If I can diffidently put myself forward as a mathematician of sorts (not a professional one): I agree with everything that Jason says here. Mathematics is the study of abstract patterns. (No doubt, this is not an original observation.) Rob Cliffe ----- Original Message ----- From: "Jason Orendorff" <jason.orendorff@gmail.com> To: "spir ☣" <denis.spir@gmail.com> Cc: "python ideas" <python-ideas@python.org> Sent: Friday, May 07, 2010 3:20 PM Subject: Re: [Python-ideas] integer dividion in R -- PS
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 _______________________________________________ Python-ideas mailing list Python-ideas@python.org http://mail.python.org/mailman/listinfo/python-ideas