[Python-ideas] integer dividion in R -- PS

[Rob Cliffe]

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 at gmail.com>
To: "spir ☣" <denis.spir at gmail.com>
Cc: "python ideas" <python-ideas at 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 at 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
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