Generally speaking, the modulus operator is defined as a mapping
from the integers to a modular ring of numbers. Hence, the result
of any operation should be a nonnegative value ranging from zero
to n-1 in canonical form.
That said, what your chosen example here is poor simply because
+1 and -1 are congruent modulo 2 (i.e, they are the same value). Hence
using 1 and 3 would yield more insightful results as -2 and +1 are
congruent modulo 3. Below is a quick example from PHP.
1% 3 == 1
-1% 3 == -1
1%-3 == 1
-1%-3 == -1
Note that +1 and -1 are not congruent modulo 3 so we gain more
insight into what is happening. Here (and in your C++ example) the
sign is a function of the operand (+-1) but not the modulus (3).
That said, the developers have chosen different ways to deal with negative
values here, and the Python results worry me. But without seeing the
results using something like the numbers above, it is hard to tell what is
happening.Also, did you intentionally include two minus signs below?
For a clear and simple mathematical definition, follow the link below.
http://mathworld.wolfram.com/Congruence.html
In essence, any pair of intergers, a and b, are said to be congruent
modulo some integer m if the value a-b is an integral multiple of m.
To get a solid grasp on the properties of modular arithmetic I would
suggest reading an introductory text on "abstract algebra" at either
the senior or graduate level. But I will not recommend that unless
you have been trained to read such texts.
As for applications using negative values in the modulus, I cannot come
up with any off the top of my head, and coding I favor not doing so as
your example illustrates the potentially inconsistent implementations
of it (that and I was trained to think of modular arithmetic in terms of
positive integers).
hope this helps,
Sean
