Does every irrational number have continued fraction representation?(Rational approximations of

polivijuye

polivijuye

Answered question

2022-06-20

Does every irrational number have continued fraction representation?(Rational approximations of e ,, and γ)

Answer & Explanation

mar1nerne

mar1nerne

Beginner2022-06-21Added 20 answers

You can calculate continued fraction representations, where you can choose to terminate after you've reached some level of accuracy:
e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , ]
Here are some of the first results:
e 19 7 , 87 32 , 106 39 ,
Another (beautiful, but non-simple) continued fraction representation of e is
e = 2 + 1 1 + 1 2 + 2 3 + 3 3 + .
Same for γ:
γ = [ 0 ; 1 , 1 , 2 , 1 , 2 , 1 , 4 , 3 , 13 , 5 , 1 , 1 , 8 , 1 , 2 , 4 , 1 , 1 , 40 , ] ,
where the first few (useful) convergents are
γ 11 19 , 15 26 , 71 123 ,

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