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Template:Cfrac/doc
The {{cfrac}} mathematical formatting template typesets either finite or infinite [generalized] continued fractions in either HTML+CSS or LaTeX.
Contents
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 lastak
blank for infinite continued fractions);
-
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}} |$}}
yields the text style LaTeX
the continued fraction is
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|HTM}} |&}}
yields the text style HTML+CSS
? = 0 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8,, 18//9,, 20//10|TEX}} |$}}
yields the text style LaTeX
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 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4,, 10//5,, 12//6,, 14//7,, 16//8|TEX}} |$$}}
yields the display style LaTeX
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4|HTM}} |&&}}
yields the display style HTML+CSS
? = 0 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; 2//1,, 4//2,, 6//3,, 8//4|TEX}} |$$}}
yields the display style LaTeX
The code
{{indent}}{{math| ? {{=}} {{cfrac|;; 2//1,, 4//2,, 6//3,, 8//|HTM}} |&&}}
yields the display style HTML+CSS
? =
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|;; 2//1,, 4//2,, 6//3,, 8//|TEX}} |$$}}
yields the display style LaTeX
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//4,, ''x''{{^|5}}//5,, <!-- -->''x''{{^|6}}//6,, ''x''{{^|7}}//7,, ''x''{{^|8}}//|HTM}} |&&}}
yields the display style HTML+CSS
? = 0 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4,, ''x''{{^|5|tex}}//5,, <!-- -->''x''{{^|6|tex}}//6,, ''x''{{^|7|tex}}//7,, ''x''{{^|8|tex}}//|TEX}} |$$}}
yields the display style LaTeX
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 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//4|TEX}} |$$}}
yields the display style LaTeX
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2}}//2,, ''x''{{^|3}}//3,, ''x''{{^|4}}//|HTM}} |&&}}
yields the display style HTML+CSS
? = 0 +
|
The code
{{indent}}{{math| ? {{=}} {{cfrac|0;; ''x''//1,, ''x''{{^|2|tex}}//2,, ''x''{{^|3|tex}}//3,, ''x''{{^|4|tex}}//|TEX}} |$$}}
yields the display style LaTeX
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 +
|
and with the $$
option, yields the LaTeX
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) =
|
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
More examples
For more examples, see
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
: {3, 22}pi , i ≥ 0 - Convergents denominators
: {1, 7}qi , i ≥ 0
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
See also
- {{nroot}} (typesets [generalized] nested radicals in either HTML+CSS or LaTeX) (finite or infinite [generalized] continued fractions are a particular case of finite or infinite [generalized] nested radicals with root indices all equal to
)− 1