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

= ⅔

⅔ 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

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 : 

⅔ ÷ ⅓ = ⅔ 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⁄d+ ᵃ⁄b

Example : 

²⁄₉ + ⁴⁄₉ = ⁶⁄₉ = ⅔

⁴⁄₉ + ²⁄₉ = ⁶⁄₉ = ⅔ 

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 = ⅑

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⁄d x ᵃ⁄b 

Example : 

⁵⁄₉ x ²⁄₉ = ¹⁰⁄₈₁

²⁄₉ x ⁵⁄₉ = ¹⁰⁄₈₁

So, 

⁵⁄₉ x ²⁄₉ = ²⁄₉ x 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⁄d ÷ ᵃ⁄b

Example :

⅔ ÷ ⅓ = ⅔ x ³⁄₁ = ²⁄₁

⅓ ÷ ⅔ = ⅓ ÷ ³⁄₂ = ½

And, 

⅔ ÷ ⅓ ≠ ⅓ ÷ ⅔

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⁄b + (ᶜ⁄d + ᵉ⁄f) = (ᵃ⁄b + ᶜ⁄d) + ᵉ⁄f

Example :

²⁄₉ + (⁴⁄₉ + ⅑) = ²⁄₉ + ⁵⁄₉ = ⁷⁄₉ 

(²⁄₉ + ⁴⁄₉) + ⅑ = ⁶⁄₉ + ⅑ = ⁷⁄₉

So, 

²⁄₉ + (⁴⁄₉ + ⅑) = (²⁄₉ + ⁴⁄₉) + ⅑

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⁄b - (ᶜ⁄d - ᵉ⁄f) ≠ (ᵃ⁄b - ᶜ⁄d) - ᵉ⁄f

Example :

²⁄₉ - (⁴⁄₉ - ⅑) = ²⁄₉ - ³⁄₉ = ⁻¹⁄₉

(²⁄₉ - ⁴⁄₉) - ⅑ = ⁻²⁄₉ - ¹⁄₉ = ⁻³⁄₉ = ⁻¹⁄₃

And,

²⁄₉ - (⁴⁄₉ - ⅑) ≠ (²⁄₉ - ⁴⁄₉) - ⅑

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⁄d x ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄d) x ᵉ⁄f

Example :

²⁄₉ x (⁴⁄₉ x ⅑) = ²⁄₉ x ⁴⁄₈₁ = ⁸⁄₇₂₉

(²⁄₉ x ⁴⁄₉) x ⅑ = ⁸⁄₈₁ x ⅑ = ⁸⁄₇₂₉

So,

²⁄₉ x (⁴⁄₉ x ⅑) = (²⁄₉ x ⁴⁄₉) x ⅑

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⁄d ÷ ᵉ⁄f) ≠ (ᵃ⁄b ÷ ᶜ⁄d) ÷ ᵉ⁄f

Example :

²⁄₉ ÷ (⁴⁄₉ ÷ ⅑) = ²⁄₉ ÷ 4 = ¹⁄₁₈

(²⁄₉ ÷ ⁴⁄₉) ÷ ⅑ = ½ ÷ ⅑ = ⁹⁄₂

And,

²⁄₉ ÷ (⁴⁄₉ ÷ ⅑) ≠ (²⁄₉ ÷ ⁴⁄₉) ÷ ⅑

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⁄b x (ᶜ⁄d + ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄d) + (ᵃ⁄b x ᵉ⁄f)

Example :

⅓ x (⅖ + ⅕) :

= ⅓ x ⁽² ⁺ ¹⁾⁄₅

= ⅓ x ³⁄₅

= ³⁄₁₅

= ⅕ ----(1)

⅓ x ⅖ + ⅓ x ⅕ :

= ²⁄₁₅ + ¹⁄₁₅

  = ⁽² ⁺ ¹⁾⁄₁₅

= ³⁄₁₅

= ⅕ ----(2)

From (1) and (2), 

⅓ x (⅖ + ⅕) = ⅓ x ⅖ + ⅓ x ⅕

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⁄b x (ᶜ⁄d + ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄d) + (ᵃ⁄b x ᵉ⁄f)

Example :

⅓ x (⅖ - ⅕) :

= ⅓ x ⁽² ⁻ ¹⁾⁄₁₅

= ²⁄₁₅ - ¹⁄₁₅

  = ⁽² ⁻ ¹⁾⁄₁₅

 = ¹⁄₁₅ ----(1)

⅓ x ⅖ - ⅓ x ⅕ :

= ²⁄₁₅ - ¹⁄₁₅

= ⁽² ⁻ ¹⁾⁄₁₅

= ¹⁄₁₅ ----(2)

From (1) and (2), 

⅓ x (⅖ - ⅕) = ⅓ x ⅖ - ⅓ x ⅕

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 ⅗ is -⅗.

Additive inverse of -⅗ is ⅗.

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 ⅔ is ³⁄₂.

The multiplicative inverse of ⅓ is 3.

The multiplicative inverse of 3 is ⅓.

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

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