I believe that the order of operations has to be a convention, since it would be possible to to calculate otherwise without violating any other mathematical rules.
Furthermore, a past colleague of mine argued that the order convention not be taught, at least in lower grades, since the context would always tell you the order that needed to be used. Your example illustrates that nicely - If the problem was instead to put together five packs of 2 peaches and 6 grapes, we would have to calculate the answer differently. But we would still be able to write it as 2+6*5, provided we all agreed that that string of symbols represented the operations we wanted. Since we DON'T agree, it is necessary to write instead (2+6)*5; but there is nothing I can see that forces us to adopt the convention except convention itself, the need to have a system that is unambiguous and succinct.
For these reasons, I don't believe that the rules are fixed in the heavens - we humans have invented the system and so have the ability/right to set the rules by which the system operates. This doesn't answer a question about the origin of numbers themselves, or their properties, of course.
Keep up the great online discussion - I'm really enjoying it.
Peter Price Christian Heritage College Brisbane, Queensland
At 09:23 31/01/2001 , you wrote:
I'm not so sure that our order of operations really is "non-universal." Certainly we've agreed on conventions, but I tend to think those conventions are more necessary than arbitrary. My favorite example is an expression like 2+6*5. If I see that as representing two one-dollar bills and six five-dollar bills, then to determine how much money I have, I have no choice but to multiply first and then add -- otherwise, I'd effectively be converting my ones into fives. (Not that I would mind doing that, come to think of it!) It's for this sort of reason that I think our order of operations conventions are really the only ones we could have arrived at.
On the other hand, it may be possible that we're only willing to write expressions like the above because we've adopted the conventions we have. If we'd adopted others, maybe we just wouldn't write expressions like I have above (or they'd be interpreted differently). So I don't necessarily disagree with Ron either. Maybe someday E.T. will come down and we can ask about the order of operations elsewhere! :-)
Mike Kenyon Yakima Valley Community College Grandview, Wash.
> -----Original Message----- > From: Ron Ferguson [mailto://email@example.com] > Sent: Tuesday, January 30, 2001 10:21 AM > To: firstname.lastname@example.org > Subject: Re: [math-learn] - new question for discussion > > > We must realize that "order of operations" are an arbitrary > convention: no > such ordering is implicit in the field postulates, hence to obtain > "universal?" agreement as to the value each expression should be > well-punctuated with grouping symbols so that precedence of > dyadic (binary) > operations are non-ambiguous. Even monadic (unary) > operations and functions > could benefit from the clarity of precise punctuation. But > few of us are > willing to bear the burden of such notational overhead. > Hence, we adopt an > order of operation convention. > >
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