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It's not correct, it's totally arbitrary.

It doesn't matter what order of operations you use, all that matters is that we all agree on the same order of operations. Otherwise two people will look at the same calculation and get different results. Neither result is more correct than any other, but we need to agree on one of them.

We didn't figure it out; we agreed to a single order so that we can all communicate math effectively. This is the same way that we generally all use the English sentence form "subject, verb, object": I ate the bagel, you saw the fox, the car sped by us, etc.

Changing that order can introduce ambiguity: "Saw, you, the fox" is unclear who is doing the seeing. However, it can also be interesting in terms of drawing attention; look no farther than how Yoda in Star Wars talks. He uses a very different sentence structure, which is clear enough to not be ambiguous, but is also different. If everyone spoke like Yoda, we could still communicate effectively, even though the form of the sentences are different. So long as we all agree on how to speak, it works.

Likewise, if we did PASEMD, so long as everyone agreed to use the same order of operations, we could still do math. We'd have to change our current math formulas to match the new system, but the math is still correct.

It's not a case of being correct or incorrect mathematically, in the same way that neither "library" nor "bibliotheque" is the correct way to refer to a building full of books you can borrow. They're different combinations of letters, but they represent the same thing.

If you make judicious use of brackets, order of operations becomes irrelevant. PEMDAS is just an agreement for how to interpret equations that don't make judicious use of brackets, and so it affects how we write them down, but not the fundamentals of maths itself - and water boils at the same temperature, whether you refer to it in Fahrenheit or Celsius or Kelvin.

PEMDAS is a convention, not a fact of mathematics. And it's not the only convention we use, it's just the most common one.

Because PEMDAS is the main convention used, people understand that 5 + 4 x 3 ^ 2 - 1 means "raise 3 to the power of 2, to get 9, then multiply 4 by that, to get 36, then add 5 and 36 to get 41, then subtract 1.

If we used "PEASMD" instead (nobody does), we'd have to write 5 + (4 x 3 ^ 2) - 1 to get the same result.

There are systems that don't need brackets, such as Polish (or Reverse Polish) notation, where you put the operator first (or last) and the things it operates on after that (or before that, for Reverse Polish).

So 4 + 9 would be + 4 9, and the expression I gave above would be

- + 5 x 4 ^ 3 2 1

Which means "subtract the result of + 5 x 4 ^ 3 2 and 1"

+ 5 x 4 ^ 3 2 

would mean "add 5 and the result of x 4 ^ 3 2"

And x 4 ^ 3 2 would mean "multiply 4 and the result of ^ 3 2"

and so on

Math is a language; written equations are like sentences.

"The man bit the dog" is just as valid as "The dog bit the man", but since we agree on the grammatical rules that when a Verb takes two cases the Object goes after it, which is "correct" depends on whoever the was bitten. If the man was bitten, the first one is incorrect and the second one is correct.

PEMDAS is the grammar of the equation. It's "correct" because it helps to all agree on what we mean. It clears up ambiguity.

It is not something we figured out is ti something we decided.

We needed to find a way to write down something like first add two and three and then multiply that result by five. and we decided that (2 + 3) x 5 would be the best way to write that down.

We could have decided on another system but that one is useful in many ways with how numbers work.

The important part is that we all agree that what we write means the same thing.

The details are far less important than that we are all on the same page.

There's a sense in which it's totally arbitrary, like the alphabet. It could be any order, but the important thing is that we settled on a particular order.

On the other hand, there's a sense in which the order we use was the only one that made sense, or maybe one of two that made sense. Let's forget about division and subtraction for a second. Multiplication is just repeated addition, and exponentiation (or "order" if you use BODMAS) is just repeated multiplication. So it really only makes sense to go power-multiplication-addition or addition-multiplication-power. Then, we get parentheses (or brackets for BODMAS). The whole point of brackets is to say "DO THIS BUT FIRST," so it has to be at the start.

Now, subtraction and division are just shorthands for adding negatives or multiplying by fractions, so that's why they go in pairs with addition and subtraction being equal and multiplication and division being equal.

So why exp-muli-add? Well, we often want to write something like 3y²+2y+6. If we used add-mult-exp, then we'd need to rewrite this as:

(3×(y²))+(2×y)+6

So we pick the format that makes the notation simpler and easier to read.

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It's an arbitrary convention, and hasn't entirely caught on for the order of MD and AS, or even MD vs AS.

For those, I recommend always using parentheses to avoid confusion regardless of the convention.

Say you were calculating the trajectory of a probe on its way to one of Jupiter's moons (Europa Clipper for a real world example). Your plan is to launch from Earth, and with the help of both Earth and Mars' gravity three years later to launch the probe into Jupiter's orbit.

The calculations for this would be complex and orders of operations would absolutely matter. Would these calculations even be possible using a different order of operations? I know if we did, our equations would have to be different, I just can't begin to imagine what that would look like.

We didn't figure it out, we made it up.

Brackets being first makes sense because all that brackets do is change the order. If they weren't first they'd be useless.

The others are fairly arbitrary but do make some sort of sense. Exponents are repeated multiplication, multiplication is repeated addition. So it does make sense to go from the "most powerful" operation to the least powerful.

But if we wanted to we could have an order of operations that does addition first and exponents last. We'd just have to rewrite all our equations if we wanted to get the same result.

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All you really need is a system for denoting "do this part before you calculate the rest" (parentheses). As long as you have that, you can do every calculation exactly right, regardless of whatever other order of operations you have.

The only issue you'd have is clarity. A dozen nested parentheses is not a recipe for clear equations.

We can write math without any "order of operations". It is just inner functions (or rather relations) all the way down.

It gets really ugly looking really fast. So you never do it unless you have to prove something involving order of operations.