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### More on Order of Operations

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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

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.

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/
```
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