Real Products
Why
We want to multiply real numbers.
Definition
The real product of two
real numbers $R$ and $S$ is defined
- if $R$ or $S$ is $\Set{q \in \Q }{q < 0_{\Q }}$, then
the $\Set{q \in \Q }{q < 0_{\Q }}$
- otherwise,
- if $R$ or $S$ is $0_{\R }$, then $0_{\R }$.
- 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 $T \cup \Set{q \in \Q }{q \leq 0_{\Q }}$
- 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 $S$.
- 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 $-S$.
- 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
$x + (y + z) = (x + y) + z$
$x + y = y + x$
The set of all rationals less than $1_{\Q }$
is the multiplicative identity.
We denote the the multiplicative identity by
$1_{\R }$.
When it is clear from context, we call
$1_{\R }$ “one”.