(Numeric) should -7 % 5 = -2 ?

[Stan Heckman]

Is the following behavior expected, or have I broken my Numeric
installation somehow?

$python
Python 2.2.2 (#1, Mar 21 2003, 23:01:54)
[GCC 3.2.3 20030316 (Debian prerelease)] on linux2
Type "help", "copyright", "credits" or "license" for more information.-2

 

[Fredrik Lundh]

Is the following behavior expected, or have I broken my Numeric
installation somehow?

$ python
-2

looks like Numeric implements C semantics, which is different
from how Python does it.

</F>

 

[Tim Roberts]

Hmmm... "remainder" makes sense. But "%" is mod, right. IIRC from my
abstract algebra days (only 30 yrs ago ) The "X mod n" function
maps onto the postive integers from 0 to n-1. So sounds like numeric
contradicts the math texts. Not good since it's a math module.

That's a bit harsh. The problem is that there is no universal agreement in
the world of computer science as to what the semantics of the modulo
operator should be when presented with a negative operand. Contradicting
Fredrik, something I do with great reluctance, the C standard specifies
that the behavior is implementation-defined, so in fact BOTH answers
"implement C semantics".

Fortran, on the other hand, defines A mod P as A-INT(A/P)*P, which is
exactly what Numeric produces. Since folks interested in numerical
programming often have a strong Fortran background, it is not terribly
surprising that Numeric should follow Fortran's lead.

 

[Richard Brodie]

Contradicting Fredrik, something I do with great reluctance, the C standard
specifies that the behavior is implementation-defined, so in fact BOTH answers
"implement C semantics".

Only if you're looking at an old version of the standard; it's not implementation
dependent in C99.

 

[Louis M. Pecora]

That's a bit harsh.

You may be right. I got to work and checked my old Abstract Algebra
book. The defintion is,

We write a=b mod m if m divides (a-b) (i.e. no remeinder).

The defintion does not say how to compute the mod, rather it is an
expression of a relationship between a and b. Hence, writing -2=-7 mod
5 appears to be OK.

The "uniqueness" comes in when we recogize that mod m defines an
equivalence relation on the integers and so for a given m every integer
falls into a unique class (or subset of integers). The set of m
subsets is equivalent to the positive integers 0 to m-1.

So it appears that the translation between math and computer science is
not as clear as I thought. In math (well, number theory) mod is a
relation, not an operation. In computer science it is an operation.

Waddayathink?

 

[Bob Gailer]

That's a bit harsh.

You may be right. I got to work and checked my old Abstract Algebra
book. The defintion is,

We write a=b mod m if m divides (a-b) (i.e. no remeinder).

The defintion does not say how to compute the mod, rather it is an
expression of a relationship between a and b. Hence, writing -2=-7 mod
5 appears to be OK.
[snip]

To quote from "Number Theory and its History" by Oystein Ore, page 213f:
"When an integer a is divided by another m, one has
a = km + r
where the remainder is some positive integer less than m. Thus for any
number a there exists a congruence
a (is congruent to) r (mod m)
where r is a unique one among the numbers 0, 2, 1, .... m-1"

Bob Gailer
(e-mail address removed)
303 442 2625

 

[Erik Max Francis]

We write a=b mod m if m divides (a-b) (i.e. no remeinder).

The defintion does not say how to compute the mod, rather it is an
expression of a relationship between a and b. Hence, writing -2=-7
mod
5 appears to be OK.

Right. Equivalences modulo m are really alternate numerical spaces in
which arithmetic is done; in mathematics, the modulo is not strictly an
operator. In those cases, you don't really have to pick a unique
residue when doing arithmetic (mod m), since it's all equivalence
relation anyway.

In computer science, where modulo is an operator that must return a
unique value, it's not really specified whether (-n % m) (m, n positive)
should be negative or not. Some languages/systems chose negative, some
don't. The choice is never "wrong" unless it's done inconsistently
within a particular language/system.

 

[Julian Tibble]

Maybe I'm misunderstanding this, but what about -7 divided by 5? We
get k=-1 and r=-2.

-7 = (-2) * 5 + 3

So actually, k = -2 and r = 3

m can be negative. Maybe your quote was for the positive integers only.

True, that definition does not make sense when m is negative. When I came
across the subject in introductory group theory they said "positive divisor".

Julian