Definition:Multiplicative Group of Positive Real Numbers

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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.

Also see