Euler's Continued Fraction Theorem for fractionsHow can I use Euler's Continued Fraction Theorem to find the continued fraction expansion for a (ordinary, finite) fraction via its (terminating or recurring) decimal expansion, rather than via the more obvious Euclid's Algorithm?The examples I'm thinking of are 5 / 7 and 3 / 8 each of which can be thought of as a series in a power of 10 5 / 7 = 0. 571428 ¯ = ∑ k = 1 ∞ 571428 ( 10 − 6 ) k and 3 / 8 = 0.375 = 3 ∗ 10 − 1 + 7 ∗ 10 − 2 + 5 ∗ 10 − 3 .I'd like to use these coefficients and expansion points together with Euler's Theorem to show the continued fraction expansions 5 / 7 = [ 0 ; 1 , 2 , 2 ]and 3 / 8 = [ 0 ; 2 , 1 , 2 ] .

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Euler&#039;s Continued Fraction Theorem for fractionsHow can I use Euler&#039;s Continued Fraction Theorem to find the continued fraction expansion for a (ordinary, finite) fraction via its (terminating or recurring) decimal expansion, rather than via the more obvious Euclid&#039;s Algorithm?The examples I&#039;m thinking of are   5      /    7 and   3      /    8 each of which can be thought of as a series in a power of   10  5      /    7  =  0.      571428    ¯    =      ∑          k      =      1        ∞    571428  (      10          −      6            )    k  and  3      /    8  =  0.375  =  3  ∗      10          −      1        +  7  ∗      10          −      2        +  5  ∗      10          −      3        .I&#039;d like to use these coefficients and expansion points together with Euler&#039;s Theorem to show the continued fraction expansions  5      /    7  =  [  0  ;  1  ,  2  ,  2  ]and  3      /    8  =  [  0  ;  2  ,  1  ,  2  ]  .

Harold Cantrell · Accepted Answer

You could use the repeating digits to come up with a non-standard continued fraction in that the terms will not be integers. With sufficient applications of equivalence transforms these can be converted to integers.in your example:         a    0    =  .571428         a    i    =      10          −      6           x  =  5      /    7  =      a    0    +      a    0        a    1    +      a    0        a    1        a    2    +  .  .  .  =            a      0              1      −                        a          1                          1          +                      a            1                    −                                    a              2                                      1              +                              a                2                            −              ⋱                                               x  =      .571428          1      −                        10                      −            6                                    1          +                      10                          −              6                                −                                    10                              −                6                                                    1              +                              10                                  −                  6                                            −              ⋱                                               =      571428          1000000      −              1000000                  1000001          −                      1000000                          1000001              −              ⋱

Euler's Continued Fraction Theorem for fractions How can I use Euler's Continued Fraction Theorem

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