I know some places use BIDMAS as well. Overall though, to someone who knows little about mathematics, why is this the correct order? What’s the exact logic behind it
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Why is this the correct order? Convention. *Why* is it convention? This gets interesting. The first time you start seeing really recognizable writing of algebra from a modern standpoint — [ ax^2 + bx + c ] and the like — is with the notational [choices of Descartes](https://www.sciencedirect.com/science/article/pii/S0315086010000285) (though as that paper argues, he was far from perfect). The implied multiplication sign between adjacent variables, with addition signs required to be explicit — [ ax + b ] — implies multiplication has [precedence](https://en.wikipedia.org/wiki/Order_of_operations) already in Descartes’s notation — i.e. he’s using PEMDAS. If you look at authors before him, it’s hard to find enough consistent notation, or consistent use of algebraic notation in general, to pin the origins of this down exactly — most algebra of the time, including in most of Descartes’s work, was done in text-only prose. However, a [global history of algebraic notation](https://www.academia.edu/7114581/A_Brief_History_of_Algebraic_Notation) (worth downloading — it’s fascinating stuff) suggests that from India to China since late antiquity there was some sense that some masters were grouping multiplicative terms. The more important development comes when it’s realized that the difficult problems people were solving — usually geometric — were difficult based mostly on the degree of polynomial equation and not on any additional terms. For example, some geometric calculations on a triangle might require solving a cubic like [ 2x^3 + 3x + 2 ], and as the mathematicians knew that the algorithm to solve this was similar to that of any [ ax^3 + cx + d ], or [ ax^(3)] in general, the “cubus” would be the key term used to describe that equation. That may indicate then how the exponent on a variable would be seen to have precedence, once somebody decided to write all these conventions out for sure.
What’s the *exact logic* behind it? Let’s take a slightly more advanced look. Addition and multiplication on the real numbers are a type of abstract structure called a [field](https://en.wikipedia.org/wiki/Field_(mathematics)). (The point of this abstraction being that you could use something other than real numbers (like, say, vectors, or arrangements of a Rubik’s Cube) and something other than addition and multiplication (like dot and cross products, or rotating sides of the Rubik’s Cube) and still have the same some of the underlying properties as the real number.)
The important thing about multiplication is that it *distributes* over addition — that is, [ a*(b+c) = a*b + a*c ]. If we did the reverse order of PEMDAS, so addition had precedence over multiplication, then an expression like [ ab + c ] would be equal to [ a(b + c) = (ab) + (bc) ]. Essentially, every time we would write an expression mixing multiplication and addition in which addition has precedence, the multiplication would be distributing like this unless we explicitly set up parentheses telling it not to. Why is this bad? I’m actually not sure — it just seems silly that the no-parentheses form for an expression would not be fully distributed — I thought I had it figured out for sure that this was not optimal, but now I’m stuck at “it’s messy”, so someone else may be able to help out.
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