# What is $\mathbb R^*$?What are its elements? [closed]

I know the * operator means the conjugate transpose but I am not sure how to get the conjugate transpose of $$\mathbb R$$

• What do you mean with R? Feb 17 '19 at 7:17
• If by R you mean the set of real numbers, $\mathbb{R}^{*}$ typically refers to the set of non-zero real numbers. Feb 17 '19 at 7:18

$$\mathbb{R^*} = \mathbb{R} - \{0\}$$
In general if $$R$$ is an arbitrary ring with $$1$$, then $$R^\ast$$ denotes the set of units, i.e. of invertible elements. Moreover, $$R^\ast$$ forms a group with respect to the ring multiplication, the so called group of units. In particular, if $$R$$ is a field, then $$R^\ast=R \setminus \{0\}$$.
Just because $$z^*$$ for a complex number $$z\in\mathbb C$$ and $$R^*$$ for a ring $$R$$ use the same symbol, you can't conclude it is the same "* operator". One is complex conjugation, the other is taking the group of units. These two operations share nothing apart from their notation.
To answer the question: $$\mathbb R^* = \mathbb R\setminus\{0\}$$.