Why Rules?Date: 08/15/2001 at 18:16:17 From: Ramila Subject: Order of Operations Why is there a need to have rules for order of operations? Date: 08/15/2001 at 22:42:27 From: Doctor Peterson Subject: Re: Order of Operations Hi, Ramila. Why are there rules for anything? Mostly they allow people to work and communicate together. The order of operations is like the grammar of a language. It lets us know how numbers and operations fit together. For example, when we say "the cat ate the rat," grammar tells us that "cat" is the subject (the eater) and "rat" is the object (the eaten). Without an understanding of how word order affects the meaning of the sentence, we couldn't tell which was which. Similarly, given an expression like 2 + 3 * 5 (I'm using "*" for multiplication), we could take it either as "add 2 and 3, then multiply by 5," or as "multiply 3 and 5, then add 2." These would give two different answers: (2 + 3) * 5 = 6 * 5 = 30 2 + (3 * 5) = 2 + 15 = 17 We have to choose some rule. Among the many rules possible, it has been found that the one that works best, which people just naturally seem to have agreed upon, is the one that makes it read like "two and three fives," where we naturally group "three fives" together as one thing, which is added to "two." That is, doing multiplication first fits our natural sense of how multiplication works, and also works well for algebra. But even if there's no reason to prefer this rule, some rule had to be chosen. The rule for parentheses is also very simple. If we want to tell everyone to do the addition first, we do it by making sort of a box around that part of the expression that says, "do this first and then treat it as a single number." It would be possible to survive in math with only this rule, by using parentheses around everything; but that would cost too much in ink and in frustration. It's easier to have rules that let us say what we want quickly and concisely. All of this allows us to write an expression and be sure that whoever reads it will take it the way we intended it. And that's what the rules are for. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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