Re: query on number theory

From: Richard Henry (rphenry_at_home.com)
Date: 07/17/04 

Date: Sat, 17 Jul 2004 13:28:09 -0700

"Rob Johnson" <rob@trash.whim.org> wrote in message
news:20040717.031328@whim.org...
>
> >3)0.9999... <> 1? Is it because we follow decimal notation. It is
> >clear that binary representation cannot represent all the fractions
> >even if the number of bits is huge as it can only represent fractions
> >as multiple of 2^(-n).
> >Can a base > 10 represent all the fractions that can be represented in
> >decimal format along with even more other precise fractions.
>
> If by .9999... you mean that the 9s repeat indefinitely, then your
> inequality is false. 0.9999... = 1. To represent a fraction exactly in
> a finite number of digits, the denominator of the fraction, in lowest
> terms, must be only divisible by primes that divide the numerical base.
>
> Certainly any base which is divisible by ten can exactly represent any
> fraction in finitely many digits that can be so represented in base ten.
> To include more fractions, the base must be divisible by other primes.
> For example, the same fractions are exactly representable in finitely
> many digits in base ten and base twenty. However, more are exactly
> representable in finitely many digits in base 30, which represents the
> same fractions as base 60.

A curious aside that I noticed while fiddling with the numbers preparing my
earlier response.

In base 30, 0.99999.... = 1/3.

Relevant Pages

  • Re: Fractions: 1/7 and others like it
    ... six digits repeat in the same order. ... Find all sets of fractions less ... It's all the ones where 1/n has a repeating decimal of period n-1, ... Pre-computers, William Shanks computed k for all primes less than 120000, and in ...
    (rec.puzzles)
  • Re: Type Question
    ... this is true of all programming languages that use floating point ... Common Lisp comes pretty close to providing that as ... Change the floating point representation. ... computationally efficient binary representation, the fractions should be ...
    (comp.lang.lisp)
  • Re: Mapping rationals to binary strings while preserving order
    ... the mapped representation as a bitstream. ... may signify the two fractions 01;011 or the only fraction 01011 or even ... Is the max. denominator 9 as per your example? ...
    (comp.theory)
  • Re: Decimal versus binary arithmetic was Re: J4 - presentation/discussion on "Future of the COBO
    ... One very good reason that pure binary representation was not used is that ... decimal fractions cannot be expressed except as a repeating ... is your meaning, Robert, then you haven't made it clear. ... 01 quotient binary pic 9v9. ...
    (comp.lang.cobol)
  • Re: query on number theory
    ... >>even if the number of bits is huge as it can only represent fractions ... must be only divisible by primes that divide the numerical base. ... > fraction in finitely many digits that can be so represented in base ten. ... My question is does all our decimal or fractional representation are ...
    (sci.math)