Pub Mar 20, 2026 Up Jun 20, 2026
Definition 1. 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.
Definition 2. Based on the construction method, $\Q$ can be embedded into $\R$. Then we can say
\[\N \subseteq \Z \subseteq \Q \subseteq \R.\]Definition 3. The set of non-zero real numbers is
\[\R^* = \R \setdif \{0\}.\]Definition 4. The set of positive real numbers is
\[\R^+ = \{x \in \R : x > 0\}.\]Definition 5. The set of negative real numbers is
\[\R^- = \{x \in \R : x < 0\}.\]