Difference between revisions of "Template:Continued fraction examples"

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imported>Cedar101
m ({{sqrt}})
imported>Cedar101
m (''ra'')
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}}
 
|-
 
|-
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| 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}}
 
|-
 
|-
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| 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}}
 
|-
 
|-
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| 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}}
 
|-
 
|-
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| −2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
| −2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 
|-
 
|-
! ra
+
! ''ra''
 
| −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}} || −{{sfrac|233|144}}
 
| −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}} || −{{sfrac|233|144}}
 
|-
 
|-
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| 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|1 393|985}} || {{sfrac|3 363|2 378}} || {{sfrac|8 119|5 741}}
 
| 1 || {{sfrac|3|2}} || {{sfrac|7|5}} || {{sfrac|17|12}} || {{sfrac|41|29}} || {{sfrac|99|70}} || {{sfrac|239|169}} || {{sfrac|577|408}} || {{sfrac|1 393|985}} || {{sfrac|3 363|2 378}} || {{sfrac|8 119|5 741}}
 
|-
 
|-
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| 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|1 393}} || {{sfrac|2 378|3 363}}
 
| 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|1 393}} || {{sfrac|2 378|3 363}}
 
|-
 
|-
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| 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}}
 
|-
 
|-
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| 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}}
 
|-
 
|-
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| 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|1 170|1 351}} || {{sfrac|2 521|2 911}} || {{sfrac|16 296|18 817}} || {{sfrac|35 113|40 545}} || {{sfrac|226 974|262 087}}   
 
| 0 || 1 || {{sfrac|6|7}} || {{sfrac|13|15}} || {{sfrac|84|97}} || {{sfrac|181|209}} || {{sfrac|1 170|1 351}} || {{sfrac|2 521|2 911}} || {{sfrac|16 296|18 817}} || {{sfrac|35 113|40 545}} || {{sfrac|226 974|262 087}}   
 
|-
 
|-
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| 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|5 429|4 309}} || {{sfrac|6 064|4 813}}
 
| 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|5 429|4 309}} || {{sfrac|6 064|4 813}}
 
|-
 
|-
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| 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|1 264|465}} || {{sfrac|1 457|536}} || {{sfrac|2 721|1 001}}
 
| 2 || 3 || {{sfrac|8|3}} || {{sfrac|11|4}} || {{sfrac|19|7}} || {{sfrac|87|32}} || {{sfrac|106|39}} || {{sfrac|193|71}} || {{sfrac|1 264|465}} || {{sfrac|1 457|536}} || {{sfrac|2 721|1 001}}
 
|-
 
|-
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| 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|103 993|33 102}} || {{sfrac|104 348|33 215}} || {{sfrac|208 341|66 317}} || {{sfrac|312 689|99 532}} || {{sfrac|833 719|265 381}} || {{sfrac|1 146 408|364 913}} || {{sfrac|4 272 943|1 360 120}}
 
| 3 || {{sfrac|22|7}} || {{sfrac|333|106}} || {{sfrac|355|113}} || {{sfrac|103 993|33 102}} || {{sfrac|104 348|33 215}} || {{sfrac|208 341|66 317}} || {{sfrac|312 689|99 532}} || {{sfrac|833 719|265 381}} || {{sfrac|1 146 408|364 913}} || {{sfrac|4 272 943|1 360 120}}
 
|}
 
|}
 
'''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>''

Revision as of 05:36, 8 November 2018

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:Sqrt 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:Sqrt 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:Sqrt 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:Sqrt 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:Sqrt2 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
³Template:Sqrt 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

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