linguistic:paraphrasing
Best rational approximations

One can choose to define a best rational approximation to a real number x as a rational number n / d, d > 0, that is closer to x than any approximation with a smaller or equal denominator. The simple continued fraction for x can be used to generate all of the best rational approximations for x by applying these three rules:

For example, 0.84375 has continued fraction. Here are all of its best rational approximations.

The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation.

The "half rule" mentioned above requires that when a is even, the halved term a /2 is admissible if and only if | x − | > | x − | This is equivalent to: Shoemake (1995).

The convergents to x are "best approximations" in a much stronger sense than the one defined above. Namely, n / d is a convergent for x if and only if | dx − n | has the smallest value among the analogous expressions for all rational approximations m / c with c ≤ d; that is, we have | dx − n | < | cx − m | so long as c < d. (Note also that | d x − n | → 0 as k → ∞.)