We introduce a new presentation for rational tangles which to
the number theory of positive regular continued fractions. This
presentation also admits a suitable extension to the contact setting,
allowing us to define a natural Legendrian embedding of a particular class
of rational tangles into the standard contact Euclidean 3-space. We will
briefly discuss how these box-dot diagrams, along with an associated
construction, can be used to determine when the Legendrian flyping
operation yields tangles which are not Legendrian isotopic, further
refining an earlier result of Traynor.