Continued fractions

Continued fractions, both finite and infinite, are a fascinating and revealing way of representing real numbers.
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[@GeoffryGifari]

oh, and since we know a series expansion of a number (like pi) and also its continued fraction, maybe we can figure out a relationship between both? maybe there is an entire set of theorems relating series and continued fractions

[@hamdaniyusuf_dani]

pi can be represented in regular patterns using generalized continued fraction. en.wikipedia.org/wiki/Generalized_continued_fraction#%CF%80

[@miraculix666]

[3;7,15,1]=355/113

[@GeoffryGifari]

maybe the coefficients of continued fractions of irrationals mentioned here are small because the those numbers themselves are small? what if we expand something like √15555401097000167000011

[@martinepstein9826]

Making the number itself large only makes the a_0 term large. To see why the terms are generally small look up Khinchin's constant.

[@GeoffryGifari]

how do you get the continued fraction of π, knowing π first?

[@martinepstein9826]

You find a rational approximation for pi (there are some very good formulas you can use), then convert that approximation into a continued fraction.

[@ExhaustedPenguin]

Is there a least irrational irrational number?

[@discovermaths]

Good question! The answer is "no" for the same reason there isn't a biggest number. The "least irrational irrational number" would be the one with the biggest partial quotients - but there's no such thing.

[@martinepstein9826]

I feel that Phi = (sqrt(5) + 1)/2 = [1;1,1,1,1,...] is the least irrational irrational number. People usually say the opposite because it's hard to approximate by rationals. But the rationals themselves are the hardest numbers to approximate by rationals (if we stipulate that you can't use the number itself) so that property actually makes Phi similar to a rational number. Also, Phi is a quadratic irrational so it's just one step removed from being rational.