The additive inverse property of addition states that for every rational number a, there exists a unique rational number −a such that when a is added to its additive inverse −a, the result is the additive identity, which is usually denoted as 0. In simpler terms, every rational number has a negative counterpart such that their sum is zero.

Formally, for any rational number 𝑎a, there exists a rational number −a such that:

a + (−a) = (−a) + a = 0

In other words, −a is the additive inverse of a.

The additive inverse property holds true for all rational numbers. For any rational number a, its additive inverse −a can be found by changing the sign of its numerator while keeping the denominator the same.

To illustrate with examples:-

Example.1)

3                                                                    3

• If a = ----------​, then its additive inverse is − a =  − ----------​.

4                                                                     4

Therefore:-

3                      3                 (3 - 3)             0

a + (− a)----------- + (− -----------) =  ------------- = ----------  =  0

4                      4                     4                 4

3                 3             (− 3 + 3)              0

(− a) + a = (− -----------) + --------- = ---------------- =  ---------- = 0

4                 4                   4                     4

So,  a + (− a) =  (− a) + a = 0            (Proven)

Example.2)

7                                                                       7

If, a = − ----------- ​, then its additive inverse is − a = ------------​.

2                                                                       2

Therefore:−

7                  7                 (− 7 + 7)               0

a + (− a)(− ----------) + ----------- =  ------------------- = ----------  =  0

2                  2                      2                       2

7                    7                (7 − 7)                0

and,    (− a) + a ---------- + (− ----------) = --------------- =  ----------- = 0

2                     2                    2                     2

So,  a + (− a) =  (− a) + a = 0            (Proven)