Left AssociativityDate: 05/27/99 at 00:45:57 From: Mike Subject: Left to right rule I'm just wondering... When we add and subtract we usually do it from left to right. For example, 5 - 3 - 2 = ?, we do 5-3 = 2, then 2-2 = 0. Is this left to right thing a law for mathematics or do we just use it for consistency? If it isn't any sort of law for math, then in the above example the answer could be 4. Thus, as a professor once told me, this question would contain no solution. Date: 05/27/99 at 08:47:58 From: Doctor Rick Subject: Re: Left to right rule Hi, Mike, thanks for your question. This rule ("left associativity") is not a law but a convention. Yes, we do this for consistency. We could have chosen to associate right, as long as we were consistent. We could also have chosen to have no such rule at all. As the professor said, _if_ the left associativity convention were annulled, expressions without parentheses would be ambiguous; they would have more than one solution, and therefore no solution. In the latter case, we would have no way to determine how to evaluate 5-3-2, and we would have to use parentheses to "disambiguate" the expression: we would write either (5-3)-2 or 5-(3-2). As it is, we do adhere to the left associativity convention, so 5-3-2 is meaningful. But you can always use parentheses to prescribe either left or right associativity explicitly. All the left-associativity rule does is to tell us where the parentheses belong by default (if they are omitted). Computer languages have precedence and associativity rules that describe exactly how the computer is to evaluate any expression you can throw at it. For instance, my C++ programming manual has this chart: Operators Associativity Type --------- ------------- ---- () left to right parentheses * / % left to right multiplicative + - left to right additive << >> left to right stream insertion/extraction < <= > >= left to right relational == != left to right equality = right to left assignment If you don't know C++ (or C), don't worry about what some of these operators mean; the point is that the precedence and associativity are prescribed. The "order of operations" is defined by the top-to-bottom order: multiplication is done before addition, for instance. If two operators are on the same precedence level, they are carried out either left to right or right to left; note that in C++, there is one operator that has right-to-left associativity. It's harder for people to keep all these rules straight than it is for computers, so it's good programming style to put in parentheses wherever a human reader might possibly get confused, even though the computer knows exactly what to do. The same goes double for math that is written solely for human readers: use parentheses wherever they might avoid confusion. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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