Definition:Multiplicative Group of Positive Rational Numbers
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Definition
The multiplicative group of positive rational numbers $\left({\Q_{> 0}, \times}\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication.
Also see
Thus rational multiplication is:
- Well-defined on $\Q_{> 0}$
- Closed on $\Q_{> 0}$
- Associative on $\Q_{> 0}$
- Commutative on $\Q_{> 0}$
- The identity of $\left({\Q_{> 0}, \times}\right)$ is $1$
- Each element of $\left({\Q_{> 0}, \times}\right)$ has an inverse.