A segment from ABS-CBN’s noontime show “It’s Showtime” has been creating a buzz lately. In the segment “Isang Tanong, Isang Milyon”, the hosts asks a contestant to evaluate the expression “1000 – 500 x 2”. The contestant answers “1000” and wins the round. Netizens, however, cry that the answer is wrong because of that thing called PEMDAS. In fact, the comments in this page reveals that netizens do not only think that the PEMDAS should have been applied, but also that anyone who argues that the answer is 0 is an idiot who was probably sleeping during math class when PEMDAS was taught. Reading the commentsmade me realize that PEMDAS has become a sort of infallible doctrine. Is Showtime wrong in giving 1000 as the right answer? Well, as far the “status quo” is concerned (i.e., of PEMDAS being the golden rule that must not be broken), “1000 – 500 x 2” equals “0”. That’s loud and clear, and there shouldn’t really be any debate about it . That’s what we Filipinos know and accept. Hence, the contestant should have lost the round.

I would like, however, to go beyond the debate on what the value of “1000 – 500 x 2” should be and consider the question: *Should we really be teaching PEMDAS to grade school students in the first place*. No scientist, accountant or computer scientist will encounter an expression such as “2 – 4 + 5 x 5 / 6 + 7” in actual practice. As a graduate student, I do not recall any instance where PEMDAS had to be used at all. The four basic operations are not the only binary operations that one encounters in mathematics, and none of them have a rule corresponding to PEMDAS. When an expression is possibly ambiguous, we use parentheses (in varying sizes!).

Ideally, mathematical expressions should be free from ambiguity. PEMDAS *appears* to remove this ambiguity. The problem is, one PEMDAS is counter-intuitive. The intuitive way to simplify an expression without parentheses is to evaluate every operation from left to right. Also PEMDAS neglects the important field properties of real numbers (See this enlightening Minute Physics video where the narrator argues that PEMDAS is “morally wrong”). Interestingly, the PEMDAS is mainly a North American construct which has been handed down to us by the Americans, much like democracy and basketball. You won’t find questions in the IMO involving long expressions where PEMDAS has to be applied. I have participated in local math olympiads in grade school and high school, and I do remember having to apply PEMDAS. I think I have also encountered PEMDAS problems in some entrance examinations.

PEMDAS, however, is prone to being misunderstood – most students will have a problem with PEMDAS even if they know what the acronym stands for. The commonly neglected fact is that multiplication does not necessarily take priority over division – if multiplication and division occur in progression, the expression is evaluated from left to right. This is the same for addition and subtraction. Hence, with PEMDAS 4 – 4 + 3 – 2 is not 4 – 7 – 2.

The good thing is, one eventually neglects PEMDAS anyway as one learns more advanced mathematics in college. An engineering, accounting or science major will eventually get a sense of when an expression is ambiguous and will use parentheses when needed. I don’t think that the use of parentheses is too advanced to be taught to grade school students, and I believe that we should be teaching it instead of PEMDAS. If anything, PEMDAS exercises give students the impression that math is about making calculations and memorizing rules. There is really nothing dogmatic about PEMDAS, and that’s because because it is not derived from any of the axioms of real numbers. Not all applications and programming languages obey it. Any careful programmer would familiarize himself with the order of operations a particular language uses.

Other criticisms of PEMDAS are found here, here and here. They are admittedly more well-written than this haphazardly-composed blog entry, and I encourage you to read them. 🙂

It should be taught. The axioms of Real Numbers follow from the PEMDAS rule. You must have been mistaken the rules of PEMDAS which is equivalent of PEMA.

Nothing in the (field) axioms tell us how to evaluate sequences of operations such at a + b * c – d + e.

The axioms of Real Numbers do NOT follow from the PEMDAS rule or any of its kissing cousins (e.g. BODMAS, GEMDAS). In fact, the axioms of Real Numbers are independent of any notational system, formal language, or natural language used to describe them.