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# Template:Cfrac/doc

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[⧼Purge⧽ Template:Cfrac/doc]

The {{cfrac}} mathematical formatting template typesets either finite or infinite [generalized] continued fractions in either HTML+CSS or LaTeX.

## Usage

For finite continued fractions, use

{{cfrac|a0 ;; b1//a1 ,, b2//a2 ,, ... ,, bk//ak|format}}

or (to omit the integer part)

{{cfrac|;; b1//a1 ,, b2//a2 ,, ... ,, bk//ak|format}}

while for infinite continued fractions, use (where the last ak is left blank)

{{cfrac|a0 ;; b1//a1 ,, b2//a2 ,, ... ,, bk//|format}}

or (to omit the integer part)

{{cfrac|;; b1//a1 ,, b2//a2 ,, ... ,, bk//|format}}

where

• a0 followed by two consecutive semi-columns is the integer part, and
• bi//ai, with 1   ≤   i   ≤   12, separated by two consecutive commas, are the partial quotients (with the expanded styles, only the first eight partial quotients are considered, the others are ignored);

and where format is from

• htm: condensed style HTML+CSS,
• HTM: expanded style HTML+CSS,
• tex: condensed style LaTeX,
• TEX: expanded style LaTeX.

## Examples

### Text style

The code

{{indent}}the continued fraction is {{math|
? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|htm}} {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//|htm}}
|&}}


yields the text style HTML+CSS

the continued fraction is
 ? = 0  + 2  /  (1  + 4  /  (2  + 6  /  (3  + 8  /  (4  + 10  /  (5  + 12  /  (6  + 14  /  (7  + 16  /  (8  + 18  /  (9  + 20  /  (10 + ⋯ ) ⋯ ) = 0  + 2  /  (1  + 4  /  (2  + 6  /  (3  + 8  /  (4  + 10  /  (5  + 12  /  (6  + 14  /  (7  + 16  /  (8  + 18  /  (9  + 20  /  ( + ⋯ ) ⋯ )

The code

{{indent}}the continued fraction is {{math|
? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|tex}} {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//|tex}}
|$}}  yields the text style LaTeX the continued fraction is${\displaystyle \textstyle {?=0\,+\,(\,2\,/\,1\,+\,(\,4\,/\,2\,+\,(\,6\,/\,3\,+\,(\,8\,/\,4\,+\,(\,10\,/\,5\,+\,(\,12\,/\,6\,+\,(\,14\,/\,7\,+\,(\,16\,/\,8\,+\,(\,18\,/\,9\,+\,(\,20\,/\,10\,+\,\cdots \,)\,\cdots \,)=0\,+\,(\,2\,/\,1\,+\,(\,4\,/\,2\,+\,(\,6\,/\,3\,+\,(\,8\,/\,4\,+\,(\,10\,/\,5\,+\,(\,12\,/\,6\,+\,(\,14\,/\,7\,+\,(\,16\,/\,8\,+\,(\,18\,/\,9\,+\,(\,20\,/\,\,+\,\cdots \,)\,\cdots \,)}}$ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|HTM}} |&}}  yields the text style HTML+CSS ? = 0 + 2 1 + 4 2 + 6 3 + 8 4 + 10 5 + 12 6 + 14 7 + 16 8 + 18 9 +  20 10 The code {{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|TEX}} |$}}


yields the text style LaTeX

${\displaystyle \textstyle {?={0+{\frac {2}{1+{\frac {4}{2+{\frac {6}{3+{\frac {8}{4+{\frac {10}{5+{\frac {12}{6+{\frac {14}{7+{\frac {16}{8+{\frac {18}{9+{\frac {20}{10}}}}}}}}}}}}}}}}}}}}}}}$

