Order of Operations: Math vs. English; Calculators

Date: 11/13/96 at 14:07:41
From: 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,

An 'order of operations' is a convention that is followed by some group 
of people who write mathematical expressions.  The idea is to allow the
expressions to be evaluated in the same way by everyone in the group, and to 
allow the use of grouping symbols to be kept to a minimum. 

As a simple example, using '*' to represent multiplication, we might interpret

  3 + 4 * 5

as meaning either

  (3 + 4) * 5

or

  3 + (4 * 5)

The PEMDAS convention currently taught in schools,

  http://mathforum.org/dr.math/faq/faq.order.operations.html   

tells us to use the second interpretation.  And as long as we're working with 
people who also follow that convention, we can leave the grouping symbols out.  
That doesn't seem like such a big benefit, but if you're writing hundreds or 
thousands of expressions each day, then those little savings can add up.  

This particular convention is optimized to make it possible to write 
polynomials without grouping symbols.  For example, instead of writing

  ((3/4)(x^3)) + ((4/5)(x^2)) + ((5/6)x) + 7

if we agree that exponents should be evaluated first, we can drop some
grouping symbols, 

  ((3/4)x^3) + ((4/5)x^2) + ((5/6)x) + 7

and if we agree that multiplications and divisions should be evaluated before 
additions and subtractions, we can drop some more, 

  (3/4)x^3 + (4/5)x^2 + (5/6)x + 7

and if we agree that multiplications and divisions should be evaluated left to 
right, we don't need any at all:

  3/4x^3 + 4/5x^2 + 5/6x + 7

Because mathematicians are interested in brevity, and because polynomials are 
so widely used, this convention is a big help - even if it seems  somewhat 
arbitrary to students who haven't yet started thinking about algebra at all. 

The thing about a convention is, it only works if everyone is aware of it, and 
agrees to use it.  So, for example, you might come to America from England, and 
decide that you're going to drive on the left-hand side of the road, because 
that's what you do back home.  Clearly, this would create some very serious 
problems for you!  But that's not because what you're doing is 'wrong' in any 
absolute sense.  It's just different from what everyone else has agreed to 
do in your changed context. 

For better or worse, calculators can't be used to resolve questions like this, 
because 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. 

Special problems arise when we mix English and mathematical notation.  For 
example, in word problems, we sometimes see things like

  Twice a number plus 5 is equal to 11.
  What is the number? 

English doesn't tell us whether this means

  (Twice a number) plus 5 is equal to 11.

or

  Twice (a number plus 5) is equal to 11. 

In your expression, 

  144 divided by 3(8 + 4)

it's not immediately clear whether the use of English is intended to provide 
some implicit grouping,

  (144) divided by (3(8 + 4))

or whether we should just replace the words 'divided by' with an obelus,

  144 ÷ 3(8 + 4)

and proceed from there. 

The particular kind of division symbol you use ('/' or '÷') shouldn't matter at 
all, since they don't imply any special grouping.  But this is NOT the case for 
the horizontal fraction bar, or 'vinculum', which DOES imply grouping symbols 
around both the numerator and denominator.

For example, 

   3 * 4 + 5
   ---------
   6 - 7 ÷ 8

means the same thing as

   (3 * 4 + 5) / (6 - 7 ÷ 8)

We use the vinculum for other kinds of grouping as well, e.g., for 
roots,
     __________
   \/ 3 + 4 * 5 = square root of (3 + 4 * 5)

and for repeating decimals, 
                         __
   0.123454545... = 0.12345

So this isn't a new idea that's introduced for fractions, but rather a natural 
extension of an old idea. 

In any case, if we follow the PEMDAS convention for your expression,

  144 divided by 3(8 + 4)

that convention tells us to do mulitiplications and divisions with equal 
priority, from left to right.  The first multiplication or division we see is 

  [144 divided by 3](8 + 4)

The next one is

  [[144 divided by 3](8 + 4)]

So that's the meaning of this expression, IF you're following the
PEMDAS convention. 

If you're following a different convention, you can come up with a different 
answer, just as depending on where you are, you can come up with different 
answers for the question 'On what side of the road should I drive?'

In some sense, the best answer to that last question is another question:  'It 
depends.  Where are you?'  In the same sense, the answer to your question is 
another question: 'It depends.  What convention are you using?'

If no one is available to answer that (e.g., in a testing situation), the 
safest course of action is to just state the convention you're using, and make 
that part of the answer.  So if given your expression to evaluate, I'd write 
something like 

  Following the PEMDAS convention,
 
    144 divided by 3(8 + 4) = (144 divided by 3)(8 + 4) 

                            = 48(12)

                            = 576

The examiner might still knock of points for not reading his mind, but I'd be 
on solid ground as far as doing the calculation is concerned.

-Doctors Donald, Rachel, and Ian  

 Check out our web site!  http://mathforum.org/dr.math/