This was an EXCELLENT question asked by one of our students, Gabrielle Welch. Dr. Math of Math Forum gives his answer below: Sal Khan talks about the Order of Operations.

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/