Laypeople seem to spend an inordinate amount of mental energy on something called “Order of Operations” that they imagine to be deeply important and fundamental in mathematics.

It’s not their fault. This is how their teachers present it to them. This is what they are trained to think comprises the nature of math and so on.

Of course, anyone reading this blog knows that reality is not so.

There are only three genuine “order of operations” notation rules:

  1. Subtractions in a string of additions and subtractions are parsed with minimal subtrahend scope.

  2. Multiplication binds more tightly than addition or subtraction.

  3. Base of exponentiation binds more tightly than any of those.

Everything else is something that either provides explicit impossible to misparse scope (bracketed expressions, superscripted exponentiation, fractions written vertically with numerator and denominator separated by a vinculum) or which cannot be relied on as a mechanical rule to clarify ambiguity in the empirical practice of how mathematicians read and write.

All of this only exists because we insist on sticking with infix notation in the first place (frankly, even prefix notation is but a hack for writing out the nonlinear expression trees that directly capture what we mean; we should just write out the trees!). A purely syntactic matter, having nothing to do with the actual mathematical concepts of arithmetic, algebra, etc. Just notational conventions, for the translation of what one means into symbols and back.