PROPERTIES OF MULTIPLICATION OF RATIONAL NUMBER
We will learn the properties of multiplication of rational numbers i.e. closure property, commutative property, associative property, existence of multiplicative identity property, existence of multiplicative inverse property, distributive property of multiplication over addition and multiplicative property of 0.
Multiplication of rational numbers follows several fundamental properties that ensure consistency and reliability in mathematical operations. Here are the key properties of multiplication of rational numbers, explained with examples:
1. Closure Property -
The closure property states that the product of any two rational numbers is also a rational number.
Example.1)
3 5
If ------- & ------ are two rational number, their product is -
4 6
3 5
Ans.) ------- X -------
4 6
3 X 5 15
= --------- = --------
4 X 6 24
5
= -------
6
5
Since ------​ is also a rational number, the closure property holds.
8
2. The Commutative Property -
The commutative property states that the order in which two rational numbers are multiplied does not affect the product.
Example.2)
2 4
If ------- & ------ are two rational number, their product is -
3 5
2 4 2 X 4 8
Ans.) ------ X ------ = ---------- = -------
3 5 3 X 5 15
5
Since ------​ is also a rational number, the closure property holds.
8
3. Associative Property -
The associative property states that when three or more rational numbers are multiplied, the grouping of the numbers does not affect the product.
Example.3)
1 2 3 1 2 3
(------ X ------) X ------- = ------- X (------- X -------)
2 3 4 2 3 4
Ans.) Calculating both sides -
1 2 3
L.H.S = (------- X -------) X -------
2 3 4
1 3 1
= ------- X ------- = -------
3 4 4
1 2 3
R.H.S = ------ X (------ X ------)
2 3 4
1 1
= --------- X ---------
2 2
1
= -------
4
Both results are the same, thus proving the associative property.
4. Multiplicative Identity Property -
The multiplicative identity property states that any rational number multiplied by 1 remains unchanged.
Example:-
7 7
------- X 1 = -------
9 9
5. Multiplicative Inverse Property -
The multiplicative inverse property states that for every non-zero rational number a/b​, there exists another rational number b/a such that their product is 1.
Example:-
5 8 40
------- X ------- = -------- = 1
8 5 40
6. Distributive Property -
The distributive property states that multiplying a rational number by a sum of two other rational numbers is the same as multiplying the number by each addend separately and then adding the products.
Example:-
1 2 4 1 2 1 4
------ X (------ + ------) = ------ X ------ + ------ X ------
3 5 7 3 5 3 7
Calculating both sides:-
1 2 4
L.H.S = ------ X (------ + ------)
3 5 7
1 (2 X 7) + (4 X 5)
= ------ X {-----------------}
3 (5 X 7)
1 (14 + 20) 34
= ------- X ----------- = --------
3 35 105
1 2 1 4
R.H.S = (------ X ------) + (------ X ------)
3 5 3 7
(1 X 2) (1 X 4)
= ---------- + -----------
15 21
2 4
= ------- + -------
15 21
(2 X 7) + (4 X 5)
= -------------------
105
(14 + 20) 34
= ------------ = -------
105 105
Both results are the same, thus proving the distributive property.
These properties ensure that multiplication of rational numbers is consistent and reliable, facilitating complex mathematical operations and problem-solving.