Definition:Multiplicative Group of Real Numbers

Definition
The multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$ is the set of real numbers without zero under the operation of multiplication.

Also see

 * Non-Zero Real Numbers under Multiplication form Abelian Group

Thus real multiplication is:


 * Well-defined on $\R_{\ne 0}$
 * Closed on $\R_{\ne 0}$
 * Associative on $\R_{\ne 0}$
 * Commutative on $\R_{\ne 0}$

and:
 * The identity of $\struct {\R_{\ne 0}, \times}$ is $1$
 * Each element of $\struct {\R_{\ne 0}, \times}$ has an inverse.