Number does not imply or assume commutativity. The Quaternions package
provides a Quaternion type which is a subtype of Number. Rational, however,
only allows integer numerators and denominators. Since integer
multiplication is commutative, rational multiplication is too. But I still
think it best to reserve \\ with an analogous meaning to //. There are
already two syntaxes for integer division, which is plenty.
On Mon, Apr 4, 2016 at 7:20 AM, Jeffrey Sarnoff <jeffrey.sarn...@gmail.com>
wrote:
> Since supertype(Number) == Any, the most abstract number-like entity that
> Julia has to offer with the current hierarchy is Number. Most packages I
> have seen that implement number-like types which are outside of the Reals
> use Number as their supertype. This is more helpful, generally, than
> having Quaternions, Dual Numbers etc be direct subtypes of Any.
>
> On Monday, April 4, 2016 at 7:06:17 AM UTC-4, Sheehan Olver wrote:
>>
>>
>> Rational is currently restricted to numbers:
>> Rational{T<:Number}. I’m assuming Number implicitly means commutative
>> number (i.e., not Quaternion).
>>
>>
>>
>> > On 4 Apr 2016, at 8:55 PM, Tim Holy <tim....@gmail.com> wrote:
>> >
>> > On Sunday, April 03, 2016 09:28:28 AM Scott Jones wrote:
>> >> With \\, would you worry about confusion from the fact that in a \\ b,
>> a is
>> >>
>> >>> in
>> >>> the denominator? Especially if it gets called the "integer division
>> >>> operator."
>> >>
>> >> It might confuse some of the mathematicians, used to a \ b as a
>> inverse
>> >> multiplication operator, however, other languages use a \ b for what
>> in
>> >> Julia is div(a, b) or a÷b,
>> >> so I really don't think a definition of \\ as div(a, b) would be that
>> much
>> >> of a problem, no worse than other things you have to remember in
>> Julia,
>> >> and it's reminiscent of the // integer division operator in Python and
>> Lua.
>> >
>> > Thinking about it a second time, there's another reason I'd recommend
>> against
>> > this choice: if you're working with non-commutative numbers, for
>> consistency
>> > \\ too should be reserved for forming Rationals rather than meaning
>> something
>> > completely different. In other words, b//a means RightRational(b, a)
>> (b*inv(a))
>> > and a\\b means LeftRational(a, b) (inv(a)*b). I suspect our current
>> Rational
>> > framework implicitly assumes commutativity, but that's a separate issue
>> and
>> > one that, with a suitable number-traits system, is cleanly resolvable.
>> >
>> > I'm not trying to argue that making rationals with non-commutative
>> numbers is
>> > a daily task, but to me consistency in the meaning of notation seems
>> like the
>> > most important guiding principle. Code gets read many more times than
>> it is
>> > written, so I don't mind a few extra keystrokes if it makes reading
>> code
>> > clearer.
>> >
>> > But once again, it seems there just aren't enough characters on the
>> keyboard.
>> >
>> > Best,
>> > --Tim
>> >
>>
>>
