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Re: Integer residue-classes [was: Questions about srfi-77 Generic Arithmetic]



About certain issues that have been raised recently on this list (and some other random observations):

1. I now understand Egner et al.'s motivation of having (div x 0) => 0 and (mod x 0) => x. I still disagree with having it in a programming language; see (2).

2. If someone passes 0 as the second argument to a division/quotient/ remainder/modulo/div/mod operator, I think it is infinitely more likely to be an error than to mean return 0 (as div) or return the first argument (as mod), because the programmer was thinking of the result of the homomorphism of mapping the integers onto the quotient ring of the integers by the zero ideal (which again happens to be isomorphic to the integers).

3. I agree with Bawden and Egner et al. that there should be a division operator associated with the modulo operation, in the same way that current quotient is related to current remainder.

4. I cannot remember any argument to say that the modulo function (mod x n) with representatives from -n/2 (inclusive) to n/2 (exclusive) is of use in programs. I can't think of any examples where it's useful. Furthermore, the only reason I can remember which end of the interval containing the result is inclusive is because of an analogy with hardware implementations of twos complement arithmetic, so I think that perhaps there is no natural choice of which endpoint is inclusive.

5. I still really dislike choosing what operation is done by the sign of the second argument.

6. I think (quotient x n) => (round-to-zero (/ x n)) is a natural operator for those who've been taught long division in school; and so I find no problem with the associated operator (remainder x n) => (- x (* n (quotient x n))).

7. When n is positive, the (mathematical) modulo operation has the definition (modulo x n) => (- x (* n (floor (/ x n)))). The current definition in Scheme of modulo retains this definition for negative n, which I have no problem with (but I'm open to considering other options). So, like I said in (3) above, I would welcome a new operator that implements the operation (/ (- x (modulo x n)) n), whatever the definition of modulo.

Brad