Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!watmath!clyde!cbosgd!gatech!seismo!brl-smoke!gwyn
From: gwyn@brl-smoke.ARPA (Doug Gwyn )
Newsgroups: net.lang
Subject: Re: Integer division: a winner declared
Message-ID: <986@brl-smoke.ARPA>
Date: Sun, 16-Feb-86 02:56:15 EST
Article-I.D.: brl-smok.986
Posted: Sun Feb 16 02:56:15 1986
Date-Received: Tue, 18-Feb-86 04:23:39 EST
References: <11610@ucbvax.BERKELEY.EDU> <5100003@ccvaxa> <548@ism780c.UUCP> <1970@peora.UUCP>
Reply-To: gwyn@brl.ARPA
Organization: /usr/local/lib/news/organization
Lines: 47

In article <1970@peora.UUCP> jer@peora.UUCP (J. Eric Roskos) writes:
>
> Donald Knuth in _Fundamental_Algorithms_ defines a "mod" function
>
>	x mod y = x - y * floor(x/y) if y!=0; x mod 0 = x
>
> and goes on to show (by dividing the RHS through by y) that
>
>	if y > 0, then 0 <= x mod y < y
>	if y < 0, then 0 >= x mod y > y
>
> He goes on to show that x mod y is the same as the remainder when x is
> divided by y.  Then, finally, he states that "mod" is *not* the same
> as the meaning of the word "modulo" in number theory.

What he actually said was, "In number theory, the abbreviation `mod'
is used in a different but related sense; ..." and then introduces
the word "modulo" for what a number theorist would call "mod".  All
he meant by this was that one can define "modulo" in terms of "mod":
We say "x == y (modulo m)" iff (x - y) mod m = 0.

Knuth also does not "show that mod is the same as remainder"; he
says (for pedagogical reasons) "the quantity x - (x mod y) is an
integral multiple of y; and so we may think of x mod y as *the
remainder when x is divided by y*."  And, indeed, that is a good
way to think about the meaning of "x mod y", so long as the exact
definition is used in technical work.

> So there you have a well known authority in computer science defining
> "mod" in the "unpopular" way.  However, Knuth's definition runs up against
> an immutable obstacle: the Ada Programming Language's definition of the
> functions.  Ada defines "rem" and division [and remember that Knuth
> said "rem"=="mod"] this way: ...

Knuth said no such thing.  He has defined "mod" as any decent
mathematician would, which is NOT the same as "rem" or C's %.

> So there you have it.  Ada defines "mod" as distinct from "rem", and defines
> division (and thus "mod") in the "intuitive" way, contradicting Knuth.

Wrong.  Ada's "rem" is like C's % with more specific semantics,
and its "mod" is like everybody's "mod" (C doesn't have "mod").

I think where you went wrong was in not reading carefully enough.
For x mod y, what does y < 0 mean?  Nobody has been arguing about
that case in this debate; the discussion has centered on what
happens when x < 0.  "x" != "y".