Write the continued fraction in the form (a_0 + frac{1}{a_1 + frac{1}{a_2 + cdots}})
Evaluate from the innermost fraction outward
For a finite continued fraction, start with the last term and work backward
Use the recurrence for convergents:
(p_{-2}=0, p_{-1}=1)
(q_{-2}=1, q_{-1}=0)
(p_n=a_n p_{n-1}+p_{n-2})
(q_n=a_n q_{n-1}+q_{n-2})
(x approx frac{p_n}{q_n})
For infinite continued fractions, compute successive convergents until the values stabilize
Convert a rational number to a continued fraction using repeated division
For (frac{a}{b}), set (a_0=lfloor a/b rfloor), then repeat with the reciprocal of the remainder
For quadratic irrational forms, use algebraic manipulation to identify repeating patterns
Check each step for arithmetic accuracy
Simplify fractions at each stage if needed
Verify the final result by converting back to the original value