The real number system $\R$ is, up to isomorphism, the unique ordered field that is Dedekind-complete.
The existence of such a field can be shown through construction. See for example Construction of the real numbers (Wikipedia) and its references.
Based on the construction method, $\Q$ can be embedded into $\R$. Then we can say
\[\N \subseteq \Z \subseteq \Q \subseteq \R.\]
The set of non-zero real numbers is
\[\R^* = \R \setdif \{0\}.\]
The set of positive real numbers is
\[\R^+ = \{x \in \R : x > 0\}.\]
The set of negative real numbers is
\[\R^- = \{x \in \R : x < 0\}.\]