Working with the set of real numbers (was: Finding size of Variable)

[Steven D'Aprano]

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