[issue1682] Move Demo/classes/Rat.py to Lib/rational.py and fix it up.

[Raymond Hettinger]

Raymond Hettinger added the comment:

> Decimal.from_rational(Rat(1, 3)) wouldn't be lossless

It should be implemented as Decimal(1)/Decimal(3) and then let the
context handle any inexactness.

> Rational.from_decimal is easier than from_float.

Yes.  Just use Decimal.as_tuple() to get the integer components.

> Then Decimal.from_rational() could rely on just numbers.
> Rational, so it would be independent of this module. 
>Is that a method you'd want on Decimal anyway? 

Instead, just expand the decimal constructor to accept a rational input.

> Regarding float->rational, I propose to refuse Rational(1.1)
> for the same reasons as Decimal(1.1) is refused,

+1 

> but to add a separate constructor (a class method?) that 
> converts a float to a rational exactly (as long as it 
> isn't nan or inf), large denominators be damned. This 
> can help people who are interested in taking floating
> point numbers apart. 

Here's how to pick the integer components out of a float:

   mant, exp = math.frexp(x)
   mant, exp = int(mant * 2.0 ** 53), exp-53


>> * Likewise, follow decimal's lead in avoiding all 
>> automatic coercions from floats:  
>>    Rational(3,10) + 0.3 for example.  The two don't mix.

> I'm reluctant to remove the fallback to float,

You will need some plan to scale-down long integers than exceed the
range of floats (right shifting the numerator and denominator until they
fit).

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