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/