Re: Square roots - decimal expansions --

bischar wrote:

Yes, probably every numberan being has its own body guard that makes secret things in the coulisses of the decimal part :)

Another question, are there numbers who have palindroms in their decimal part (other than '0,base of a palindrom') or is it a caracteristic of the integer part ?

Same question for the repeating sequences in the decimal part.


In this metaphor rational numbers may or may not have a body guard and may or may not have secret things of a finite size lurking in the coulisses.

Assuming the notation p/q (p integer, q positive integer) then

if q=1 then there is no body guard and no secret to hide.

If q contains only the prime factors 2 and/or 5 then there is a body guard that has no secret to protect: the decimal expansion terminates.

If q>1 and q contains neither a prime factor 2 nor a prime factor 5, then there is no body guard, and the decimal expansion is periodical and the period begins right after the decimal point.

If q contains a prime factor 2 and/or a prime factor 5, and moreover one or more prime factors different from 2 and 5, then p/q has a body guard as well as a periodic secret in the coulisses, i.e. its decimal expansion starts with some digits that do not repeat themselves in that sequence, and eventually shows up a period.

Carl Friedrich Gauss found out about all this in his boyhood time. His discovery of the constructibility of the regular 17-gon by compass and unmarked ruler only finally made him prefer a mathematics study to a study in law.

BTW, for palindromic decimal expansions just take a palindromic string of digits, and interpret is as a decimal expansion or as the period of a decimal expansion.
It equals some rational number p/q, but there is no general rule to predict from given p and q if p/q has a palindromic period.

Johan E. Mebius
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