#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: patch
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: patch
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: patch
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: patch
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: patch
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: new
Priority: normal | Milestone: Not GHC
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: new
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: new
Priority: normal | Milestone: 7.2.1
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec | Owner:
Type: proposal| Status: closed
Priority: normal | Milestone: Not GHC
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec|Owner:
Type: proposal | Status: new
Priority: normal|Milestone: Not GHC
#3304: define gcd 0 0 = 0
--+-
Reporter: stevec| Owner:
Type: proposal | Status: new
Priority: normal| Milestone
#3304: define gcd 0 0 = 0
-+--
Reporter: stevec| Owner:
Type: proposal | Status: new
Priority: normal| Component: libraries/base
PROTECTED]]On
Behalf Of Jan de Wit
Sent: den 19 december 2001 01:15
To: [EMAIL PROTECTED]
Subject: Re: gcd 0 0 = 0
Why not define gcd a b as the largest (in 'normal' order)
integer d such
that the set of sums of
multiples of a and b {na+mb | n - Z, m - Z} is equal to the set of
multiples of d
Ch. A. Herrmann ([EMAIL PROTECTED]) wrote:
: In contrast, 0*x=0, thus 0 divides 0 (somehow).
: But I have problems with gcd being the greatest positive integer ...
[snip]
: - 0 is not positive, it is non-negative or natural
: - 2 also divides 0 and 2 is a greater integer than 0
: (0 is the
if the report just
: says:
:
:In order to make the non-negative integers into a lattice under `gcd'
:and `lcm', we define `gcd 0 0 = 0'.
It would surely make things a lot less accessible to people (including
me) who do not have any (or limited) knowledge about lattices. Why not
make
Simon == Simon Peyton-Jones [EMAIL PROTECTED] writes:
Simon Christoph does not like this
It's OK if the definition is clear; it wasn't using
the words positive or greatest integer.
Stating gcd 0 0 = 0 explicitly is a good thing,
even if it could be expressed verbatim;
people may think
'
:and `lcm', we define `gcd 0 0 = 0'.
It would surely make things a lot less accessible to people
(including me) who do not have any (or limited) knowledge about
lattices. Why not make it more accessible and use the following rule
(ore something similar)?
The greates common divison (gcd) of two
Lars Henrik Mathiesen ([EMAIL PROTECTED]) wrote:
: Alan Bawden ([EMAIL PROTECTED]) wrote:
: : Indeed, that's a nice way of putting it. How about if the report just
: : says:
: :
: :In order to make the non-negative integers into a lattice under `gcd'
: :and `lcm', we define `gcd
(in the ordinary sense)
divisor of 0. Indeed, +infinity is a much larger divisor of 0...
I'm not in favour of using a very special-purpose order, not used for
anything else, and that isn't even an order but a preorder, just to
motivate gcd 0 0 = 0. Even if using this very special-purpose preorder
they are defined (with gcd 0 0 = 0).
As I said, I was surprised; to me, the definiton with all a and b is the
more natural one. I still recommend using the full domain (especially since
exceptions are awkward to deal with in Haskell), but I'm not as certain.
Best,
Dylan Thurston
, there is a natural choice for d (e.g., in
the integers, the non-negative d; in the ring of polynomials over a
field, the monic d (having leading coeff. 1)). In some UFDs there is no
canonical choice (e.g. in the Gaussian integers, a + ib for a, b integers).
gcd(0, 0) = 0.
Cheers,
Michael Ackerman
Why not define gcd a b as the largest (in 'normal' order) integer d such
that the set of sums of
multiples of a and b {na+mb | n - Z, m - Z} is equal to the set of
multiples of d
{nd | n - Z}? Easy to understand, no talk of division, lattices, rings,
ideals etcetera, and it covers the cases with
, and that the
quotient identifies x and -x).
The only thing that is lacking to make it a lattice on the
non-negative integers, is that gcd 0 0 = 0 . All other cases
involving zero (i.e., gcd 0 x = x for non-zero x, and lcm 0 x = 0
for all x) are consistent with 0 being the maximal element
I've reconsidered my earlier position and think now that the Prelude is wrong to make
gcd 0 0 an error, and should return 0. It probably doesn't make much difference to
anyone, but it's like 1 not being a prime; it may be slightly harder to explain, but it
makes the maths come out nicer
George == George Russell [EMAIL PROTECTED] writes:
George I've reconsidered my earlier position and think now that the
George Prelude is wrong to make gcd 0 0 an error, and should return
George 0. It probably doesn't make much difference to anyone, but
George it's like 1
, that's a nice way of putting it. How about if the report just
says:
In order to make the non-negative integers into a lattice under `gcd'
and `lcm', we define `gcd 0 0 = 0'.
___
Haskell mailing list
[EMAIL PROTECTED]
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Marc van Dongen ([EMAIL PROTECTED]) wrote:
: An integer $a$ divides integer $b$ if there exists an integer
: $c$ such that $a c= b$.
[snip]
: gcd 0 0 = 0; and
: gcd 0 0 /= error Blah
To make clear why $0$ (and not any other non-zero integer) is the
gcd of $0$ and $0$ I should have added
From: Simon Peyton-Jones [EMAIL PROTECTED]
Date: Fri, 14 Dec 2001 01:18:56 -0800
...
If someone could write a sentence or two to explain why gcd 0 0 = 0,
(ideally, brief ones I can put in the report by way of explanation),
I think that might help those of us who have
greatest (positive) integer that divides both n and m
but debate seems to have swirled round whether (gcd 0 0) should
be 0 or an error. Currently in H98 it's an error; but it is the kind
of thing I'm willing to change IF there is a consensus, because it
will only make more programs work
Simon Peyton Jones ([EMAIL PROTECTED]) wrote:
: If someone could write a sentence or two to explain why gcd 0 0 = 0,
: (ideally, brief ones I can put in the report by way of explanation),
: I think that might help those of us who have not followed the details
: of the discussion.
Division
People write on gcd 0 0 :
Alan Bawden [EMAIL PROTECTED]
If you take the point-of-view that gcd is actually an operation on
ideals, then gcd(0, 0) is 0. I.e. define gcd(x, y) to be the smallest
z = 0 such that {m*x + n*y | m, n in Z} = {n*z | n in Z}. This is
probably the most natural
From: S.D.Mechveliani [EMAIL PROTECTED]
Date: Thu, 13 Dec 2001 12:53:32 +0300
Further, the definintion
gcd(x, y) to be the smallest
z = 0 such that {m*x + n*y | m, n in Z} = {n*z | n in Z}
is not natural. In particular, how does it generalize to gcd X Y
for
People write about the Report definition of
gcd x y
as of greatest integer that divides x and y,
and mention
gcd 0 0 = 0
But 2 also divides 0, because 2*0 = 0.
Does the Report specify that gcd 0 0 is not defined?
For an occasion
S.D.Mechveliani wrote
Does the Report specify that gcd 0 0 is not defined?
Yes. The Report definition says
gcd :: (Integral a) = a - a - a
gcd 0 0 = error Prelude.gcd: gcd 0 0 is undefined
gcd x y = gcd
From: George Russell [EMAIL PROTECTED]
Date: Tue, 11 Dec 2001 18:18:31 +0100
...
Yes. The Report definition says
gcd :: (Integral a) = a - a - a
gcd 0 0 = error Prelude.gcd: gcd 0 0 is undefined
gcd x y = gcd' (abs x) (abs y
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