EXISTENCE OF ADDITIVE INVERSE PROPERTY OF ADDITION OF RATIONAL NUMBERS -
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
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)