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/
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