1
vote
3answers
95 views

Convergents of square root of 2

On wikipedia I read about the continued fraction of the square root of 2: $$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{...}}}}$$ The first convergents are ...
3
votes
2answers
64 views

Why is the coefficient in front of $\sqrt n$ always 1 in the intermediate terms for finding the continued fraction expansion of $\sqrt n$?

After playing around on paper for a bit, I came up with a short python generator to find the continued fraction expansion of $\sqrt n$. I understand why it gets the right answer when it gets an ...
12
votes
1answer
179 views

Request for a proof of the following continued-fraction identity

I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close): $$ ...
4
votes
2answers
493 views

Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...