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Topic: Reason for operator precedence
Replies: 15   Last Post: Mar 15, 2006 8:56 AM

 Messages: [ Previous | Next ]
 Bill Dubuque Posts: 1,739 Registered: 12/6/04
Re: Reason for operator precedence
Posted: Mar 14, 2006 1:09 PM

matt271829-news@yahoo.co.uk wrote:
>briggs@encompasserve.org wrote:
>>matt271829-news@yahoo.co.uk writes:
>>>Tony wrote:
>>>>
>>>> Hope this isn't a silly question.
>>>> I was wondering what the reason is for having multiple levels
>>>> of operator precedence? Phrased another way, why is it that
>>>> we don't just evaluate everything from left to right? Having
>>>> multiple levels of precedence obviously adds complexity, so I
>>>> assume there must be some payback. However, I don't see what it is.

>>>
>>> As far as addition/subtraction vs multiplication/division is concerned,
>>> one reason is to ensure that the distributive property of
>>> multiplication works sensibly. For example, we want 3*(4 + 6) = 3*4 +
>>> 3*6 = 3*(6 + 4) = 3*6 + 3*4.

>>
>> Remember that what we're talking about here is merely a notational
>> convention. It has nothing whatsoever to do with the distributive
>> property of multiplication over addition.

>
> And, to elaborate a bit more, I venture to disagree and suggest that
> the convention *does* have to do with this property. I suggest that the
> distributive property of * over + is one of the reasons - possibly the
> main reason - why it is "natural" to view multiplication as "tighter"
> than addition, and to want to interpret, say, 3*4 + 3*6 as (3*4) +
> (3*6) rather than as ((3 * 4) + 3) * 6 or whatever.

Expanding (X+Y)*Z -> X*Z+Y*Z is easy and leads to normal forms.

Factoring (X+Y)*Z <- X*Z+Y*Z is hard and leads to non-normal forms.

Therefore X+Y*Z = X+(Y*Z) not (X+Y)*Z (i.e. expanded, not factored).

E.g. compare representational forms for integers and polynomials.
It's easy to compute the normal expanded form of polynomials and
but its much harder to compute factorizations. Further, poly
factorizations needn't be unique if the coef ring isn't a UFD.
Performing arithmetic operations in expanded form is easy but
is much harder in factored form, e.g. the sum of prime powers
p^m + q^n must be completely factored to retain factored form.

Given such a preference for expanded vs. factored representations
it is only natural to specify a notation that is more efficient
at representing the preferred expanded form.

--Bill Dubuque