Posted by Joel on November 09, 2002 at 12:14:58:
In Reply to: Order of operations posted by jebmath on November 08, 2002 at 23:25:04:
: This has to do with what I always called implied multiplication, but don't see that phrase anywhere. Presenting two problems: (DS stands for the traditional division symbol that I do not know how to make with out equation editor) 105 DS ab where a=3 and b=5 Do you multiple 5 by 3 and then divide 105 by that product or do you divide 105 by 3 and then multiply by 5? SEcond problem is 12 DS 2(6-3)+3^2-1. do you multiply 2 by the difference of 6 and 3 first and then divide 12 by that product or do you divide 12 by 2 and then multiply by the 3? I need references, not just opinions. We already have lots of those.
Here, we offer help based on our accumulated knowledge. This is not a free research service. You want references, find them yourself.
Here's what I can offer you.
1. "/" is the accepted symbol for division
2. Multiplication and division have equal precedence.
3. Parentheses have a higher precedence than any arithmetic operator, so expressions in parentheses are always evaluated first.
4. SOME TEXTS (and some calculators) adopt their own rule that binds ab together more tightly than a*b, in effect treating ab as being equivalent to (a*b).
5. By CONVENTION (not by "law") the associativity of multiplication, division, addition and subtraction is left to right. Therefore, in the absence of parentheses, BY CONVENTION, these operations are simply evaluated sequentially from left to right. There is no "mathematical law" that requires this; it is simply a convention that is (nearly?) universally accepted. Without this convention, expressions like 105 / 3 * 5 would simply be ambiguous. There would be no way for people to agree as to how to evaluate such expressions in the absence of parentheses.
6. There is NO SUCH CONVENTION concerning "implied multiplication". SOME PEOPLE choose to say ab => (a*b). Most people would say ab => a*b.
IN MY OPINION, those who take it upon themselves to adopt a "new convention" (which is not a convention because it is not generally agreed upon) do us all a disservice by introducing ambiguity where there was none before. Perhaps a better convention is this: don't use implied multiplication because it is ambiguous.
But unfortunately, that's not a convention either. It's just my opinion.
So (again, in my opinion) "105 DS ab where a=3 and b=5"
=> 105/3*5 = (105/3)*5 = 35*5 = 175
And, "12 DS 2(6-3)+3^2"
=> ((12/2)*(6-3)) + 3^2