Real Multiplicative Inverses

Why

What is the multiplicative inverse in the reals?

Result

We can show the following.¹

The multiplicative inverse of $R \in \R $, $R \neq 0_{\R }$,

if $0_{\Q } \in R$, then
\[ S = \Set*{q \in \Q }{q \leq 0_{\Q }} \union \Set*{r^{-1}}{\exists s < r, (r \not\in R)} \]
is a multiplicative inverse of $R$.
if $0_{\Q } \not\in R$, then case (1) applies to $-R$. Let $S$ be the multiplicative inverse of $-R$. Then the additive inverse of $S$, i.e., $-S$ is a multiplicative inverse of $R$.

Notation

We denote the multiplicative inverse of $r \in \R $ by $\inv{r}$. We denote $q \cdot (\inv{r})$ by $q/r$.

Division

We call the operation $(a, b) \mapsto a/b$ real division. We call the product of $a$ and the multiplicative inverse of $b$ the (real) quotient of $a$ and $b$.

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