Definition:Multiplicative Group of Positive Real Numbers

Definition
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.

Also see

 * Positive Real Numbers under Multiplication form Abelian Group

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.