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{center [[rationals]] | [[rationals2]] | [[fractions]] | fractions2 | [[habiter]]}

_img https://wikimedia.org/api/rest_v1/media/math/render/svg/247535cef4b9b94eabeb16908cf72436cd01d0c9

_h1 continued fractions

{pre

F = [a{sub 0};a{sub 1},a{sub 2},...,a{sub n}]
  = a{sub 0} + 1
         {{bt}a{sub 1} + 1}
              {{bt}a{sub 2} + 1}
                   {{bt} ... 1}
                       {{bt}a{sub n}} 
}

_h2 algorithm

{pre 
a = [a{sub 0}, a{sub 1}, ..., a{sub n-1}]
u(0) = a[0]
u(n) = 1 / (a[n] + u(n-1))
}

_h2 code

{pre
'{def frac
 
{def frac.r
 {lambda {:f :u :i}
  {if {= :i 0}
   then {+ {:f 0} :u}  
   else {frac.r :f
                {/ 1 {+ {:f :i} :u}}
                {- :i 1}}}}}
 
{lambda {:f :n}
  {frac.r :f 1 :n}}} 
->  {def frac
 {def frac.r
 {lambda {:f :u :i}
  {if {= :i 0}
   then {+ {:f 0} :u}  
   else {frac.r :f
                {/ 1 {+ {:f :i} :u}}
                {- :i 1}}}}}
 {lambda {:f :n}
  {frac.r :f 1 :n}}}
}

_h2 computing φ, √2, √3, e, π, ...

{prewrap
φ = [1 1 1 1 1 ...]
'{frac {lambda {:n} 1}
 40}
-> {frac {lambda {:n} 1}
 40}
   {/ {+ 1 {sqrt 5}} 2}

√2 = [1 2 2 2 2 ...]
'{frac {lambda {:n}
        {if {= :n 0} then 1 else 2}}
 20}
-> {frac {lambda {:n}
        {if {= :n 0} then 1 else 2}}
 20}
   {sqrt 2}

√3 = [1 1 2 1 2 1 2 ...]
'{frac {lambda {:n}
        {if {= :n 0}
         then 1
         else {if {= {% :n 2} 0}
         then 2
         else 1}}}
 30}
-> {frac {lambda {:n}
        {if {= :n 0}
         then 1
         else {if {= {% :n 2} 0}
         then 2
         else 1}}}
 30}
   {sqrt 3}

e = [2 1 2 1 1 4 1 1 6 1  1  8 ...]
'{frac {lambda {:n}
         {A.get :n {A.new 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 14}}}
 10} 
-> {frac {lambda {:n}
         {A.get :n {A.new 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 14}}}
 20}
   {E}

π = [3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2]
'{frac {lambda {:n}
         {A.get :n {A.new 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2}}}
 15}
-> {frac {lambda {:n}
         {A.get :n {A.new 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2}}}
 15}
   {PI}

}

°°°
{pre

√2 = [1;2,2,2,2,...]
   = 1 + 1
         {{bt}2 + 1}
             {{bt}2 + 1}
                 {{bt}2 + 1}
                     {{bt}2 + ...}

'{frac 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2}
-> {frac 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2} 
   {sqrt 2}  // √2

√3 = [1;1,2,1,2,1,2...]  // repeat 1 2
   = 1 + 1
         {{bt}1 + 1}
             {{bt}2 + 1}
                 {{bt}1 + 1}
                     {{bt}2 + ...}

'{frac 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2}
-> {frac 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2}
   {sqrt 3}  // √3

e = [2;1,2,1,1,4,1,1,6,1,1,8...]   // repeat 1 1 2n
  = = 2 + 1
          {{bt}1 + 1}
              {{bt}2 + 1}
                  {{bt}1 + 1}
                      {{bt}1 + 1}
                          {{bt}4 + 1} 
                              {{bt}1 + ...}

