Difference between revisions of "Template:Continued fraction examples"

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imported>Anita5192
(Corrected a0 term in sqrt(3))
imported>Colonies Chris
m (minor fixes)
 
(4 intermediate revisions by one other user not shown)
Line 7: Line 7:
 
| 123
 
| 123
 
|-
 
|-
! ra
+
! ''ra''
 
| 123
 
| 123
 
|-
 
|-
Line 15: Line 15:
 
| 12 || 3 || 3
 
| 12 || 3 || 3
 
|-
 
|-
! ra
+
! ''ra''
 
| 12 || {{sfrac|37|3}} || {{sfrac|123|10}}
 
| 12 || {{sfrac|37|3}} || {{sfrac|123|10}}
 
|-
 
|-
Line 23: Line 23:
 
| 1 || 4 || 2 || 1 || 7
 
| 1 || 4 || 2 || 1 || 7
 
|-
 
|-
! ra
+
! ''ra''
 
| 1 || {{sfrac|5|4}} || {{sfrac|11|9}} || {{sfrac|16|13}} || {{sfrac|123|100}}
 
| 1 || {{sfrac|5|4}} || {{sfrac|11|9}} || {{sfrac|16|13}} || {{sfrac|123|100}}
 
|-
 
|-
Line 31: Line 31:
 
| 0 || 8 || 7 || 1 || 2 || 5
 
| 0 || 8 || 7 || 1 || 2 || 5
 
|-
 
|-
! ra
+
! ''ra''
 
| 0 || {{sfrac|1|8}} || {{sfrac|7|57}} || {{sfrac|8|65}} || {{sfrac|23|187}} || {{sfrac|123|1 000}}
 
| 0 || {{sfrac|1|8}} || {{sfrac|7|57}} || {{sfrac|8|65}} || {{sfrac|23|187}} || {{sfrac|123|1 000}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|''&#981;'' =<br /> {{sfrac|&#8730;5 + 1|2}}
+
| rowspan="2"|''&#981;'' =<br /> {{sfrac|{{radic|5}} + 1|2}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| 1 || 2 || {{sfrac|3|2}} || {{sfrac|5|3}} || {{sfrac|8|5}} || {{sfrac|13|8}} || {{sfrac|21|13}} || {{sfrac|34|21}} || {{sfrac|55|34}} || {{sfrac|89|55}} || {{sfrac|144|89}}
 
| 1 || 2 || {{sfrac|3|2}} || {{sfrac|5|3}} || {{sfrac|8|5}} || {{sfrac|13|8}} || {{sfrac|21|13}} || {{sfrac|34|21}} || {{sfrac|55|34}} || {{sfrac|89|55}} || {{sfrac|144|89}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|&#8722;''&#981;'' =<br /> &#8722;{{sfrac|&#8730;5 + 1|2}}
+
| rowspan="2"|&#8722;''&#981;'' =<br /> &#8722;{{sfrac|{{radic|5}} + 1|2}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| &#8722;2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
| &#8722;2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| &#8722;2 || &#8722;{{sfrac|3|2}} || &#8722;{{sfrac|5|3}} || &#8722;{{sfrac|8|5}} || &#8722;{{sfrac|13|8}} || &#8722;{{sfrac|21|13}} || &#8722;{{sfrac|34|21}} || &#8722;{{sfrac|55|34}} || &#8722;{{sfrac|89|55}} || &#8722;{{sfrac|144|89}} || &#8722;{{sfrac|233|144}}
 
| &#8722;2 || &#8722;{{sfrac|3|2}} || &#8722;{{sfrac|5|3}} || &#8722;{{sfrac|8|5}} || &#8722;{{sfrac|13|8}} || &#8722;{{sfrac|21|13}} || &#8722;{{sfrac|34|21}} || &#8722;{{sfrac|55|34}} || &#8722;{{sfrac|89|55}} || &#8722;{{sfrac|144|89}} || &#8722;{{sfrac|233|144}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|&#8730;2
+
| rowspan="2"|{{radic|2}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 1 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
 
| 1 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
 
|-
 
|-
! ra
+
! ''ra''
 
| 1 || {{sfrac|3|2}} || {{sfrac|7|5}} || {{sfrac|17|12}} || {{sfrac|41|29}} || {{sfrac|99|70}} || {{sfrac|239|169}} || {{sfrac|577|408}} || {{sfrac|1&thinsp;393|985}} || {{sfrac|3&thinsp;363|2&thinsp;378}} || {{sfrac|8&thinsp;119|5&thinsp;741}}
 
| 1 || {{sfrac|3|2}} || {{sfrac|7|5}} || {{sfrac|17|12}} || {{sfrac|41|29}} || {{sfrac|99|70}} || {{sfrac|239|169}} || {{sfrac|577|408}} || {{sfrac|1&thinsp;393|985}} || {{sfrac|3&thinsp;363|2&thinsp;378}} || {{sfrac|8&thinsp;119|5&thinsp;741}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|{{sfrac|1|&#8730;2}}
+
| rowspan="2"|{{frac|1|{{radic|2}}}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 0 || 1 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
 
