|Subject:||Three old questions about the feasibility of encoding any number with decimal expansions (not a proposal for a magical compression algorithm, just the plain questions I am curious about)|
|Posted by:||dario.mx (dario.…@gmail.com)|
|Date:||Sun, 12 Feb 2012|
I thought this would be a good place to ask these questions, given
that they are often involved in flawed arguments about magical
compression, involving decimal expansion of irrationals.
I am just curious whether these questions have a positive answer or
not, regardless of any practical usage for compression. Actually, I am
not even asking for a constructive proof (one that shows how to
calculate the numbers in question) ... a proof that shows they do
exist or not would be enough to bring peace to my soul.
Thanks in advance.
PS1: Feel free to suggest other groups, forums or places that are more
appropriate for this matter.
PS2: These questions may have been asked and answered already
somewhere, a link to the reference would be very welcome here.
1) For any finite sequence of digits, is there a rational number whose
decimal expansion period matches precisely that sequence?
2) For any finite sequence of digits, is there an irrational number
whose decimal expansion contains precisely that sequence?
3) For any finite sequence of digits, is there an irrational number
whose decimal expansion starts precisely with that sequence?