COMMUTATIVE PROPERTY OF MULTIPLICATION OF RATIONAL NUMBER -
The commutative property of multiplication states that changing the order of the factors does not change the product. In other words, for any two rational
a c
numbers ------ & ------, the following equation holds: -
b d
a c c a
------- X ------- = ------- X ------
b d d b
Explanation -
a c
Let's consider two rational numbers ------ and ------
b d
where a,b,c, and d are integers and bβ 0, dβ 0,
a c
When we multiply ------ & ------ we get:-
b d
a c a X c
------ X ------ = ----------
b d b X d
c a
Similarly, when we multiply ------ & ------ we get:-
d b
c a c X a
------- X -------- = ----------
d b d X b
Since multiplication of integers is commutative (i.e. a X c = c X a, and b X d = d X b) the two expressions are equal:-
Therefore,
a X c c X a
----------- = -----------
b X d d X b
a c c a
So, ------ X ------ = ------ X ------
b d d b
Examples Conclusion -
1) Example with Positive Rational Numbers:-
2 4
Let's take two rational numbers, ------ and ------
3 5
Ans.)
2 4 8
------- X ------- = -------
3 5 15
βSwitching the order:-
4 2 8
------- X ------- = --------
5 3 15
βBoth products are the same, demonstrating the commutative property.
2) Example with Negative Rational Numbers:-
- 3 5
Consider the rational numbers ------- & ------
7 3
- 3 5 - 15
------- X ------- = -------
7 3 21
βSwitching the order:-
5 - 3 - 15
------- X ------- = --------
3 7 21
Again, both products are the same, demonstrating the commutative property.
The commutative property of multiplication for rational numbers ensures that the order in which two rational numbers are multiplied does not affect the product. This property is fundamental in ensuring the consistency and reliability of mathematical operations involving rational numbers.