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.

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

• Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group