History and Background of the Order of Operations
We can't say that any one person invented the rules of the Order of Operations. They have
grown gradually over several centuries and still continue to evolve.
Here are a few notes on the how the Order of Operations became how what they are today:
1. The 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. It may also be that the concept existed before the symbolism, perhaps
just reflecting the natural structure of problems.
2. Some of the specific rules were not yet established in the 1920's.
Math historian, Florian Cajori, 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. However, there has not been found any twentieth-century declarations to
resolved these issues. Therefore, it is hard to say exactly how they were resolved.
3. Many 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. It is believed it has been more important to text authors
than to mathematicians, who have just informally agreed without
needing to state anything officially.
4. There is still some development in the order of operations, as it is frequently heard
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,the rules can 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 such as 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.
(Peterson, 2000)
To see an example of early notation and to learn more about the "Earliest Uses of Grouping Symbols" click HERE