Re: Integers and Mathematical Correctness

[csampson@inetworld.net (Charles Sampson)]

chris.danx <chris.danx@ntlworld.com> wrote:


> According to Euclid, the "division algorithm" (which isn't an algorithm)
> asserts that for a/b
> 
> a = kb + r
> 
> where r >= 0
>       k <- Integers

     Disclaimer: Working from long dormant memory.

     Yes, this is also the definition used when you develop numbers be-
ginning with the Peano postulates (with the correction, noted elsewhere,
of 0 <= r < b).  However, in both cases only non-negative integers are 
being considered.  So the claim "according to Euclid" is incorrect in 
the case of negative integers.

     The mathematical issue is, how do you extend the definition of di-
vision to include negative integers?  The extension I'm familiar with is
the same as the definition used in most programming languages, including
Ada.

     I'm not even sure what the mathematical implications would be if 
you adopted the above "Euclid definition" of a/b if a < 0 and b > 0.  I
suspect that things would become very sloppy along the way.  They cer-
tainly would be in programming.

                                Charlie