[Python] Integer math question

"Frank" <mersmann at szut.uni-bremen.de> wrote in message
news:3987e01c.0401030832.114c6f2a at posting.google.com...
That's a feature. Integer division is explicitly defined in
Python to do exactly that.

The basic thing to remember is that the *correct*
mathematical result of dividing one integer by
another integer is a rational. Python does not
have a rational data type, so it has to pick one
of the multitude of possible ways of rounding a
non-integral result to an integer.

There is no universally right answer to this process:
the "right" answer to any rounding problem is
what the customer wants it to be.

John Roth

"Frank" <mersmann at szut.uni-bremen.de> wrote in message
news:3987e01c.0401030832.114c6f2a at posting.google.com...

Using the division algorithm:

Let a,b be integers with b>0. Then there exist unique integers q and r such
that:

a = bq + r and 0<=r<b [1]

If we let a=5 and b, then a/b is represented by [1] as

a = bq + r
5 = (10) q + r

... skipping some steps, we find that q=0 and r=5
5 = 10(0) + 5
5 = 0 + 5
5 = 5
LHS = RHS

so, 5/10 = 0 in integer division (with no remainder).

Now, let a = -5 and b = 10

-5 = (10)q + r

If we have q = -1, then

-5 = (10)(-1) + r
-5 = -10 + r

Recall [1] above, 0 <= r < b. r must be nonnegative.
In this case r=5,

-5 = -10 + 5
-5 = -5
LHS = RHS

So, q = -1, and r=5, in which case -5/10 = -1 in integer division (without
remainder).

Suppose we let q=0 for the second problem above. Then

-5 = (10)(0) + r
-5 = 0 + r
-5 = r
or
r = -5

But, by [1], r must be nonnegative. So, we have a contradiction. In which
case we can say that q cannot equal 0 in the algorithm above.

So, I suppose the answer to your question would be,

"This is a feature - Python does it properly, where the others do not."

HTH
Sean

"Sean Ross" <sross at connectmail.carleton.ca> wrote in message
news:11EJb.20461$Vl6.3782481 at news20.bellglobal.com...
'represented' is the wrong word here, but hopefully, you get the idea ...

5
-5
-1
5
-5
True
>>>

But 0 <= r < b is a contradiction when b < 0.

"Rainer Deyke" <rainerd at eldwood.com> wrote in message
news:kTEJb.724242$HS4.5376202 at attbi_s01...
Right. But, the division algorithm states "Let a,b be integers with b>0"
(which I mentioned in that post).

"Rainer Deyke" <rainerd at eldwood.com> wrote in message
news:kTEJb.724242$HS4.5376202 at attbi_s01...
Hmm....
-1
-5
>>>

Here, the division algorithm does not apply (b is a negative integer).
Perhaps there's some other theorem for this case?
b<r<=0, when b < 0? I don't know.

"Sean Ross" <sross at connectmail.carleton.ca> wrote in message
news:5sFJb.20923$Vl6.3818930 at news20.bellglobal.com...
I think you're supposed to do something like this
a = bq + r, 0<= r < |b|
5 = (-10)q + r
-5 = -(-10)q - r
-5 = 10q - r

But, then, q would be 0 and r would be 5. <shrug>

It is, indeed, 0 <= r < abs(b)

"If a and b are integers such that b != 0, then there exist unique
integers r and q such that a = q*b + r and 0 <= r < abs(b)"

For non-mathematical spectators, the divmod() function is defined so that
q, r = divmod(a, b) by the definition above.

Sam Walters

"Samuel Walters" <swalters_usenet at yahoo.com> wrote in message
news:pan.2004.01.05.17.54.22.894833 at yahoo.com...
Right. The only thing that was still mildly interesting to me was why does
divmod() return a negative remainder?
-1
-5

If divmod() where defined based on the definition above, then divmod(5, -10)
should return (0, 5).
Well ... that's too strong. The above is a theorem - it doesn't say
remainders have to be nonnegative,
it only says that there exist yada yada yada ... whatever, I'm not that
interested.

C rounds toward the nearest integer and Python rounds down. The behavior is
consistent in each case.

"Frank" <mersmann at szut.uni-bremen.de> wrote in message
news:3987e01c.0401030832.114c6f2a at posting.google.com...

Sorry, I should have said "C rounds toward zero". In other words, the result of
casting x to integer type is sgn(x)*floor(abs(x)).

