\begin{definition}[Definition:Multiplicative Group of Positive Real Numbers]
The '''multiplicative group of positive real numbers''' $\left({\R_{> 0}, \times}\right)$ is the set of (strictly) positive real numbers under the operation of multiplication.
Thus real multiplication is:
: Well-defined on $\R_{> 0}$
: Closed on $\R_{> 0}$
: Associative on $\R_{> 0}$
: Commutative on $\R_{> 0}$
: The identity of $\left({\R_{> 0}, \times}\right)$ is $1$
: Each element of $\left({\R_{> 0}, \times}\right)$ has an inverse.
\end{definition}