Definition:Multiplication of Positive Cuts
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Definition
Let $0^*$ denote the rational cut associated with the (rational) number $0$.
Let $\alpha$ and $\beta$ be positive cuts, that is, cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts.
Let the operation of multiplication be defined on $\alpha$ and $\beta$ as:
- $\gamma := \alpha \beta$
where $\gamma$ is the set of all rational numbers $r$ such that either:
- $r < 0$
or
- $\exists p \in \alpha, q \in \beta: r = p q$
where $p \ge 0$ and $q \ge 0$.
In this context, $\gamma$ is known as the product of $\alpha$ and $\beta$.
Also see
- Product of Positive Cuts is Positive Cut which proves existence and uniqueness of $\alpha \beta$
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.23$. Definition