  We use the classical interpretation of the braid group $B_3$ as a central
extension of the modular group $\text{PSL}_2\left(\mathbb{Z}\right)$ to
establish new and fundamental properties of $B_3$ using the theory of continued
fractions. In particular, we give simple and natural linear time algorithms to
solve the word and conjugacy problems in $B_3$. The algorithms introduced in
this paper are easy to implement and are the most efficient algorithms in the
literature to solve these problems in the braid group $B_3$.
