  This paper gives two new combinatorial topological proofs of the
classification of rational tangles. Each proof rests on an elegant lemma
showing that rational tangles are isotopic to canonical alternating rational
tangles. The first proof defines the tangle fraction from the canonical form
and uses flyping to prove invariance. The second proof defines the fraction of
a rational tangle via integral coloring of the tangle. The coloring method is
then used to prove the Kauffman-Harary coloring conjecture for alternating
knots and links in the case of rational knots and links (closures of rational
tangles). This paper forms the basis for a sequel on the classification of
rational knots and links.
