Real Products

Why

We want to multiply real numbers.¹

Definition

The real product of two real numbers $R$ and $S$ is defined

1. if $R$ or $S$ is $\Set{q \in \Q }{q < 0_{\Q }}$, then the $\Set{q \in \Q }{q < 0_{\Q }}$
2. otherwise,
   1. if $R$ or $S$ is $0_{\R }$, then $0_{\R }$.
   2. if $R,S \neq 0_{\R }$ and $0_{\R } \in R, S$, let $T$ be

      \[ \Set{t \in \Q }{r \in R, s \in S, r, s \geq 0_{\Q }, t = r\cdot s} \]

      then $T \cup \Set{q \in \Q }{q \leq 0_{\Q }}$²
   3. If $R, S \neq 0_{\R }$, $0_{\R } \in R$ and $0_{\R } \not\in S$, then the additive inverse of the product of $-R$ with $S$.
   4. If $R, S \neq 0_{\R }$, $0_{\R } \not\in R$ and $0_{\R } \in S$, then the additive inverse of the product of $R$ with $-S$.
   5. If $R, S \neq 0_{\R }$, and $0_{\R } \not\in R,S$, then the product of $-R$ with $-S$.

Notation

We denote the product of two real numbers $x$ and $y$ by $x \cdot y$.

Properties

We denote the the multiplicative identity by $1_{\R }$. When it is clear from context, we call $1_{\R }$ “one”.

1. Future editions will expand. ↩︎
2. We use $\geq$ in the usual way, it will be defined earlier in future editions. ↩︎