Template:Cfrac/doc

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The {{cfrac}} mathematical formatting template typesets either finite or infinite [generalized] continued fractions in either HTML+CSS or $L a T e X$ .

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);
- bi are the partial numerators;
- ai are the partial denominators (leave the last ak blank for infinite continued fractions);

and where format is from

htm: condensed style HTML+CSS,
HTM: expanded style HTML+CSS,
tex: condensed style $L a T e X$ ,
TEX: expanded style $L a T e X$ .

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 $L a T e X$

the continued fraction is $\textstyle {?=0\,+\,(\,2\,/\,1\,+\,(\,4\,/\,2\,+\,(\,6\,/\,3\,+\,(\,8\,/\,4\,+\,(\,10\,/\,5\,+\,(\,12\,/\,6\,+\,(\,14\,/\,7\,+\,(\,16\,/\,8\,+\,(\,18\,/\,9\,+\,(\,20\,/\,10\,+\,\cdots \,)\,\cdots \,)=0\,+\,(\,2\,/\,1\,+\,(\,4\,/\,2\,+\,(\,6\,/\,3\,+\,(\,8\,/\,4\,+\,(\,10\,/\,5\,+\,(\,12\,/\,6\,+\,(\,14\,/\,7\,+\,(\,16\,/\,8\,+\,(\,18\,/\,9\,+\,(\,20\,/\,\,+\,\cdots \,)\,\cdots \,)}$

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 $L a T e X$

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

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 $L a T e X$

${\begin{array}{l}\displaystyle {?={0+{\cfrac {2}{1+{\cfrac {4}{2+{\cfrac {6}{3+{\cfrac {8}{4+{\cfrac {10}{5+{\cfrac {12}{6+{\cfrac {14}{7+{\cfrac {16}{8}}}}}}}}}}}}}}}}}}\end{array}}$

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 $L a T e X$

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

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 $L a T e X$

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

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 $L a T e X$

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

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 $L a T e X$

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

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 $L a T e X$

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

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 $L a T e X$

${\begin{array}{l}\displaystyle {\pi ={3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\cfrac {1}{292+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}\end{array}}$

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 $L a T e X$

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

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

Template:Cfrac/doc

Contents

Usage

Examples

Text style

Display style

Simple continued fraction for π

Continued fraction with minus signs

Two-dimensional continued fractions

More examples

Convergents

Tests

See also

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