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# The Order of Operations (PEMDAS/BODMAS) that you learned in elementary school... IS WRONG!!!

Well, okay, it's not incorrect, but it's flawed and by a mathematician's standards: morally wrong.

I'm sure at one point you boringly learned the order of operations. These are the set of rules that tell you whether you should do multiplication before division or addition before subtraction to get the correct answer on your math problem.
1) Parentheses (brackets)
2) Exponents x^x
3) Multiplication 2*2
4) Division 2/2
6) Subtraction 2-2
7) Get the right answer :)

Except, you don't always get the right answer.

For example: 8-2+1. Is it 5 because 8-3=5? Or is it 7 because 6+1=7?

Is 6/3/3 equal to 2/3 or 6/1?

The issue here is that focusing on the order of operations can lead to ambiguity and obscures the real beauty of mathematics.

A mathematician will tell you that 8-2+1 is actually 8+(-2)+1, which is unambiguously equal to 7 even though the standard order of operations (PEMDAS) taught in the US tell you to add first and gives you 5. But in reality, if you want 5 to be your answer, then you need some parentheses like so: 8-(2+1).

But why is the ambiguity even possible? It's because fundamentally, all these operations are different procedures that start with two numbers and in some way combine them to make a single number. Each operation takes two number, no more, as an input and gives you an output. If you want to be entirely unambiguous then you would have to put parentheses around everything.
It would take something like 1+2+3+4*5-18/3 and make it look like ((1+2)+(3+((4*5)-(18/3)))).
Then, there would be no need to know any order of operations. You would just evaluate the innermost parentheses first and always get the same answer.

But this isn't the only way. You can actually change the order of these operations by changing where the parentheses are like in (1+(2+3)) and ((1+2)+3). But the only way to do this is if you know what the underlying mathematical is.

For example, if you want to multiply the results of 3+4 by 5 or just (3+4)*5 you can either do the addition first and get 7 and multiply that by 5 and get 35. OR you can multiply first as long as you multiply the 5 by both the 3 and the 4 which would yield 15+20. The latter example is distribution and in both cases you get 35.

This even works for exponents like ((3*2)^2) which simplifies to 6^2 and ultimately 36. But you can also square both of them before you multiply which becomes 3^2 * 2^2 or 9*4 and finally 36.

So, the TRUE Order of Operations is this:

1) Parentheses first
2) Learn Math (basically what multiplication, division, exponentiation, and the rest are really doing)
3) Do whatever you want.

All this doesn't mean that we don't have a conventional order of operation in mathematics, but deciding to do multiplication before addition helps us get rid of LOTS of redundant parentheses. Also, learning things like the Associative Property(ies) of Multiplication/Addition helps get rid of a lot more. The parentheses are still there, but they're just implied.

The order of operations learned in school is very different. It's just a set of mechanical instructions that dictate just one of the MANY ways you can use algebra. It locks you in a single path in the beautiful landscape of mathematics. Like a computer, it gives you the right answer but cannot actually give you any insight on what it is that you're actually doing.

So, the order of operations isn't technically wrong, since it generally gives you the right answer, but it is morally wrong because it turns you into a robot.

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