational number.

(iv) Division :

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

If ᵃ⁄b and ᶜ⁄d are two rational numbers, such that b ≠ 0 and c/d ≠ 0, then (ᵃ⁄b ÷ ᶜ⁄d) is always a rational number. 

Example : 

 ÷  =  x ³⁄₁ 

= ²⁄₁

²⁄₁ is a rational number.

Commutative Property

(i) Addition : 

Addition of two rational numbers is commutative.

If ᵃ⁄b + ᶜ⁄d are any two rational numbers, then

ᵃ⁄b + ᶜ⁄d = ᶜ⁄dᵃ⁄b

Example : 

²⁄₉ + ⁴⁄₉ = ⁶⁄₉ = 

⁴⁄₉ + ²⁄₉ ⁶⁄₉ =  

So,

²⁄₉ + ⁴⁄₉ = ⁴⁄₉ + ²⁄₉

Therefore, Commutative Property is true for addition of two rational numbers.

(ii) Subtraction : 

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 

²⁄₉ ⁵⁄₉ = ⁻³⁄₉ ⁻¹⁄₉

And,

⁵⁄₉ - ²⁄₉ ≠ ²⁄₉ ⁵⁄₉

Therefore, Commutative property is NOT true for subtraction of two rational numbers.

(ii) Multiplication :

Multiplication of two rational numbers is commutative.

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

ᵃ⁄b x ᶜ⁄d = ᶜ⁄ᵃ⁄b 

Example : 

⁵⁄₉ x ²⁄₉ = ¹⁰⁄₈₁

²⁄₉ ⁵⁄₉ = ¹⁰⁄₈₁

So, 

⁵⁄₉ x ²⁄₉ = ²⁄₉ ⁵⁄₉

Therefore, Commutative property is true for multiplication of two rational numbers.

(iv) Division :

Division of two rational numbers is NOT commutative.

If ᵃ⁄b and ᶜ⁄d are two rational numbers, then

ᵃ⁄b ÷ ᶜ⁄d ≠ ᶜ⁄÷ ᵃ⁄b

Example :

 ÷  =  x ³⁄₁ ²⁄₁

 ÷  =  ÷ ³⁄₂ = ½

And, 

 ÷  ≠  ÷

Therefore, Commutative property is NOT true for division two rational numbers.

Associative Property

(i) Addition :

Addition of rational numbers is associative.

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

ᵃ⁄+ (ᶜ⁄ᵉ⁄f) = (ᵃ⁄ᶜ⁄d) + ᵉ⁄f

Example :

²⁄₉ + (⁴⁄₉ + ) = ²⁄₉ + ⁵⁄₉ = ⁷⁄₉ 

(²⁄₉ + ⁴⁄₉) +  = ⁶⁄₉ + ⁷⁄₉

So, 

²⁄₉ + (⁴⁄₉ + ) = (²⁄₉ + ⁴⁄₉) +

Therefore, Associative Property is true for addition of rational numbers.

(ii) Subtraction :

Subtraction of rational numbers is NOT associative.

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

ᵃ⁄- (ᶜ⁄ᵉ⁄f) ≠ (ᵃ⁄ᶜ⁄d) - ᵉ⁄f

Example :

²⁄₉ - (⁴⁄₉ - ) = ²⁄₉ - ³⁄₉ = ⁻¹⁄₉

(²⁄₉ - ⁴⁄₉) -  = ⁻²⁄₉ ¹⁄₉ = ⁻³⁄₉ = ⁻¹⁄₃

And,

²⁄₉ - (⁴⁄₉ - ) ≠ (²⁄₉ - ⁴⁄₉) -

Therefore, Associative property is NOT true for subtraction of rational numbers.

(iii) Multiplication :

Multiplication of rational numbers is associative.

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

ᵃ⁄b x (ᶜ⁄x ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄dx ᵉ⁄f

Example :

²⁄₉ x (⁴⁄₉ x ) = ²⁄₉ x ⁴⁄₈₁ = ⁸⁄₇₂₉

(²⁄₉ x ⁴⁄₉) x  = ⁸⁄₈₁ x  = ⁸⁄₇₂₉

So,

²⁄₉ x (⁴⁄₉ x ) = (²⁄₉ x ⁴⁄₉) x

Therefore, Associative property is true for multiplication of rational numbers.

(iii) Division :

Division of rational numbers is NOT associative.

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

ᵃ⁄b ÷ (ᶜ⁄÷ ᵉ⁄f) ≠ (ᵃ⁄b ÷ ᶜ⁄d) ÷ ᵉ⁄f

Example :

²⁄₉ ÷ (⁴⁄₉ ÷ ) = ²⁄₉ ÷ 4 = ¹⁄₁₈

(²⁄₉ ÷ ⁴⁄₉) ÷  = ½ ÷  = ⁹⁄₂

And,

²⁄₉ ÷ (⁴⁄₉ ÷ ) ≠ (²⁄₉ ÷ ⁴⁄₉) ÷ 

Therefore, Associative property is NOT true for division 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⁄x (ᶜ⁄d ᵉ⁄f) = (ᵃ⁄x ᶜ⁄d) + (ᵃ⁄x ᵉ⁄f)

Example :

⅓ x ( + ) :

⅓ x ⁽² ⁺ ¹⁾⁄₅

⅓ x ³⁄₅

³⁄₁₅

= ⅕ ----(1)

⅓ x  + x  :

= ²⁄₁₅ + ¹⁄₁₅

  = ⁽² ⁺ ¹⁾⁄₁₅

³⁄₁₅

= ⅕ ----(2)

From (1) and (2), 

⅓ x ( + ) = ⅓ x  + x

Therefore, Multiplication is distributive over addition.

(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⁄x (ᶜ⁄d ᵉ⁄f) = (ᵃ⁄x ᶜ⁄d) + (ᵃ⁄x ᵉ⁄f)

Example :

⅓ x ( - ) :

⅓ x ⁽² ⁻ ¹⁾⁄₁₅

²⁄₁₅ - ¹⁄₁₅

  = ⁽² ⁻ ¹⁾⁄₁₅

 = ¹⁄₁₅ ----(1)

⅓ x  -  x  :

= ²⁄₁₅ - ¹⁄₁₅

⁽² ⁻ ¹⁾⁄₁₅

¹⁄₁₅ ----(2)

From (1) and (2), 

⅓ x ( - ) = ⅓ x  -  x

Therefore, Multiplication is distributive over subtraction.

Identity Property

(i) Additive Identity :

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 = ²⁄₇

(ii) Multiplicative Identity :

The product of any rational number and 1 is the rational number itself. ‘1’ 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 = 1 x ⁵⁄₇ = ⁵⁄₇

Inverse Property

(i) Additive Inverse :

-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  is -.

Additive inverse of - is .

Additive inverse of 0 is 0 itself.

(ii) Multiplicative Inverse or Reciprocal :

For every rational number ᵃ⁄b, b ≠ 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 multiplicative inverse of  is ³⁄₂.

The multiplicative inverse of  is 3.

The multiplicative inverse of 3 is .

The multiplicative inverse of 1 is 1.

The multiplicative inverse of 0 is undefined.

Multiplication by Zero

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 = 0 x ⁵⁄₇ = 0

properties-of-rational-numbers.png

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