### Display style

The code

{{indent}}{{math|
? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|HTM}}
|&&}}


yields the display style HTML+CSS

? = 0 +
2
1 +
4
2 +
6
3 +
8
4 +
10
5 +
12
6 +
14
7 +
 16 8

The code

{{indent}}{{math|
? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|TEX}}
|$$}}  yields the display style LaTeX ${\displaystyle {\begin{array}{l}\displaystyle {?={0+{\cfrac {2}{1+{\cfrac {4}{2+{\cfrac {6}{3+{\cfrac {8}{4+{\cfrac {10}{5+{\cfrac {12}{6+{\cfrac {14}{7+{\cfrac {16}{8}}}}}}}}}}}}}}}}}}\end{array}}}$ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4|HTM}} |&&}}  yields the display style HTML+CSS ? = 0 + 2 1 + 4 2 + 6 3 +  8 4 The code {{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4|TEX}} |$$}}


yields the display style LaTeX

${\displaystyle {\begin{array}{l}\displaystyle {?={0+{\cfrac {2}{1+{\cfrac {4}{2+{\cfrac {6}{3+{\cfrac {8}{4}}}}}}}}}}\end{array}}}$

The code

{{indent}}{{math|
? {{=}} {{cfrac|;; 2//1,, 4//2,, 6//3,, 8//|HTM}}
|&&}}


yields the display style HTML+CSS

? =
2
1 +
4
2 +
6
3 +
 8 ⋱

The code

{{indent}}{{math|
? {{=}} {{cfrac|;; 2//1,, 4//2,, 6//3,, 8//|TEX}}
|$$}}  yields the display style LaTeX ${\displaystyle {\begin{array}{l}\displaystyle {?={\cfrac {2}{1+{\cfrac {4}{2+{\cfrac {6}{3+{\cfrac {8}{\ddots }}}}}}}}}\end{array}}}$ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//4,, ''x''{{^|5}}//5,, <!-- -->''x''{{^|6}}//6,, ''x''{{^|7}}//7,, ''x''{{^|8}}//|HTM}} |&&}}  yields the display style HTML+CSS ? = 0 + x 1 + x 2 2 + x 3 3 + x 4 4 + x 5 5 + x 6 6 + x 7 7 +  x 8 ⋱ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4,, ''x''{{^|5|tex}}//5,, <!-- -->''x''{{^|6|tex}}//6,, ''x''{{^|7|tex}}//7,, ''x''{{^|8|tex}}//|TEX}} |$$}}


yields the display style LaTeX

${\displaystyle {\begin{array}{l}\displaystyle {?={0+{\cfrac {x}{1+{\cfrac {x^{2}}{2+{\cfrac {x^{3}}{3+{\cfrac {x^{4}}{4+{\cfrac {x^{5}}{5+{\cfrac {x^{6}}{6+{\cfrac {x^{7}}{7+{\cfrac {x^{8}}{\ddots }}}}}}}}}}}}}}}}}}\end{array}}}$

The code

{{indent}}{{math|
? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//4|HTM}}
|&&}}


yields the display style HTML+CSS

? = 0 +
x
1 +
x 2
2 +
x 3
3 +
 x 4 4

The code

{{indent}}{{math|
? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4|TEX}}
|$$}}  yields the display style LaTeX ${\displaystyle {\begin{array}{l}\displaystyle {?={0+{\cfrac {x}{1+{\cfrac {x^{2}}{2+{\cfrac {x^{3}}{3+{\cfrac {x^{4}}{4}}}}}}}}}}\end{array}}}$ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//|HTM}} |&&}}  yields the display style HTML+CSS ? = 0 + x 1 + x 2 2 + x 3 3 +  x 4 ⋱ The code {{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//|TEX}} |$$}}


yields the display style LaTeX

${\displaystyle {\begin{array}{l}\displaystyle {?={0+{\cfrac {x}{1+{\cfrac {x^{2}}{2+{\cfrac {x^{3}}{3+{\cfrac {x^{4}}{\ddots }}}}}}}}}}\end{array}}}$

