More on Order of Operations

Date: 02/13/2000 at 13:59:53
From: Jerome Breitenbach
Subject: Order of Arithmetic Operations

Hi,

I'm a professor in the field of electrical engineering. Occasionally I 
remind my students of the precedence order regarding the four 
arithmetic operations: addition, subtraction, multiplication, and 
division. Apparently though, based upon viewing numerous Web sites and 
the messages of various on-line discussion groups, there seems to be 
some controversy regarding these simple rules! For example, compare

  Mathnerds' Archive
  http://www.mathnerds.com/archive/DetailedAnswer.asp?index=12768   

with

  Order of Operations, Electronics Mathematics (ELET141)
  http://www.csi.edu/ip/ti/elec/math1-5e.htm   

Alas, my search for an "authority" on this matter has been nearly 
fruitless. The closest thing I have found is the convention used by 
the _Mathematical Reviews_ of the American Mathematical Society (AMS), 
at

  Mathematical Reviews Database - Guide for Reviewers
  http://www.ams.org/authors/guide-reviewers.html   

that "multiplication indicated by juxtaposition is carried out before 
division." Thus, in general, for any variables a, b and c, we would 
have  a/bc = a/(bc) (assuming, of course, that b and c are nonzero). 
Indeed, this convention is consistent with what I have seen in many 
mathematical books at various levels; for example, on p. 84 of 
Allendoerfer and Oakley, _Principles of Mathematics_, 1969 (my 
pre-college math book), we find:

     (a / b) x (c / d) = a c / b d

which is generally true only if the right side is interpreted as:

     (a c) / (b d)

Notably, the above equality would *not* be generally true were we to 
interpret the right side as:

     [(a c) / b] d

per the first Web page above (and many others), which states that one 
should "do multiplication and division as they come." However, perhaps 
this page is tacitly ignoring "implicit multiplication" (by 
juxtaposition) and only considering "explicit multiplication" (via 
some multiplication sign) - a distinction is made at:

  Order of Operations - Dr. Math Archives
  http://mathforum.org/dr.math/problems/wuandheil.05.19.99.html   

Unfortunately, in every instance where I have seen someone assert the 
rule that one should first perform multiplication and division as they 
occur (from left to right), I have yet to see them give an example 
that *really* puts this rule to the test. Specifically, how would they 
evaluate:

     6 / 2 x 3

According to their rule, we would obtain:

     6 / 2 x 3 = (6 / 2) x 3 = 9

But, wouldn't it be less confusing to follow the AMS convention for
*all* multiplications (implicit and explicit) thereby obtaining:

     6 / 2 x 3 = 6 / (2 x 3) = 1

- just as we would obtain:

     a/bc = 1

when   a = 6, b = 2 and c = 3?

For when dealing with numerals rather than variables, juxtaposition is 
not an option for indicating multiplication (here, "23" would be read 
as "twenty-three" rather than "2 times 3").

This approach also makes practical sense, since it frequently happens 
that one has a series of multiplications divided by another series of 
multiplications (e.g., consider a binomial coefficient); for example, 
one might desire to write the fraction:

     5 x 4 x 3
     ---------
       2 x 1

more compactly (and without parentheses) as:

     5 x 4 x 3 / 2 x 1

especially if this is to be written in-line (i.e., *within* the 
surrounding text) rather than separately displayed as above.

Or, consider the convenience obtained when dealing with quantities 
expressed in scientific notation. For example, without resorting to 
parentheses, we would interpret:

     6 x 10^9 / 3 x 10^5

as

     (6 x 10^9)/(3 x 10^5) = 2 x 10^4

rather than:

     [(6 x 10^9)/3] x 10^5 = 2 x 10^14

Surely the former is typically the intended interpretation.

As I remember them being taught to me, the rules giving the precedence 
order for the four arithmetic operations are:

  (1) all multiplication (in any order)
  (2) all division, as they occur from left to right
  (3) all addition and subtraction, as they occur from left to right

Moreover, even though an expression containing successive divisions 
such as

     4/2/1

evaluates unambiguously by these rules, I would view such an 
expression as poor form. Based upon inquiries I have made of my math 
colleagues, I am not the only one who remembers multiplication as 
being given *sole* top precedence.

Please comment.

Sincerely,
Jerome Breitenbach

P.S. Some people argue about arithmetic-operation precedence by 
referring to what this or that calculator or programming language 
does. However, I believe all such references are irrelevant; for what 
may be syntactically convenient for some computing device need not be 
convenient (or traditional) for human mathematical writing.

Date: 02/13/2000 at 23:05:28
From: Doctor Peterson
Subject: Re: Order of Arithmetic Operations

Hi, Jerome.

You made some good points. On the whole, I suppose I agree with you 
that it would be easier and perhaps more consistent to give 
multiplication precedence over division everywhere; but of course 
there is no authority to decree this, so the more prudent approach is 
probably just to recognize that there really isn't any universal rule. 
I ran across the same AMS reference that you found while trying to see 
if any societies had made official statements on the rules of 
operations in general; the fact that they took note of this one rule 
alone demonstrates only that this is the one rule on which there is 
not universal agreement at the present time, but it probably is 
growing in acceptance.

I've been continuing to research the history of Order of Operations, 
and one of the references in our FAQ now includes a mention of 
something I had also discovered, that the multiplication-division rule 
has never really been fully accepted:

  Earliest Uses of Symbols of Operation - Jeff Miller
  http://jeff560.tripod.com/operation.html   

As a result, I'm not entirely surprised that you learned a different 
rule than I think I did. (I'm not sure I didn't first learn the 
equal-precedence rule in a programming class, however.)

When algebraic notation was first being developed, it was common for 
each writer to begin by explaining his own notation. If we could 
convince enough writers to follow your rule and state it at the 
beginning of whatever they wrote, maybe we could get it accepted. But 
even then, I'd rather continue to do as we do now; especially with the 
development of publishing software, mathematicians can easily avoid 
in-line expressions of the sort you refer to in published works, and 
in e-mail it's safest to use all the parentheses you need so that no 
one can misunderstand you, whether they remember the rules or not.

I think this is far preferable to making detailed rules that are 
likely to trick people. Sometimes one rule seems natural, and 
sometimes another, so people will forget any rule we choose to teach 
in this area. I've heard from too many students whose texts do "give 
an example that really puts this rule to the test," but do so by 
having them evaluate an expression like:

     6/2(3)

that is too ambiguous for any reasonable mathematician ever to write. 
And no matter what the rule, we would still constantly see students 
write things like "1/2x" meaning half of x, so we'd still have to make 
reasonable guesses rather than stick to the rules.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/