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/
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