Like us!

Have a Look!

WhatsApp Message

Share Link

Properties of Rational Numbers

Rational Numbers II - Concepts

Identity Property

Commutative Property

Multiplicative Inverse

Division of Rational Numbers

Multiplication of Rational Numbers

Rational Numbers between Two Rational Numbers

Properties of Rational Numbers

Simplification of Rational Numbers

Class - Navodaya Entrance 9th Subjects

Concept Explanation

Properties of Rational Numbers

Properties of rational numbers are closure property, commutative property and associative property. All these properties have been discussed earlier. Let us briefly describe these properties on the four binary operations (Addition, subtraction, multiplication and division) in mathematics.

Addition

(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. Example : 2/9 + 8/9 = 10/9 is a rational number

(ii) Commutative Property : Addition of two rational numbers is commutative. If a/b and c/d are any two rational numbers, then (a/b) + (c/d) = (c/d) + (a/b) Example : 2/9 + 4/9 = 6/9 = 2/3 and 4/9 + 2/9 = 6/9 = 2/3

(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

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

(iv) Additive Identity : The sum of any rational number and zero is the rational number itself. Example : 5/7 + 0 = 0 + 5/7 = 5/7

(v) Additive Inverse : (-a/b) is the negative or additive inverse of (a/b). Example : Additive inverse of 3/5 is (-3/5).

Subtraction

(i) Closure Property : The difference between any two rational numbers is always 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 whereas 2/9 - 5/9 = -3/9 = -1/3 And, 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 whereas (2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9

And, 2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9 . Therefore, Associative property is not true for subtraction.

Multiplication

(i) Closure Property : The product of two rational numbers is always a rational number. Example : 5/9 x 2/3 = 10/27 is a rational number.

(ii) Commutative Property : Multiplication of rational numbers is commutative. Example : 5/9 x 2/9 = 10/81 and 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.

(iii) Associative Property : Multiplication of rational numbers is associative. 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

So, 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. Example : 5/9 x 1 = 1 x 5/9 = 5/9

(v) Multiplication by 0 : Every rational number multiplied with 0 gives 0. Example : 5/7 x 0 = 0 x 5/7 = 0

(vi) 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. Example : The multiplicative inverse of 2/3 is 3/2.

Division

(i) Closure Property : The collection of non-zero rational numbers is closed under division . Example : 2/3 ÷ 1/4 = 2/3 x 4/1 = 8/3 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 whereas 1/3 ÷ 2/3 = 1/3 x 3/2 = 1/2 And, 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 whereas (2/9 ÷ 4/9) ÷ 1/9 = 1/2 - 1/9 = 7/18 .And, 2/9 ÷ (4/9 ÷ 1/9) ≠ (2/9 ÷ 4/9) ÷ 1/9 . Therefore, Associative property is not true for division.

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)

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

From (1) and (2), 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 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 ------- (3)

1/3 x 2/5 - 1/3 x 1/5 = 2/15 - 1/15 = (2 - 1)/15 = 1/15 -----(4)

From (3) and (4),

1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5

Therefore, Multiplication is distributive over subtraction.

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Which of the following is example of the Associative Property of Multiplication?
A. (1/3 + 1/2 ) + 1/4 = 1/3 + (1/4 + 1/2)	B. 5/2 x 4 = 4 x 5/2	C. (1/3 x 1/4) x 1/5 = 1/3 x (1/4 x 1/5)	D. 1/3 x 1/2 = 1/6
Right Option : C
View Explanation

Question : 2

Mohan rearranged an expression to make it easier to solve. He changed 1/2 + (2/3 + 1/5) to (1/2 + 2/3) + 1/5. Which of the following properties tells us that it is okay for Mohan to arrange the problem that way? Also simplify the expression.
A. Associative property of addition , $large frac{41}{30}$	B. Distributive property of multiplication over addition $large frac{40}{13}$	C. Commutative property $large frac{41}{15}$	D. Associative property of multiplication $large frac{41}{30}$
Right Option : A
View Explanation

Question : 3

1/9 x (1/3 x 1/2) = (1/9 x 1/3) x 1/2 , is explained by which property?
A. Associative over multiplication	B. Commutative property	C. Distributive property	D. Associative over addition
Right Option : A
View Explanation