Re: rational to floating point

On Sun, 9 Mar 2003, Thant Tessman wrote:
>
> This is a math question, but one I hope is relevant to this newsgroup.


Possibly also comp.programming. Cross-posted.


> I've been writing an interpreter for a functional programming language
> in C++. One of the datatypes it supports is an arbitrary-precision
> rational number type. Currently the interpreter displays rational
> numbers as the ratio of two integers. I'd like it to display rational
> numbers as floating-point numbers whenever the conversion to floating
> point won't produce an infinite (repeating) stream of digits.
>
> The question is: Under what conditions will a rational number produce
> an infinite stream of digits for a given base? What I've come up with
> is this:
>
> Converting a rational number to a floating point value is equivalent
> to multiplying the numerator and denominator by some number that
> converts the denominator to a whole-number power of the base. That is:
> b^n = c * d where 'b' is the base, 'd' is the denominator, and 'n' and
> 'c' are some whole numbers that satisfy the equation.


This is absolutely correct. But for interfacing to real computer
languages like C++, you also have the constraint that 'b' is 2 (or
FLT_RADIX) and 'n' is no larger than some fixed value (the number of bits
in the mantissa of a floating-point number; I forget what C++ calls it
exactly).
    But I see from your website that you probably want to implement your
own floating-point class also, and you don't care about sizeof(double).
So that constraint isn't as relevant as it might be.


> I think there is no solution to the above equation if the denominator
> of the original rational number and the base contain no prime factors
> in common.


More generally, there is a solution c to (b^n = c*d) *if and only
if* d has no prime factors which are not prime factors of b. (For
example, 1/12 is not terminating in base 4, because 3 does not divide
4. But 1/4 is terminating in base 12, because 2 divides 12.)


> And I think that this in turn implies that if and only if
> gcd(d,b) is 1 and 'd' is not 1, then the original rational number can
> only be represented by an infinite stream of digits.


I don't think that's quite equivalent to what I said above, but you
were on the right track.


> Is my reasoning sound? Is there a simpler test?


See above.


> For the curious, a description and source for my interpreter can be
> found at:
>
> http://www.standarddeviance.com
>
> My gratitude goes to folks in this newsgroup who have helped improve my
> understanding of concepts I put to use building this.
>
> -thant


Hope this helps.
-Arthur

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