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
@GeoffryGifari Жыл бұрын
how do you get the continued fraction of π, knowing π first?
@martinepstein9826 Жыл бұрын
You find a rational approximation for pi (there are some very good formulas you can use), then convert that approximation into a continued fraction.
@GeoffryGifari Жыл бұрын
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 Жыл бұрын
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.
@hamdaniyusuf_dani Жыл бұрын
pi can be represented in regular patterns using generalized continued fraction. en.wikipedia.org/wiki/Generalized_continued_fraction#%CF%80
@ExhaustedPenguin Жыл бұрын
Is there a least irrational irrational number?
@discovermaths Жыл бұрын
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 Жыл бұрын
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.