Math Question: My understanding of the order of operations is that it is a arbitrary order decided upon a long time ago by convention, with no underlying mathematical justification. Given this, if I solve and equation using a non-conventional order of operations, for example: 2 + (3 × 4) = 14, as opposed to (2 + 3) × 4 = 20, is my answer objectively "wrong"? Or is my answer just not subjectively accepted?

There is a justification for the Order of Operations, but it mostly isn't taught or gets lost in memorizing PEDMAS. You see, all arithmetic operations are built on addition, I guess it's somewhat arbitrary that we agree that 1 thing plus 1 thing is two things, and zero is nothing, and so on. Anyway building on that, subtraction, or more correctly, adding a negative, balances addition. Multiplication is a Group of addition (3+3+3) = 9 and exponents is a Group of multiplying (3^2 is 3*3 is 3+3+3) so, you should do those Groups first, plus any groups defined by a division line, a radical sign, or (sigh) parenthesis. Division, by the way, is merely the regrouping of the repeated addition, so it's the same Group, I don't teach PEDMAS to students in my classes.
Answered On 04/23/2020 14:00

Thus, in the two problems above, if you just had the expression 2+3X4 what you have is two plus three groups of four or 2+3+3+3+3, which is 14. You can use the parenthesis to clearly define which group you operate on first, so saying (2+3)X4 says add the group before repeating the addition, or take the group (2+3)+(2+3)+(2+3)+(2+3) or 5+5+5+5 , which is 20.
Answered On 04/25/2020 23:29