Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Everett, from Illinois, writes, "I was born in 1937 in Southern Illinois, Franklin County. We were not taught PEMDAS in school. When did they start teaching this order of operations. Is it taught all over the state now?"
Hi Everett, since the beginning of people writing down mathematical ideas, mathematicians have had to develop conventions for indicating the meaning of expressions on paper. Those rules have not always been consistent. For instance, as a precursor to parentheses, some mathematicians used to write expressions with a horizontal bar over the top to indicate that things should be grouped.
Consider the expressions 7 x 2 + 3. This looks like it should be 14 + 3 = 17. But if the author of a text wanted the addition to be done first, he would write:
_____ 7 x 2 + 3
Typographically, this was cumbersome, and eventually led to the idea that parentheses could be used, and would be simpler to write. This was first suggested in the early 1600s, so it was in vogue as a mathematical ordering notation long before you were born.
Exponents were first used much earlier than the 1600s, but there was no "official" way to write them. Some used Roman Numerals for exponents, some wrote a number after a variable, but not raised (thus, 3x meant 3 times x, but x3 meant x to the third power). It was probably Rene Descartes who made the raised notation popular, and it became the standard. Descartes was doing this in the mid 1600s (not too long after the parentheses started becoming popular). And because of the way it was written, outside the normal baseline of the equation, it had an implicit ordering to it. You would never see the following expression:
23 + 1
and interpret it as though the 3 and the 1 should be combined first, and then 2 raised to the 4th power.
So there you have the first two elements of PEMDAS - parentheses and exponents - that have been implicitly understood since the 1600s.
As for multiplication and division coming before addition and subtraction, I'm not clear on exactly when these became part of the standard ordering notation. However, if you took an Algebra class in high school, you used this ordering of operations, even if no one taught you a name for what you were doing. Consider the following examples:
Example One: Multiplication before Addition
5 + 3x
If multiplication was not evaluated before addition, this would be evaluated (as some of my Algebra students like to do) as:
5 + 3x = 5 + 3 · x = (5 + 3)x = 8x.
The ordering of multiplication and addition was implictly given in the rules for how we can manipulate terms in an expression.
Example Two: Division before Addition
5 + 1/x
If division was not evaluated before addition, this would be evaluated as (5 + 1)/x or 6/x. Again, this order of operations was inherent in the rules for manipulating terms of an algebraic expression.
Which brings me to the answer to your question. First, I do think it's possible that you were never taught PEMDAS as a specific notational convention, but all the rules of the PEMDAS mnemonic were in existence long before you were born. Secondly, even if you weren't taught it specifically, you still used it when you did Algebra, and if you passed your algebra class, then you were doing it correctly! Finally, yes, PEMDAS (or BODMAS*) is taught consistently worldwide. There are still arguments that arise about some ambiguous notations (you may have read one or more posts about those), but PEMDAS is the standard convention for interpreting expressions worldwide.
* BODMAS is used (I believe) in Britain, and perhaps other places as well. B = Brackets, O = Order. Brackets are parentheses, of course, and Order is the word used for Exponents, so even though it looks different, it's the same convention.
Educators can get a free membership simply by sharing an original lesson plan on our Articles for Educators page!