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### History of the Order of Operations

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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
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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
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

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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Associated Topics:
Elementary Math History/Biography
High School History/Biography
Middle School History/Biography

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