Mathematical Analysis
Exercise 1-1
2026-01-04
Prove that \((-1) \cdot (-1) = 1\).
This was already proven for the integers as a lemma in Exercise 1.2.5 for Katznelson and Katznelson (2024), and for general fields in the Section 1.3.2 notes for Katznelson and Katznelson (2024). Nevertheless, the argument for a general field can be repeated here in the specific case of \(\symbb{R}\).
By Theorem 1.2 part 4 from Schröder (2008, 2–3), \((-1) \cdot (-1) = -(-1)\) so \((-1) \cdot (-1)\) is an additive inverse of \(-1\). On the other hand, \(1 + (-1) = 0\) so \(1 \in \symbb{R}\) is also an additive inverse of \(-1\). By Theorem 1.2 part 3 from Schröder (2008, 2–3) additive inverses are unique, whence \((-1) \cdot (-1) = 1\).