−5.1852% || −1.2346% || +0.28490% || −0.19493% || 0% |} 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 is even, the halved term /2 is admissible if and only if This is equivalent to: . . The convergents to are "best approximations" in a much stronger sense than the one defined above. Namely, / is a convergent for if and only if has the smallest value among the analogous expressions for all rational approximations / with ; that is, we have so long as . (Note also that as .) Best rational within an interval A rational that falls within the interval , for , can be found with the continued fractions for and . When both and are irrational and where and have identical continued fraction expansions up through , a rational that falls within the interval is given by the finite continued fraction, This rational will be best in the sense that no other rational in will have a smaller numerator or a smaller denominator. If is rational, it will have two continued fraction representations that are finite, and , and similarly a rational will have two representations, and . The coefficients beyond the last in any of these representations should be interpreted as ; and the best rational will be one of , , , or . For example, the decimal representation 3.1416 could be rounded from any number in the interval . The continued fraction representations of 3.14155 and 3.14165 are and the best rational between these two is Thus, is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator. Interval for a convergent A rational number, which can be expressed as finite continued fraction in two ways, will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between (see this proof) and The numbers and are formed by incrementing the last coefficient in the two representations for . It is the case that when is even, and when is odd. For example, the number has the continued fraction representations =