Additive inverses in \(\symbb{Z}\) are unique
An Introduction to Real Analysis
2025-10-24
Show that if \(n + m = 0\), then \(n = -m\). This shows that for every \(m \in \symbb{Z}\), \(-m\) is unique.
Since \[\begin{align*} n + m &= 0\\[0.75em] \implies (n + m) + (-m) &= 0 + (-m)\\[0.75em] \implies n + \underbrace{(m + (-m))}_{= 0} &= -m\\[0.75em] \implies n + 0 &= -m\\[0.75em] \implies n &= -m, \end{align*}\] additive inverses in \(\symbb{Z}\) are unique.
Remark 1. Although Katznelson and Katznelson (2024) only ask for a proof that additive inverses in the integers are unique, the proof works just as well for inverse elements in any group \(G\). The proof uses only the existence of the inverse element \(-m\), associativity of the binary operation \(+\), and the definition of the additive identity when concluding that \(n + 0 = n\). All of these properties are available in any group. \(\blacksquare\)