Re: Integers and Mathematical Correctness

[AdaMagica <christ-usch.grein@t-online.de>]

On Saturday, September 27, 2014 8:20:32 PM UTC+2, Adam Beneschan wrote:
> It may choose it for *one* of the operands, depending on whether these functions are directly visible:
>
>     function "+" (Left : Rational; Right : Integer) return Rational;
>     function "+" (Left : Integer; Right : Rational) return Rational;
>
> Actually, if either of those is visible, then there is more than one way to
> choose the meanings of "+" and "/" to form an acceptable interpretation.
> Because of this, the expression would be treated as ambiguous, and the
> compiler would reject it.

Exactly that's my point, the compiler will reject it. (You're right, the result is not 0, the compiler rejects the code. My error!)

With Integer, there's always visible with preference

function "/" (Left, Right: Integer) return Integer;

which will ruin 1/2 + 2/3, because your two functions above are there. Thus these interpretations are possible, which the compiler cannot resolve:

Rational + Rational = Rational
Integer  + Rational = Rational
Rational + Integer  = Rational

So you have to remove the integer division, which is only possible for a new type Whole:

overriding function "/" (Left, Right: Whole) return Whole is abstract;

Now everything can work. There is of course no

function "+" (Left, Right: Whole) return Rational;

So in short: The crux is not rational numbers combined with themselves, but mixed arithmetics (i.e. rational and whole).

So I still do not believe that you can do mixed arithmetics with rationals constructed from Integer:

  function "+" (Left: Rational; Right: Integer ) return Rational;
  function "+" (Left: Integer ; Right: Rational) return Rational;
  function "+" (Left: Rational; Right: Rational) return Rational;
  "/" (Left, Right: Integer) return Rational;  -- constructor
  "/" (Left, Right: Integer) return Integer;   -- from Standard

You cannot hide the last function!

[OK, you can make 1/2 + 1/3 work by qualifying the functions, but that's not the point (and extremely ugly.]