Definition:Multiplicative Group of Positive Real Numbers
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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.