(related to Proposition: Uniqueness of Rational Zero)

We will show that the rational number zero \(0\) is unique, i.e. there can be only one such number, for which

\[x=x+0\text{ for all }x\in\mathbb Q\quad\quad ( * )\]

Suppose, \(0^{\ast}\) is any (other) rational number, for which

\[x=x+0^{\ast}\text{ for all }x\in\mathbb Q\quad\quad ( * * )\]

Applying the commutativity law for adding rational numbers, we get

\[\begin{array}{rcll} 0^{\ast}&=&0^{\ast}+0&\text{ by }( * )\\ &=&0+0^{\ast}&\text{ by commutativity of adding rational numbers}\\ &=&0&\text{ by }( * * ) \end{array} \]

Thus, both zeros are equal.

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