Order of operations - arbitrary of fact?

The order of operations is nothing more than an accepted convention in order to avoid the need to write parentheses all the time. So, instead of (5\*3)+5 we allow ourselves to write 5\*3+5, and agree that the multiplication takes precedence over addition, i.e., there is an implied set of parentheses given by the agreed-upon operation precedence.

There is a good reason for having the operation precedence be as it is. The reason is that the convention we use is the one which minimizes the number of parentheses needed in the expressions we commonly encounter. So, it's the matter of practicality.

Do note that changing the operation precedence would not change mathematics in any way at all. The only thing that would change is the way we write down certain expressions. For example, instead of (a+b)^(2) = a^(2)\+2ab+b^(2) we might have to write something like (a+b)^(2) = ((a^(2))+(2(ab)))+(b^(2)), but as you can see, the square of a binomial remains the same; there are just some extra parentheses there.
Think of the order of operations as equivalent to punctuation and grammar.

If you follow "official grammar, then "I ain't no fool" means that you are saying you are a fool. There's a double negative. Unofficial grammar will lead to the opposite interpretation.

Now, which interpretation of the sentence is "correct"? Well, the correct interpretation is whatever the speaker actually meant to say.

That's where the trouble begins with BIDMAS. While it is very broadly accepted as the "correct way" to write mathematics and while it does have some benefits, the correct interpretation of an equation is what the writer intended to write. If you are writing mathematics you should do what you can to make your intentions clear.

Please know that many people disagree with my opinion on this, but I would argue that, when devoid of any of context, the expression

> 5+7x3

Is meaningless. It is EASILY interpreted as (5+7)x3 OR 5+(7x3). BIDMAS will ensure that most people will interpret the expression as the later, but the former interpretation is by no means unreasonable.

In scenario's like this, I try to include brackets, to ensure that people don't even need to consider BIDMAS. My intended meaning should be the first thing that comes to any readers mind. Just like when I am speaking, I am careful to word the sentence so that my meaning is clear.

This does not mean that you need to over whelm of your maths with a million parentheses. Just as it does not mean that you should write sentences with loads of punctuation and clarifying statements. There is clearly a healthy balance between brevity and clarity.

For example:

You'd be hard pressed to see 3 + 5y and think that it means (3+5)y. But 3+5x7 is less obvious and it costs you nothing to write 3+(5x7) or 3+5(7) to make the intended meaning clear despite the readers familiarity with BIDMAS.

I (like most mathematicians) understand BIDMAS, but even I find 3+5(7) more pleasant to read than 3+5x7.
Yeah, it's just a mostly arbitrary convention. I like to think of it as which side of the road people drive on. As a society, people could collectively decide to drive on either the right or the left about equally well. However, once one side becomes convention, it's best that everyone follows it.
3 people have 5 apples, there are 5 more apples in the fridge. How many total apples are there?

Order of operations is factual.
5x3 can be seen as the total number of objects, given three packages each containing five objects (or the other way around b/c of commutativity, doesn’t matter here)

So 5x3+5 is that with 5 more objects. Addition precedence is essentially saying that 5 objects is the same thing as 5 packages.
>Throughout primary school/high school everybody learns the order of operations. Is this order arbitrary or factual?

It's more practical

>If we had instead decided that addition comes before multiplication, answered that we would today call correct would now be incorrect.

We would need to write things differently. If we have 5×3+5, we have a few choices

|Decision|Expression|Result|
|:-|:-|:-|
|× before +|5×3+5|20|
||5×(3+5)|40|
|+ before ×|5×3+5|40|
||(5×3)+5|20|
|OOO doesn't exist at all|5×3+5|???|
||(5×3)+5|20|
||5×(3+5)|40|

No matter what, parentheses let you get either 20 or 40 anyway. So technically, the decision doesn't matter - we just chose × before + because it's the most practical (see other good comments!)

But this decision was already made, so we (usually) don't change it unless there's a really good reason. So 5×3+5 results in 20. If you want to + first, you use parentheses
Yes, it's correct because we agreed upon it. But that's also true for just about everything in math.
Arbitrary. Just like 2 ^ 3 ^ 4 = 2 ^ (3 ^ 4).
When you consider that exponentiation is "repeated multiplication" and multiplication is "repeated addition," then it makes sense to do it in that order -- the same way we do hundreds on the left, then tens, then ones.
Operator precedence and the distributive property are important, and things like PEMDAS become meaningless past elementary school.  It's just a nice teaching tool early on.

The  ➗ symbol is not used and everything is expressed as a rational number which makes order of operations no longer needed and the first two items I mentioned are of importance.

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