the sum of a rational number and its additive inverse is always
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a rational number.....
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The thing fundamentally wrong with your question is that the additive inverse of a number a is DEFINED as the number b such that a + b = 0. You can't possibly hope to justify a definition, you can just illustrate it.
example = The additive inverse of 2 is (-2) as 2 + (-2) = 0.
The additive inverse of 1/2 is (-1/2) as 1/2 + (-1/2) = 0.
example = The additive inverse of 2 is (-2) as 2 + (-2) = 0.
The additive inverse of 1/2 is (-1/2) as 1/2 + (-1/2) = 0.
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