Left Associativity

Date: 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/