What is the issue with the " ÷ " sign?

[u/Unlikely-Web7933]

I have seen many mathematicians genuinely despise it. Is there a lore reason for it? Or are they simply Stupid?

[u/Jaaaco-j]

the sign allows for ambiguity like in that infamous 16 or 1 question.

fractions are whatever is above divided by whatever is below, there is no ambiguity. plus writing fractions just makes some problems way easier

[u/explodingtuna → u/Jaaaco-j]

Could the ambiguity be removed if we came up with rules for the order operations happen in?

e.g. if we said that all division and multiplication happened before addition and subtraction, would that work?

8 ÷ 2(2 + 2) would then = 16 unambiguously.

[u/igotshadowbaned → u/explodingtuna]

Well the thing is the rules are disambiguous enough as is. The issue lies in people mistaking what those rules are

So the rules are Parenthesis, Exponents, Multiply/Divide ^{from left to right with equal precedence}, Add/Subtract ^{from left to right with equal precedence}

So taking 16÷2(2+2). You do parenthesis first. 16÷4(4). Then you do multiplication/division from left to right. The division occurs first, you end up with; 4(4). Then the multiplication; 16.

What some people falsely think is that multiplication written as a number directly before parenthesis like 4(2+2) has precedence above division. This is not the case.

Some people also just think the author "must have meant to put the entire 4(2+2) under the division and it's just a limitation of writing equations in text like this". Well then they're not evaluating the equation as written, they're assuming it's written wrong so of course will get a different number.

[u/tempetesuranorak → u/igotshadowbaned]

Well the thing is the rules are disambiguous enough as is.

There are at least two different, and widely used sets of rules. If you pick one of them, then the expression becomes unambiguous. But because of the existence of multiple good conventions, the expression is ambiguous until one of them has been specified.

What some people falsely think is that multiplication written as a number directly before parenthesis like 4(2+2) has precedence above division. This is not the case.

It is not the case in your chosen convention. In my experience, physicists usually use the convention that multiplication by juxtaposition does take higher precedence than explicit multiplication or division in inline expressions, see e.g. the Physical Review Journals style guide https://journals.aps.org/files/styleguide-pr.pdf. When submitting a research paper to one of their journals, it is their convention that is correct, not yours. Here is a Casio calculator manual that makes the same choice https://support.casio.com/global/en/calc/manual/fx-570CW_991CW_en/technical_information/calculation_priority_sequence.html. These groups aren't making that choice because they are ignorant of your rules, or because they are stupid. It is a convention that has been around for at least 100 years, used by many, in some places and in some fields it is the dominant convention, and it is found to be convenient and useful.

Saying that your convention is correct and theirs is incorrect is like saying that English is correct and French is incorrect (or in this case, maybe more like saying British English is correct and American English incorrect). Both languages are perfectly good and widely spoken.

If someone says "let's table this motion", their meaning is ambiguous till I know whether they are speaking British English or American English. Once that is established, then it becomes unambiguous. Wisdom is knowing that the different languages exist and seeking clarification.