2010/5/7 spir ☣ <denis.spir@gmail.com>:
I searched the def of int division in R. I could not find it in the english wikipedia,
On page 4 of Gallian, "Contemporary Abstract Algebra", we have: Division Algorithm Let a and b be integers with b > 0. Then there exist unique integers q and r with the property that a = bq + r where 0 <= r < b. This is, of course, the definition Python uses. I think this is pretty standard. What might mathematicians like about this definition? Well, I think the fundamentally important thing about integer division (or any mathematical object really) is the patterns it makes: >>> [a % 5 for a in range(20)] [0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4] >>> [a // 5 for a in range(20)] [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3] Those patterns show up in both C and Python. Do the patterns continue as you go into the negative numbers? In Python they do: >>> [a % 5 for a in range(-10, 10)] [0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4] >>> [a // 5 for a in range(-10, 10)] [-2, -2, -2, -2, -2, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1] In C, the patterns change as you pass 0. That is, the Python definition satisfies these mathematical properties, and the C definition doesn't: (a + b) // b == a // b + 1 (a + b) % b == a % b The Python definition agrees with modulo arithmetic: -3 ≡ 2 (mod 5) http://en.wikipedia.org/wiki/Modular_arithmetic In Python, -3 % 5 == 2 % 5 is true. In C it is false.
But: that -4/3 != -(4/3) looks simply wrong for me.
You can either have the mirror symmetry about 0 that you want, or you can have the translational symmetry shown above. I think translational symmetry is the defining thing about integer division and therefore more important. Of course for a programming language the question of which definition to use is a practical one: which is more useful? Ultimately practicality beats purity. But as far as purity goes (and it goes along with practicality a good long way) I think Python's integer division wins by a wide margin. Cheers, -j