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- 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)

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 |

Dear Sirs,

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.

Questions:

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?