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

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Bill Dubuque

Posts: 1,739
Registered: 12/6/04
Re: Reason for operator precedence
Posted: Mar 14, 2006 1:09 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote:
> wrote:
>> 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
the radix form of integers (= expanded polynomial in the radix)
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

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