Existence of Unique Inverse Element for Multiplication of Cuts

Theorem

Let $0^*$ be the rational cut associated with the (rational) number $0$:

$0^* = \set {r \in \Q: r < 0}$

Let $\alpha$ be a cut such that $\alpha \ne 0^*$.

Then for every cut $\beta$, there exists a unique cut $\gamma$ such that:

$\alpha \gamma = \beta$

where $\alpha \gamma$ denotes the operation of product of $\alpha$ and $\gamma$.

In this context, $\gamma$ can be expressed as $\beta / \alpha$.

Proof

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Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.27$. Theorem