Order of Operations: Math vs. English; Calculators

Date: 11/13/96 at 14:07:41
From: Eaton,William
Subject: Order of operations

The following problem was looked at by the math department, and we 
could not come to an agreement, especially after several scientific 
and graphing calculators didn't show the same answer.  

Here's the problem:	144 divided by 3(8 + 4)

We couldn't find the division symbol on the keyboard and we thought 
that using it would make the problem too obvious.

Please advise us on the correct answer and give a brief explanation of 
how you found the answer.

Thank you from the Cincinnatus math and science educators.

Date: 01/11/97 at 16:15:07
From: Doctor Donald
Subject: Re: Order of operations

Dear Cincinnatus Math and Science Teachers,

The order of operations is a convention that is followed by everybody 
who writes or solves mathematical expressions.  You can say that you 
don't subscribe to the convention but that doesn't make it any less of 
a convention. Briefly stated, the order of operations means that 
given a mathematical expression, first you evaluate any expressions in 
parentheses (using the order of operations on the stuff inside the 
parentheses), then you evaluate exponents. After that you do division 
and multiplication, which have the same precedence, which means that 
you compute them from left to right. Finally you do any addition and 
subtraction, which have the same precedence and are also computed from 
left to right. If you have any other questions regarding the order of 
operations, consult our FAQ on this subject at:
 
http://mathforum.org/dr.math/faq/faq.order.operations.html   

What calculators may or may not do is either a matter of 
experimentation or finding a description in the manual. Calculator 
manufacturers do not see fit to subscribe to any universal convention 
and do what they please, sometimes, but not always, explaining in the 
manual what they are doing. Because they don't subscribe to all 
conventions, we have a kind of chaos in the classroom, with students 
being unsure of which convention applies.  The upshot is that just 
about any expression you write down has to be interpreted in the 
context that applies to where you intend to use it. If the expression 
contains only mathematical notation, you can use the order of 
operations. If it is a combination of mathematical notation and 
English, then the order of operations can only be applied once you 
have translated the expression into something containing only 
mathematical notation. When English is involved, things tend to get a 
little confusing because it is difficult to tell where extra 
parentheses might be placed since English is very good at implying 
parentheses without ever writing them down.  

In fact I know of no calculator permitting the use of expressions 
(i.e., graphing calculators and the like) that does not subscribe to 
the conventional order of operations. But there are 'primitive' 
calculators that will evaluate the keystrokes (for example) 4 + 3 * 7 
as 49, when this expression in conventional evaluation means 4 + 21 = 
25. I am not sure what your calculators did with your expression.  
Perhaps you could be more explicit as to the different answers you 
actually found using your calculators.

The problem with finding the answer to "144 divided by 3(8+4)" lies in 
the fact that this expression contains both English and mathematical 
notation. Another issue is what kind of division symbol you were 
looking for on the keyboard (and how this influenced your use of 
English in your question). Regardless of whether you use the fraction 
bar or the minus sign with dots on the top and bottom of it, the 
answer to the expression is the same. Using the fraction bar 
translates "144 divided by 3(8+4)" into   

           144
          -----
          3(8+4)

which means "144 divided by the quantity 3(8+4)."  Using the 
traditional division sign (expressed as a slash on the computer), the 
original expression is translated into 144/[3(8+4)], whose English 
translation is the same as the expression using the fraction bar.  The 
brackets are used in the expression to make it absolutely clear that 
144 is not divided by just 3.

Using either of these translations, you first compute 3(8+4) because 
parentheses come first in the order of operations. This is equal to 
3(12).  Next comes multiplication, which produces 3(12) = 36. Finally 
comes division, which leaves us with the answer 144/36 = 4. This is 
the conventional interpretation of the expression that you gave.   

Another interpretation of your expression is that it means "144 
divided by 3 times the quantity (8+4)." In mathematical notation, 
this is written as (144/3)(8+4).  Following the order of operations, 
you compute the value of each of the expressions in parentheses and 
then multiply those two numbers. This means that your simplified 
expression will read (48)(12) = 576. This answer is obviously quite 
different from the first one and is not the standard answer to "144 
divided by 3(8+4)," but it is a possible one since the use of English 
in the question leads to the existence of a good deal of ambiguity.

The "correct" answer to your problem is 4 because this answer results 
from the conventional interpretation of the wording in your 
expression.

-Doctors Donald and Rachel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/