#### Simple continued fraction for π

The code

{{indent}}{{math|{{Gr|pi}} {{=}} {{cfrac|3;; 1//7,, 1//15,, 1//1,, 1//292,, 1//1,, 1//1,, 1 //|HTM}}
|tex = \pi = {{cfrac|3;; 1//7,, 1//15,, 1//1,, 1//292,, 1//1,, 1//1,, 1 //|TEX}}|&&}}


yields the display style HTML+CSS

π = 3 +
1
7 +
1
15 +
1
1 +
1
292 +
1
1 +
1
1 +
 1 ⋱

and with the $$ option, yields the LaTeX ${\displaystyle {\begin{array}{l}\displaystyle {\pi ={3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\cfrac {1}{292+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\end{array}}}$ #### Continued fraction with minus signs The code {{indent}}{{math|''G''{{sub|{''a''{{sub|''n''|80%}}{{sp|1}}, ''n''{{rel|ge}}0} }}(''x'') {{=|sp}} {{cfrac|;; 1 // 1 ,-, ''x'' // 1 ,, ''x''{{^|2}} // 1 ,-, ''x''{{^|3}} // 1 ,, ''x''{{^|4}} //|HTM}} |tex = G_{ \{ a_n,\, n \ge 0 \} }(x) = {{cfrac|;; 1 // 1 ,-, x // 1 ,, x^2 // 1 ,-, x^3 // 1 ,, x^4 //|TEX}}<!-- \cfrac{1}{1 - \cfrac{x}{1 + \cfrac{x^2}{1 - \cfrac{x^3}{1 + \cfrac{x^4}{\ddots} } } } } -->,|&&}}  yields the display style HTML+CSS (NOT RIGHT: should be minus signs in front of odd powers of x only; template needs more work...) G{an , n ≥ 0}(x) = 1 1 − x 1 − x 2 1 − x 3 1 −  x 4 ⋱ ### Two-dimensional continued fractions Example from (with commas added to separate the two-dimensional subscript indices: for testing purposes): http://www.sciencedirect.com/science/article/pii/S0377042799000308 The code {{indent}}{{math| ''f''{{sub|2}} {{=}} <!-- -->{{cfrac| ;; ''a''{{sub|0}} // <!-- -->''b''{{sub|0}} + <!-- -->{{cfrac|;; ''a''{{sub|1,0}}//''b''{{sub|1,0}} ,, ''a''{{sub|2,0}}//''b''{{sub|2,0}} ,, ''a''{{sub|3,0}}//|HTM}} + <!-- -->{{cfrac|;; ''a''{{sub|0,1}}//''b''{{sub|0,1}} ,, ''a''{{sub|0,2}}//''b''{{sub|0,2}} ,, ''a''{{sub|0,3}}//|HTM}} + <!-- -->{{cfrac|;; ''a''{{sub|1}}//''b''{{sub|1}} ,, <!-- -->{{tfrac|''a''{{sub|2,1}}|''b''{{sub|2,1}}|HTM}} + {{tfrac|''a''{{sub|1,2}}|''b''{{sub|1,2}}|HTM}} + {{tfrac|''a''{{sub|2}}|''b''{{sub|2}}|HTM}} |HTM}} |HTM}} |&&}}  yields the display style HTML+CSS  Error: String exceeds 10,000 character limit. The code {{indent}}{{math| ''f''{{sub|2|tex}} {{=}} <!-- -->{{cfrac| ;; ''a''{{sub|0|tex}} // <!-- -->''b''{{sub|0|tex}} + <!-- -->{{cfrac|;; ''a''{{sub|1,0|tex}}//''b''{{sub|1,0|tex}} ,, ''a''{{sub|2,0|tex}}//''b''{{sub|2,0|tex}} ,, ''a''{{sub|3,0|tex}}//|TEX}} + <!-- -->{{cfrac|;; ''a''{{sub|0,1|tex}}//''b''{{sub|0,1|tex}} ,, ''a''{{sub|0,2|tex}}//''b''{{sub|0,2|tex}} ,, ''a''{{sub|0,3|tex}}//|TEX}} + <!-- -->{{cfrac|;; ''a''{{sub|1|tex}}//''b''{{sub|1|tex}} ,, <!-- -->{{tfrac|''a''{{sub|2,1|tex}}|''b''{{sub|2,1|tex}}|TEX}} + {{tfrac|''a''{{sub|1,2|tex}}|''b''{{sub|1,2|tex}}|TEX}} + <!-- -->{{tfrac|''a''{{sub|2|tex}}|''b''{{sub|2|tex}}|TEX}} |TEX}} |TEX}} |$$}}


