Difference between revisions of "Template:Continued fraction examples"

From blackwiki
Jump to navigation Jump to search
imported>Cmglee
(Add trigonometric function values for common numbers)
imported>Cmglee
(Add rational numbers)
Line 1: Line 1:
 
{| class="wikitable" style="text-align:center;"
 
{| class="wikitable" style="text-align:center;"
 
! Number !! ''r'' !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
 
! Number !! ''r'' !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
 +
|-
 +
|- style="font-weight:bold;"
 +
| rowspan="2"|123
 +
! ''a<sub>r</sub>''
 +
| 123
 +
|-
 +
! ra
 +
| 123
 +
|-
 +
|- style="font-weight:bold;"
 +
| rowspan="2"|12.3
 +
! ''a<sub>r</sub>''
 +
| 12 || 3 || 3
 +
|-
 +
! ra
 +
| 12 || {{sfrac|37|3}} || {{sfrac|123|10}}
 +
|-
 +
|- style="font-weight:bold;"
 +
| rowspan="2"|1.23
 +
! ''a<sub>r</sub>''
 +
| 1 || 4 || 2 || 1 || 7
 +
|-
 +
! ra
 +
| 1 || {{sfrac|5|4}} || {{sfrac|11|9}} || {{sfrac|16|13}} || {{sfrac|123|100}}
 +
|-
 +
|- style="font-weight:bold;"
 +
| rowspan="2"|0.123
 +
! ''a<sub>r</sub>''
 +
| 0 || 8 || 7 || 1 || 2 || 5
 +
|-
 +
! ra
 +
| 0 || {{sfrac|1|8}} || {{sfrac|7|57}} || {{sfrac|8|65}} || {{sfrac|23|187}} || {{sfrac|123|1&thinsp;000}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
Line 9: Line 41:
 
! 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;"
 +
| rowspan="2"|&#8722;''&#981;'' =<br /> &#8722;{{sfrac|&#8730;5 + 1|2}}
 +
! ''a<sub>r</sub>''
 +
| &#8722;2 || 2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 +
|-
 +
! 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}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"
Line 49: Line 89:
 
! 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;"
 +
| rowspan="2"|<sup>&#179;</sup>&#8730;2
 +
! ''a<sub>r</sub>''
 +
| 1 || 3 || 1 || 5 || 1 || 1 || 4 || 1 || 1 || 8 || 1
 +
|-
 +
! 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}}
 
|-
 
|-
 
|- style="font-weight:bold;"
 
|- style="font-weight:bold;"

Revision as of 13:57, 21 August 2014

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
√2 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
Template:Sfrac 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
√3 ar 2 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
Template:Sfrac 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:Sfrac 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
³√2 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