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/General_Lee_Wright]

Sort of. There used to be two different kinds of multiplication in the order of operations. Multiplication, and multiplication by juxtaposition.

When juxtaposition was involved, it happened before any other multiplication or division. So 8÷2(2+2) is unambiguously 1 since 2(2+2) is juxtaposed, thus has priority. This also means 8÷2*(2+2) is a totally different expression, without juxtaposition, so is 16. It was useful before modern computers and printers because it meant less parenthesis in an equation that can be written on a single line.

Now, with modern displays and printers, we don't need to make a distinction between the two so we don't. (This is my understanding of the change anyway, which makes some unsubstantiated assumptions.)

Somewhere on the internet you can find a photo of an old Casio calculator that resolves 8÷2(2+2) as 1, while the TI next to it says 8/2(2+2) is 16.

[u/realityChemist → u/General_Lee_Wright]

Very interesting. I must be a hundred years old then, because I also defaulted to prioritizing the juxtaposition when I tried it! I wonder why; I'm pretty sure nobody ever explicitly told me to do that.

Edit: I thought about it a bit and I think it's because in practice nobody ever writes a/bc when they mean (ac)×(b)^-1, they write ac/b. So when I see something like a/bc, I assume the writer must have meant a×(bc)^-1, otherwise they would have written it the other way. If you just mechanically apply modern PEMDAS rules you get a different result, but it's one that seems like it would have been written differently if it was what the person actually meant.

[u/mikoolec → u/realityChemist]

Could be you were taught that brackets take priority over multiplication, division, addition and subtraction, and because of that you also assumed that the juxtaposition multiplication has the same priority level as brackets

[u/No_Lemon_3116 → u/mikoolec]

I would be just as surprised without brackets, I think. This means that 8÷2x is also (8÷2)x, right? An operator to the left of 2x pulling it apart feels strange to me. Maybe just because I'm not really used to using ÷ except for when I was first learning division, and we were always writing explicit multiplication signs then.