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

seupeljewj

seupeljewj

Answered question

2022-06-22

Euler's Continued Fraction Theorem for fractions
How 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 ] .

Answer & Explanation

Harold Cantrell

Harold Cantrell

Beginner2022-06-23Added 21 answers

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

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