Difference between revisions of "Template:Continued fraction examples"
Jump to navigation
Jump to search
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"|''ϕ'' =<br /> {{sfrac| | + | | rowspan="2"|''ϕ'' =<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"|−''ϕ'' =<br /> −{{sfrac| | + | | rowspan="2"|−''ϕ'' =<br /> −{{sfrac|{{radic|5}} + 1|2}} |
! ''a<sub>r</sub>'' | ! ''a<sub>r</sub>'' | ||
| −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}} | ||
|- | |- | ||
|- style="font-weight:bold;" | |- style="font-weight:bold;" | ||
| − | | rowspan="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|1 393|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}} | ||
|- | |- | ||
|- style="font-weight:bold;" | |- style="font-weight:bold;" | ||
| − | | rowspan="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|1 393}} || {{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}} | ||
|- | |- | ||
|- style="font-weight:bold;" | |- style="font-weight:bold;" | ||
| − | | rowspan="2"| | + | | 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"|{{ | + | | 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"|{{ | + | | 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|1 170|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}} | ||
|- | |- | ||
|- style="font-weight:bold;" | |- style="font-weight:bold;" | ||
| − | | rowspan="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|5 429|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}} | ||
|- | |- | ||
| 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|1 264|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}} | ||
|- | |- | ||
|- style="font-weight:bold;" | |- style="font-weight:bold;" | ||
| − | | rowspan="2"|''& | + | | rowspan="2"|''π'' |
! ''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|103 993|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}} | ||
| + | |+ | ||
| + | ! 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
ra: rational approximant obtained by expanding continued fraction up to ar