This rule used together with fraction bars and a high discipline when using brackets avoids the need for any "left to right" rule.
> and "division by x" is defined as "multiplying by the inverse of x".
Exactly!
Just like subtraction is defined as adding the inverse element, so - must only be defined as unary operation to get the inverse element.
With this definition you can use "a - b" as a simplified writing convention for "a + (-b)" and use commutativity to prove "a - b - c" = "(a - b) - c" = "(a - c) - b".
This proves that you can work from left to right but also out of order.
It all turns down to "how is the symbol / defined".
Do you define "a/bc" as "a (1/b) c" or "a (1/bc)", respectively "a (b^(-1)) c" or "a (bc)^(-1)"
Instead of / you could do the same with : or
÷ ... I think in practice no one really distinguishes, but the symbol that was available for typesetting was used.
And I already gave historic sources for different usage of /
and another source indicating that this is disputed and not understood consistently when read by different people.
Remember: My point here is "should you use brackets in mixed / * or multiple / expressions, because it will avoid misunderstanding"
--> Definitely yes, because there are definitely people reading this in a different way.