"Elaine Jackson" <elainejackson7355 at home.com> wrote in message
news:uEOJb.933787$pl3.753391 at pd7tw3no...

On Sat, 3 Jan 2004 8:32:07 -0800, Frank wrote
(in message <3987e01c.0401030832.114c6f2a at posting.google.com>):
Actually this is implementation defined in C89 and standard C++. Either
-5/10 == 0 and -5%10 == -5, as in your implementation, or -5/10 == -1
and -5%10 == 5, like Python. In C99, and possibly a future version of
C++, it's always done the first way (rounding towards zero).

mersmann at szut.uni-bremen.de (Frank) wrote in message news:<3987e01c.0401030832.114c6f2a at posting.google.com>...
The problem is that you aren't using "from __future__ import division"
;-) (This causes the results to be 0.5 and -0.5, and will be the
default division semantics in Python 3.0. If you want integer
division, use the // operator.)
It's a feature. The advantage of defining x // y as floor(x / y) is
that x % y is always nonnegative.

As as example of why this is useful, consider the
datetime.date.weekday method. This could be implemented as

def weekday(self):
return (self.fromordinal() - 1) % 7

If the datetime module was changed to allow BC(E) dates (which have
nonpositive Rata Die numbers), the weekday method would still work.
In C++, you'd have to treat such dates as a special case.

"Dan Bishop" <danb_83 at yahoo.com> wrote in message
news:ad052e5c.0401041659.68460bd at posting.google.com...
Sorry, but x%y can be negative in Python:
-5

It is a feature. Python does the more useful thing IMO.
If you look on / (or now //) as a denominator-defined mapping
of integer intervals to integers, it is clearer.

I.e., the mappings that python implements are

[denom*k, denom*k+denom) => k for denom >0
and
[denom*k+denom, denom*k) => k for denom <0


The C version is ugly, because it maps a unique extra-sized interval
around zero to zero, i.e., for denom>0

[-denom+1, denom) => 0

which contains 2*denom-1 source integers, and all the rest of the
intervals go symmetrically in both directions from there, containing
denom integers. Python's source intervals are all the same size.


====< showintdiv.cpp >=====================================
#include <stdio.h>
#include <stdlib.h>
void main(int argc, char* argv[]){
if(argc<4){ printf("Usage: %s xlo xhi den\n", argv[0]); return; }
int xlo = atoi(argv[1]);
int xhi = atoi(argv[2]);
int den = atoi(argv[3]);
int x;
for(x=xlo; x<xhi; ++x) printf("%3d", x); printf("\n");
for(x=xlo; x<xhi; ++x) printf("%3d", x/den); printf("\n");
}
===========================================================

We'll do a weird printf JFTHOI and to match program lines better:

====< showintdiv.py >======================================
printf = (lambda wso, fmt, *args: wso(fmt%args)).__get__(
__import__('sys').stdout.write)

def main(argv):
if len(argv)<4: printf("Usage: %s xlo xhi den\n" % argv[0]); return
xlo = int(argv[1])
xhi = int(argv[2])
den = int(argv[3])
for x in xrange(xlo, xhi): printf("%3d", x)
printf("\n");
for x in xrange(xlo, xhi): printf("%3d", x/den)
printf("\n");

if __name__== '__main__':
import sys
main(sys.argv)
===========================================================


Python maps successive equal sized intervals to successive integers from -inf to +inf

[10:38] C:\pywk\clp>showintdiv.py -15 16 5
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-3 -3 -3 -3 -3 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3

But C maps symmetrically across zero, causing 2*denom-1 points to map to zero

[10:38] C:\pywk\clp>showintdiv.exe -15 16 5
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-3 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3

With a negative denominator, python still maps successive intervals, but going the other
direction from (and including) zero.

[10:38] C:\pywk\clp>showintdiv.py -15 16 -5
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 -1 -1 -1 -1 -1 -2 -2 -2 -2 -2 -3 -3 -3 -3 -3

Whereas C is still symmetric with the 2n-1 points going to zero:

[10:38] C:\pywk\clp>showintdiv.exe -15 16 -5
-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -2 -2 -2 -2 -2 -3

The extra-sized interval across zero makes for a hiccup in the use of the mapping as a function.
E.g., you can't translate the input by k*denom and get a uniformly translated (by k) output
unless you stay away from the zero interval.

Ok, back to the grind...

Regards,
Bengt Richter