| 0 || 1 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2 || 2
 
|-
 
|-
! ra
+
! ''ra''
 
| 0 || 1 || {{sfrac|2|3}} || {{sfrac|5|7}} || {{sfrac|12|17}} || {{sfrac|29|41}} || {{sfrac|70|99}} || {{sfrac|169|239}} || {{sfrac|408|577}} || {{sfrac|985|1&thinsp;393}} || {{sfrac|2&thinsp;378|3&thinsp;363}}
 
| 0 || 1 || {{sfrac|2|3}} || {{sfrac|5|7}} || {{sfrac|12|17}} || {{sfrac|29|41}} || {{sfrac|70|99}} || {{sfrac|169|239}} || {{sfrac|408|577}} || {{sfrac|985|1&thinsp;393}} || {{sfrac|2&thinsp;378|3&thinsp;363}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|&#8730;3
+
| rowspan="2"|{{radic|3}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2
 
| 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2
 
|-
 
|-
! ra
+
! ''ra''
 
| 1 || 2 || {{sfrac|5|3}} || {{sfrac|7|4}} || {{sfrac|19|11}} || {{sfrac|26|15}} || {{sfrac|71|41}} || {{sfrac|97|56}} || {{sfrac|265|153}} || {{sfrac|362|209}} || {{sfrac|989|571}}
 
| 1 || 2 || {{sfrac|5|3}} || {{sfrac|7|4}} || {{sfrac|19|11}} || {{sfrac|26|15}} || {{sfrac|71|41}} || {{sfrac|97|56}} || {{sfrac|265|153}} || {{sfrac|362|209}} || {{sfrac|989|571}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|{{sfrac|1|&#8730;3}}
+
| rowspan="2"|{{frac|1|{{radic|3}}}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 0 || 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2 || 1
 
| 0 || 1 || 1 || 2 || 1 || 2 || 1 || 2 || 1 || 2 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| 0 || 1 || {{sfrac|1|2}} || {{sfrac|3|5}} || {{sfrac|4|7}} || {{sfrac|11|19}} || {{sfrac|15|26}} || {{sfrac|41|71}} || {{sfrac|56|97}} || {{sfrac|153|265}} || {{sfrac|209|362}}
 
| 0 || 1 || {{sfrac|1|2}} || {{sfrac|3|5}} || {{sfrac|4|7}} || {{sfrac|11|19}} || {{sfrac|15|26}} || {{sfrac|41|71}} || {{sfrac|56|97}} || {{sfrac|153|265}} || {{sfrac|209|362}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|{{sfrac|&#8730;3|2}}
+
| rowspan="2"|{{frac|{{radic|3}}|2}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 0 || 1 || 6 || 2 || 6 || 2 || 6 || 2 || 6 || 2 || 6
 
| 0 || 1 || 6 || 2 || 6 || 2 || 6 || 2 || 6 || 2 || 6
 
|-
 
|-
! ra
+
! ''ra''
 
| 0 || 1 || {{sfrac|6|7}} || {{sfrac|13|15}} || {{sfrac|84|97}} || {{sfrac|181|209}} || {{sfrac|1&thinsp;170|1&thinsp;351}} || {{sfrac|2&thinsp;521|2&thinsp;911}} || {{sfrac|16&thinsp;296|18&thinsp;817}} || {{sfrac|35&thinsp;113|40&thinsp;545}} || {{sfrac|226&thinsp;974|262&thinsp;087}}   
 
| 0 || 1 || {{sfrac|6|7}} || {{sfrac|13|15}} || {{sfrac|84|97}} || {{sfrac|181|209}} || {{sfrac|1&thinsp;170|1&thinsp;351}} || {{sfrac|2&thinsp;521|2&thinsp;911}} || {{sfrac|16&thinsp;296|18&thinsp;817}} || {{sfrac|35&thinsp;113|40&thinsp;545}} || {{sfrac|226&thinsp;974|262&thinsp;087}}   
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|<sup>&#179;</sup>&#8730;2
+
| rowspan="2"|{{radic|2|3}}
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 1 || 3 || 1 || 5 || 1 || 1 || 4 || 1 || 1 || 8 || 1
 
| 1 || 3 || 1 || 5 || 1 || 1 || 4 || 1 || 1 || 8 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| 1 || {{sfrac|4|3}} || {{sfrac|5|4}} || {{sfrac|29|23}} || {{sfrac|34|27}} || {{sfrac|63|50}} || {{sfrac|286|227}} || {{sfrac|349|277}} || {{sfrac|635|504}} || {{sfrac|5&thinsp;429|4&thinsp;309}} || {{sfrac|6&thinsp;064|4&thinsp;813}}
 
