EXISTENCE OF MULTIPLICATIVE IDENTITY OF RATIONAL NUMBER -
The existence of the multiplicative identity property states that there is a unique number, called the multiplicative identity, which when multiplied by any rational number, leaves the number unchanged. For rational numbers, the multiplicative identity is 1.
Explanation:-
For any rational number a,
a × 1 = 1 × a = a
Examples: -
1) Rational Number as a Fraction:-
2
Let, a = ------
5
2 2 1 2
a X 1 = ------ X 1 = ------ X ------ = -------
5 5 1 5
2 1 2 2
1 X a = 1 X ------ = ------- X ------ = -------
5 1 5 5
So, a X 1 = 1 X a = a
2) Whole Number as a Rational Number:-
5
Let, a = 5 [Which can be written as -----]
1
5 1 5
a X 1 = ------ X ----- = ------ = 5
1 1 1
1 5 5
1 X a = ----- X ----- = ------ = 5
1 1 1
So, a X 1 = 1 X a = a
3) Negative Rational Number
- 3
Let, a = ------
7
- 3 1 - 3
a X 1 = ------ X ----- = ------
7 1 7
1 - 3 - 3
1 X a = ----- X ------ = ------
1 7 7
So, a X 1 = 1 X a = a
In all these cases, multiplying by the multiplicative identity 1 leaves the rational number unchanged. This demonstrates the existence of the multiplicative identity property for rational numbers.