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History of the Order of Operations


Date: 11/22/2000 at 10:56:37
From: Brian Huffine
Subject: History of Order of Operations

I was teaching a computer class and the history of order of operations 
came up. Where, when and with whom did the order of operations first 
originate? Was it the Greeks or Romans?

Thank you! There is a whole class waiting to hear the answer.

Brian Huffine


Date: 11/22/2000 at 12:12:26
From: Doctor Peterson
Subject: Re: History of Order of Operations

Hi, Brian.

The Order of Operations rules as we know them could not have existed 
before algebraic notation existed; but I strongly suspect that they 
existed in some form from the beginning - in the grammar of how people 
talked about arithmetic when they had only words, and not symbols, to 
describe operations. It would be interesting to study that grammar in 
Greek and Latin writings and see how clearly it can be detected.

At the other end, I think that computers have influenced the subject, 
so that it is taught more rigidly now than it used to be, since 
programming languages have had to define how every expression is to be 
interpreted. Before then, it was more acceptable to simply recognize 
some forms, like x/yz, as ambiguous and ignore them - something I 
think we should do more often today, considering some of the questions 
we get on such issues.

I spent some time researching this question, because it is asked 
frequently, but I have not found a definitive answer yet. We can't say 
any one person invented the rules, and in some respects they have 
grown gradually over several centuries and are still evolving.

Here are my conclusions, perhaps in more detail than you want:

1. The basic rule (that multiplication has precedence over addition) 
appears to have arisen naturally and without much disagreement as 
algebraic notation was being developed in the 1600s and the need for 
such conventions arose. Even though there were numerous competing 
systems of symbols, forcing each author to state his conventions at 
the start of a book, they seem not to have had to say much in this 
area. This is probably because the distributive property implies a 
natural hierarchy in which multiplication is more powerful than 
addition, and makes it desirable to be able to write polynomials with 
as few parentheses as possible; without our order of operations, we 
would have to write

     ax^2 + bx + c
as
     (a(x^2)) + (bx) + c

It may also be that the concept existed before the symbolism, perhaps 
just reflecting the natural structure of problems such as the 
quadratic.

You can see an example of early notation in "Earliest Uses of Grouping 
Symbols" at:

   http://jeff560.tripod.com/grouping.html   

where the use of a vinculum (an early version of parentheses) shows, 
both in its presence (around an additive expression) and its absence 
(around the multiplicative term "B in D") that the rules were 
implicitly followed:
                                                 ________________
   In Van Schooten's 1646 edition of Vieta, B in D quad. + B in D
   is used to represent B(D^2 + BD). 

2. There were some exceptions early in this development; in 
particular, math historian Florian Cajori quotes many writers for 
whom, in the special case of a factorial-like expression such as

     n(n-1)(n-2)

the multiplication sign seems to have had some of the effect of an 
aggregation symbol; they would write

     n * n - 1 * n - 2

(using a dot or cross where I have the asterisks) to express this. Yet 
Cajori points out that this was an exception to a rule already 
established, by which "nn-1n-2" would be taken as the quadratic 
"n^2 - n - 2." 

There was also an early notation in which a multiplication would be 
replaced by a comma to indicate aggregation: 

     n, n - 1 

would mean
     
     n (n - 1) 

whereas 
     
     nn-1 

meant

     n^2 - 1.

3. Some of the specific rules were not yet established in Cajori's own 
time (the 1920s). He points out that there was disagreement as to 
whether multiplication should have precedence over division, or 
whether they should be treated equally. The general rule was that 
parentheses should be used to clarify one's meaning - which is still 
a very good rule. I have not yet found any twentieth-century 
declarations that resolved these issues, so I do not know how they 
were resolved. You can see this in "Earliest Uses of Symbols of 
Operation" at:

   http://jeff560.tripod.com/operation.html   

4. I suspect that the concept, and especially the term "order of 
operations" and the "PEMDAS/BEDMAS" mnemonics, was formalized only in 
this century, or at least in the late 1800s, with the growth of the 
textbook industry. I think it has been more important to text authors 
than to mathematicians, who have just informally agreed without 
needing to state anything officially.

5. There is still some development in this area, as we frequently hear 
from students and teachers confused by texts that either teach or 
imply that implicit multiplication (2x) takes precedence over 
explicit multiplication and division (2*x, 2/x) in expressions 
such as a/2b, which they would take as a/(2b), contrary to the 
generally accepted rules. The idea of adding new rules like this 
implies that the conventions are not yet completely stable; the 
situation is not all that different from the 1600s.

In summary, I would say that the rules actually fall into two 
categories: the natural rules (such as precedence of exponential over 
multiplicative over additive operations, and the meaning of 
parentheses), and the artificial rules (left-to-right evaluation, 
equal precedence for multiplication and division, and so on). The 
former were present from the beginning of the notation, and probably 
existed already, though in a somewhat different form, in the geometric 
and verbal modes of expression that preceded algebraic symbolism. The 
latter, not having any absolute reason for their acceptance, have had 
to be gradually agreed upon through usage, and continue to evolve.

You can see a briefer answer in our archives at:

   Ordering the Operations
   http://mathforum.org/library/drmath/view/58237.html   

and some of the current debates here:

   More on Order of Operations
   http://mathforum.org/library/drmath/view/57021.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Math History/Biography
High School History/Biography
Middle School History/Biography

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