Definition:Multiplication of Positive Cuts

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