On Tue, 04 Mar 2014 14:46:25 +1100, Chris Angelico wrote:
> That's neat, didn't know that. Is there an efficient way to figure out,
> for any integer N, what its sqrt's CF sequence is? And what about the
> square roots of non-integers - can you represent √π that way? I suspect,
> though I can't prove, that there will be numbers that can't be
> represented even with an infinite series - or at least numbers whose
> series can't be easily calculated.
Every rational number can be written as a continued fraction with a
finite number of terms[1]. Every irrational number can be written as a
continued fraction with an infinite number of terms, just as every
irrational number can be written as a decimal number with an infinite
number of digits. Most of them (to be precise: an uncountably infinite
number of them) will have no simple or obvious pattern.
[1] To be pedantic, written as *two* continued fractions, one ending with
the term 1, and one with one less term which isn't 1. That is:
[a; b, c, d, ..., z, 1] == [a; b, c, d, ..., z+1]
Any *finite* CF ending with one can be simplified to use one fewer term.
Infinite CFs of course don't have a last term.
--
Steven
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