\begin{definition}[Definition:Isometric Isomorphism/Normed Division Ring]
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.
Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively.
Let $\phi:R \to S$ be a bijection such that:
:$(1): \quad \phi: \struct {R, d_R} \to \struct {S, d_S}$ is an isometry
:$(2): \quad \phi: R \to S$ is a ring isomorphism.
Then $\phi$ is called an '''isometric isomorphism'''.
The normed division rings $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ are said to be '''isometrically isomorphic'''.
\end{definition}