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

• If a = ----------​, then a + 0 = --------- + 0 = ---------​.

4                                  4                    4

3                3

and,   0 + a = 0 + ----------- = ----------

4                4

So, a + 0 = 0 + a = a       (Proven)

Example.2)

7                                      7                         7

• If a = − --------- , then a + 0 − ---------- + 0 = − ----------.

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.