Editing Simple continued fraction (section)

=== Best rational within an interval ===
A rational that falls within the interval {{open-open|''x'', ''y''}}, for {{math|0 < {{mvar|x}} < {{mvar|y}}}}, can be found with the continued fractions for {{mvar|x}} and {{mvar|y}}. When both {{mvar|x}} and {{mvar|y}} are irrational and
:{{math|''x'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, ''a''{{sub|''k''}}, ''a''{{sub|''k'' + 1}}, ...]}}
:{{math|''y'' {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, ''b''{{sub|''k''}}, ''b''{{sub|''k'' + 1}}, ...]}}
where {{mvar|x}} and {{mvar|y}} have identical continued fraction expansions up through {{math|''a''<sub>''k''−1</sub>}}, a rational that falls within the interval {{open-open|''x'', ''y''}} is given by the finite continued fraction,
:{{math|''z''(''x'',''y'') {{=}} [''a''{{sub|0}}; ''a''{{sub|1}}, ''a''{{sub|2}}, ..., ''a''{{sub|''k'' − 1}}, min(''a''{{sub|''k''}}, ''b''{{sub|''k''}}) + 1]}}
This rational will be best in the sense that no other rational in {{open-open|''x'', ''y''}} will have a smaller numerator or a smaller denominator.<ref>{{cite web
 | last = Gosper | first = R. W. | author-link = Bill Gosper
 | title = Appendix 2: Continued Fraction Arithmetic
 | url = https://perl.plover.com/yak/cftalk/INFO/gosper.ps
 | year = 1977}} See "simplest intervening rational", pp. 29–31.</ref><ref>{{cite journal
 | last = Murakami | first = Hiroshi
 | date = February 2015
 | doi = 10.1145/2733693.2733711
 | issue = 3/4
 | journal = ACM Communications in Computer Algebra
 | pages = 134–136
 | title = Calculation of rational numbers in an interval whose denominator is the smallest by using FP interval arithmetic
 | volume = 48}}</ref>

If {{mvar|x}} is rational, it will have ''two'' continued fraction representations that are ''finite'', {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}, and similarly a rational&nbsp;{{mvar|y}} will have two representations, {{math|''y''<sub>1</sub>}} and {{math|''y''<sub>2</sub>}}. The coefficients beyond the last in any of these representations should be interpreted as {{math|+∞}}; and the best rational will be one of {{math|''z''(''x''<sub>1</sub>, ''y''<sub>1</sub>)}}, {{math|''z''(''x''<sub>1</sub>, ''y''<sub>2</sub>)}}, {{math|''z''(''x''<sub>2</sub>, ''y''<sub>1</sub>)}}, or {{math|''z''(''x''<sub>2</sub>, ''y''<sub>2</sub>)}}.

For example, the decimal representation 3.1416 could be rounded from any number in the interval {{closed-open|3.14155, 3.14165}}. The continued fraction representations of 3.14155 and 3.14165 are
:{{math|3.14155 {{=}} [3; 7, 15, 2, 7, 1, 4, 1, 1] {{=}} [3; 7, 15, 2, 7, 1, 4, 2]}}
:{{math|3.14165 {{=}} [3; 7, 16, 1, 3, 4, 2, 3, 1] {{=}} [3; 7, 16, 1, 3, 4, 2, 4]}}
and the best rational between these two is
:{{math|[3; 7, 16] {{=}} {{sfrac|355|113}} {{=}} 3.1415929....}}
Thus, {{sfrac|355|113}} 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.