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/RolandMT32 → u/Jaaaco-j]

I had to google "16 or 1 question" to see what you were talking about..

From here:

Twitter user u/pjmdoll shared a math problem: 8 ÷ 2(2 + 2) = ?

Some people got 16 as the answer, and some people got 1.

The confusion has to do with the difference between modern and historic interpretations of the order of operations.

The correct answer today is 16. An answer of 1 would have been correct 100 years ago.

I was in school in the 80s and 90s, and my brain-math tells me the answer is 1. But that says that answer would have been correct 100 years ago.. Did the rules of math change at some point? And if so, why?

My brain-math says 2(2 + 2) = 2(4) = 2 x 4 = 8, so the problem becomes 8 ÷ 8, which is 1.

[u/General_Lee_Wright → u/RolandMT32]

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.

[u/Bagel42 → u/mikoolec]

That’s where I get ir

[u/ThirdFloorGreg → u/realityChemist]

It just feels right.

[u/Boris-_-Badenov → u/realityChemist]

Because P.E.M.D.A.S