yields the display style LaTeX

${\displaystyle {\begin{array}{l}\displaystyle {f_{2}{\!\,\!}={\cfrac {a_{0}{\!\,\!}}{b_{0}{\!\,\!}+{\cfrac {a_{1,0}{\!\,\!}}{b_{1,0}{\!\,\!}+{\cfrac {a_{2,0}{\!\,\!}}{b_{2,0}{\!\,\!}+{\cfrac {a_{3,0}{\!\,\!}}{\ddots }}}}}}+{\cfrac {a_{0,1}{\!\,\!}}{b_{0,1}{\!\,\!}+{\cfrac {a_{0,2}{\!\,\!}}{b_{0,2}{\!\,\!}+{\cfrac {a_{0,3}{\!\,\!}}{\ddots }}}}}}+{\cfrac {a_{1}{\!\,\!}}{b_{1}{\!\,\!}+{\cfrac {{\tfrac {a_{2,1}{\!\,\!}}{b_{2,1}{\!\,\!}}}+{\tfrac {a_{1,2}{\!\,\!}}{b_{1,2}{\!\,\!}}}+{\tfrac {a_{2}{\!\,\!}}{b_{2}{\!\,\!}}}}{\ddots }}}}}}}\end{array}}}$

## More examples

For more examples, see

## Convergents

See Continued fractions#Generalized continued fractions convergents.

The code

: Convergents numerators {{math|''p''{{sub|''i''}}{{sp|1}}, ''i'' {{rel|ge}} 0|tex = p_i,\, i \ge 0|&}}: {{mathfont|{{set|{{cfrac|3;; 1//7,, 1//15,, 1//1,, 1//292,, 1//1,, 1//1,, 1 //|convnums}}}}}}
: Convergents denominators {{math|''q''{{sub|''i''}}{{sp|1}}, ''i'' {{rel|ge}} 0|tex = q_i,\, i \ge 0|&}}: {{mathfont|{{set|{{cfrac|3;; 1//7,, 1//15,, 1//1,, 1//292,, 1//1,, 1//1,, 1 //|convdens}}}}}}


yields

Convergents numerators
 pi , i   ≥   0
: {3, 22}
Convergents denominators
 qi , i   ≥   0
: {1, 7}

## Tests

The following tests show

• why we have to use ;; instead of ; between the integer part and the partial quotients;
• why we have to use // instead of / for the partial quotients;
• and why (since some CSS use comma separated values) we better use ,, instead of , to separate the partial quotients.

The code

{{pre|
{{#explode: ''x''{{^|2}};2|;|0}} ''and'' {{#explode: ''x''{{^|2}};2|;|1}}{{nl}}
{{#explode: ''x''{{^|2}};;2|;;|0}} ''and'' {{#explode: ''x''{{^|2}};;2|;;|1}}{{nl}}
{{#explode: ''x''{{^|2}}/2|/|0}} ''and'' {{#explode: ''x''{{^|2}}/2|/|1}}{{nl}}
{{#explode: ''x''{{^|2}}//2|//|0}} ''and'' {{#explode: ''x''{{^|2}}//2|//|1}}
|/pre}}


yields (see how the first and third lines inside {{pre|...|/pre}} failed to parse properly)

''x''<span style="position: relative ''and''  vertical-align: 1.25ex<br />
''x''<span style="position: relative; vertical-align: 1.25ex; line-height: 0.75ex; font-size: 0.75em;"><span style="position: relative; vertical-align: middle;"> 2</span></span> ''and'' 2<br />
''x''<span style="position: relative; vertical-align: 1.25ex; line-height: 0.75ex; font-size: 0.75em;"><span style="position: relative; vertical-align: middle;"> 2< ''and'' span><<br />
''x''<span style="position: relative; vertical-align: 1.25ex; line-height: 0.75ex; font-size: 0.75em;"><span style="position: relative; vertical-align: middle;"> 2</span></span> ''and'' 2