EXISTENCE OF ADDITIVE IDENTITY PROPERTY OF ADDITION OF RATIONAL NUMBERS -
The additive identity property of addition states that there exists a unique element, known as the additive identity, which when added to any other number leaves the number unchanged. In the context of rational numbers, this means that there exists a rational number 0 such that when added to any other rational number a, it results in a itself.
Formally, for any rational number a, there exists a rational number 0 such that:
a + 0 = 0 + a = a
In other words, 0 is the identity element for addition of rational numbers.
The additive identity property holds true for all rational numbers. The rational number 0 serves as the additive identity because when you add 0 to any rational number, the result is the same rational number.
The rational numbers include any number that can be expressed as the quotient or fraction p/q where p and q are integers and q ≠ 0.
To illustrate with an example:-
Example.1)
3 3 3
4 4 4
3 3
and, 0 + a = 0 + ----------- = ----------
4 4
So, a + 0 = 0 + a = a (Proven)
Example.2)
7 7 7
2 2 2
7 7
and, 0 + a = 0 + (−) ---------- = − -----------
2 2
So, a + 0 = 0 + a = a (Proven)
In each case, adding 0 to a rational number a does not change its value, confirming the Additive Identity Property of Addition.