'{frac 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 14 1 1 16}
-> {frac 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 14 1 1 16}
   {E}   // e

π = [3;7,15,1,292,1,1,1,...]
  = 3 + 1
        {{bt}7 + 1}
            {{bt}15 + 1}
                 {{bt}1 + 1}
                     {{bt}292 + ...}

'{frac 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2}
-> {frac 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2}
   {PI}   // π
}

_h2 generalized continuous fractions

_p From [[https://en.wikipedia.org/wiki/Continued_fraction|https://en.wikipedia.org/wiki/Continued_fraction]] " The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern. However, several generalized continued fractions for π have a perfectly regular structure, such as:

;; _img https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdee0680f07399324081b832e47d597ceabdff6

;; _p The first two of these are special cases of the arctangent function with π = 4 arctan (1).
 
{pre
π = 4
    {{bt}1 + 1{sup 2}}
        {{bt}2 + 3{sup 2}}
            {{bt}2 + 5{sup 2}}
                {{bt}2 + 7{sup 2}}
                    {{bt}2 + 9{sup 2}}
                        {{bt}2 + ...}
}

_p {i alain marty | 2002/0417}

_h2 refs

_ul [[https://en.wikipedia.org/wiki/Continued_fraction|https://en.wikipedia.org/wiki/Continued_fraction]]
_ul [[villemin frac cont|http://villemin.gerard.free.fr/Wwwgvmm/Nombre/FracCont.htm]]
_ul [[villemin frac cont constants|http://villemin.gerard.free.fr/Wwwgvmm/Nombre/FCvaleur.htm#top]]
_ul [[t-richez|http://t-richez.pagesperso-orange.fr/ressources/recherche/memoire_fractions_continues.pdf]]
_ul [[michel.waldschmidt|https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/FractionsContinuesVI.pdf]]

°°°

;; _img http://villemin.gerard.free.fr/Wwwgvmm/Nombre/FCvaleur_fichiers/image011.jpg 
;; _img http://villemin.gerard.free.fr/Wwwgvmm/Nombre/FCvaleur_fichiers/image013.jpg

{{hide}
{def bt span {@ style="border-top:1px solid;"}}
}

rationals | rationals2 | fractions | fractions2 | habiter

continued fractions

F = [a₀;a₁,a₂,...,a_n]
  = a₀ + 1
         a₁ + 1
              a₂ + 1
                   ... 1
                       a_n

algorithm

a = [a₀, a₁, ..., a_n-1]
u(0) = a[0]
u(n) = 1 / (a[n] + u(n-1))

code

{def frac
 
{def frac.r
 {lambda {:f :u :i}
  {if {= :i 0}
   then {+ {:f 0} :u}  
   else {frac.r :f
                {/ 1 {+ {:f :i} :u}}
                {- :i 1}}}}}
 
{lambda {:f :n}
  {frac.r :f 1 :n}}} 
->  frac

computing φ, √2, √3, e, π, ...

φ = [1 1 1 1 1 ...]
{frac {lambda {:n} 1}
 40}
-> 1.618033988749895
   1.618033988749895

√2 = [1 2 2 2 2 ...]
{frac {lambda {:n}
        {if {= :n 0} then 1 else 2}}
 20}
-> 1.4142135623730954
   1.4142135623730951

√3 = [1 1 2 1 2 1 2 ...]
{frac {lambda {:n}
        {if {= :n 0}
         then 1
         else {if {= {% :n 2} 0}
         then 2
         else 1}}}
 30}
-> 1.7320508075688772
   1.7320508075688772

e = [2 1 2 1 1 4 1 1 6 1  1  8 ...]
{frac {lambda {:n}
         {A.get :n {A.new 2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 14}}}
 10} 
-> 2.7182818284590455
   2.718281828459045 

π = [3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2]
{frac {lambda {:n}
         {A.get :n {A.new 3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2}}}
 15}
-> 3.141592653589793
   3.141592653589793