(i) What is meant by the continued fraction expansion of a real number $\theta$ ? Suppose that $\theta$ has continued fraction $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ to $\theta$ and give the recurrence relations satisfied by the $p_{n}$ and $q_{n}$. Show that the convergents $p_{n} / q_{n}$ do indeed converge to $\theta$.

[You need not justify the basic order properties of finite continued fractions.]

(ii) Find two solutions in strictly positive integers to each of the equations

$$x^{2}-10 y^{2}=1 \quad \text { and } \quad x^{2}-11 y^{2}=1$$