| 1 || {{sfrac|4|3}} || {{sfrac|5|4}} || {{sfrac|29|23}} || {{sfrac|34|27}} || {{sfrac|63|50}} || {{sfrac|286|227}} || {{sfrac|349|277}} || {{sfrac|635|504}} || {{sfrac|5&thinsp;429|4&thinsp;309}} || {{sfrac|6&thinsp;064|4&thinsp;813}}
 
|-
 
|-
Line 103: Line 103:
 
| 2 || 1 || 2 || 1 || 1 || 4 || 1 || 1 || 6 || 1 || 1
 
| 2 || 1 || 2 || 1 || 1 || 4 || 1 || 1 || 6 || 1 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| 2 || 3 || {{sfrac|8|3}} || {{sfrac|11|4}} || {{sfrac|19|7}} || {{sfrac|87|32}} || {{sfrac|106|39}} || {{sfrac|193|71}} || {{sfrac|1&thinsp;264|465}} || {{sfrac|1&thinsp;457|536}} || {{sfrac|2&thinsp;721|1&thinsp;001}}
 
| 2 || 3 || {{sfrac|8|3}} || {{sfrac|11|4}} || {{sfrac|19|7}} || {{sfrac|87|32}} || {{sfrac|106|39}} || {{sfrac|193|71}} || {{sfrac|1&thinsp;264|465}} || {{sfrac|1&thinsp;457|536}} || {{sfrac|2&thinsp;721|1&thinsp;001}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
| rowspan="2"|''&#960;''
+
| rowspan="2"|''&pi;''
 
! ''a<sub>r</sub>''
 
! ''a<sub>r</sub>''
 
| 3 || 7 || 15 || 1 || 292 || 1 || 1 || 1 || 2 || 1 || 3
 
| 3 || 7 || 15 || 1 || 292 || 1 || 1 || 1 || 2 || 1 || 3
 
|-
 
|-
! ra
+
! ''ra''
 
| 3 || {{sfrac|22|7}} || {{sfrac|333|106}} || {{sfrac|355|113}} || {{sfrac|103&thinsp;993|33&thinsp;102}} || {{sfrac|104&thinsp;348|33&thinsp;215}} || {{sfrac|208&thinsp;341|66&thinsp;317}} || {{sfrac|312&thinsp;689|99&thinsp;532}} || {{sfrac|833&thinsp;719|265&thinsp;381}} || {{sfrac|1&thinsp;146&thinsp;408|364&thinsp;913}} || {{sfrac|4&thinsp;272&thinsp;943|1&thinsp;360&thinsp;120}}
 
| 3 || {{sfrac|22|7}} || {{sfrac|333|106}} || {{sfrac|355|113}} || {{sfrac|103&thinsp;993|33&thinsp;102}} || {{sfrac|104&thinsp;348|33&thinsp;215}} || {{sfrac|208&thinsp;341|66&thinsp;317}} || {{sfrac|312&thinsp;689|99&thinsp;532}} || {{sfrac|833&thinsp;719|265&thinsp;381}} || {{sfrac|1&thinsp;146&thinsp;408|364&thinsp;913}} || {{sfrac|4&thinsp;272&thinsp;943|1&thinsp;360&thinsp;120}}
 +
|+
 +
! Number !! ''r'' !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
 
|}
 
|}
 
'''ra''': rational approximant obtained by expanding continued fraction up to ''a<sub>r</sub>''
 
'''ra''': rational approximant obtained by expanding continued fraction up to ''a<sub>r</sub>''

Latest revision as of 15:09, 31 August 2020

Number r 0 1 2 3 4 5 6 7 8 9 10
123 ar 123
ra 123
12.3 ar 12 3 3
ra 12 Template:Sfrac Template:Sfrac
1.23 ar 1 4 2 1 7
ra 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
0.123 ar 0 8 7 1 2 5
ra 0 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
ϕ =
Template:Sfrac
ar 1 1 1 1 1 1 1 1 1 1 1
ra 1 2 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
ϕ =
Template:Sfrac
ar −2 2 1 1 1 1 1 1 1 1 1
ra −2 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
Template:Radic ar 1 2 2 2 2 2 2 2 2 2 2
ra 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
1Template:Radic ar 0 1 2 2 2 2 2 2 2 2 2
ra 0 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
Template:Radic ar 1 1 2 1 2 1 2 1 2 1 2
ra 1 2 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
1Template:Radic ar 0 1 1 2 1 2 1 2 1 2 1
ra 0 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
Template:Radic2 ar 0 1 6 2 6 2 6 2 6 2 6
ra 0 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
Template:Radic ar 1 3 1 5 1 1 4 1 1 8 1
ra 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
e ar 2 1 2 1 1 4 1 1 6 1 1
ra 2 3 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
π ar 3 7 15 1 292 1 1 1 2 1 3
ra 3 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
Number r 0 1 2 3 4 5 6 7 8 9 10

ra: rational approximant obtained by expanding continued fraction up to ar