Math problem that is driving the Internet crazy

[A Toy Windmill]

This isn't a "maths problem". It's a "maths communication problem". And in this case, the communication spins on very shallow conventions.

There wouldn't be a big issue if we decided to change the convention tomorrow and say that division now takes priority over multiplication. Changing that convention doesn't change division. It doesn't change multiplication. It doesn't change arithmetic. It doesn't change the mathematics at all. This isn't a matter of mathematics.

Who cares about the conventions we use to denote the original calculation? We might as well be arguing over whether the Earth is 93 million miles from the sun or whether it's actually 150 million kilometers from the sun.

You can use whatever convention suits you, as long as you're consistent with that. But the point is that, if we're going strictly with the current PEMDAS/BODMAS rule ... not with some outdated convention that some people still use today and not with some future convention, but with the PEMDAS rule as it is ... then the parenthesis must be resolved first and then you divide and multiply from left to right. DIstributing the 2 in to the (2+2) before dividing violates that.

And this is after all grade/high school mathematics, so whatever convention engineers or physicists or advanced mathematicians use is not relevant here.

I think there is a more interesting point here, which is that there is a difference between mathematics and how we choose to express mathematics. It's one to keep in mind, since I think it diagnoses what so often goes wrong in the interminable 0.999... = 1 debate.

If we wanted to, we could write everything out in full:

Take 8 and divide it by 2. Multiple the resulting product by the result of adding 2 to itself.

This is the mathematics problem: finding out the value of a particular computation. It's an unwieldy way to set the problem (but there was a time when all problems were set in full prose), and nowadays, I take advantage of operators to write

(8/2)(2 + 2)

But I'm still expressing the exact same mathematical problem. The notation here depends on conventions, as does any notation or language in general. But once you understand the notation as popularly understood, you see it's the same problem I wrote above in full.

What is driving the internet crazy then is the fact that, it seems, many of them never really got to grips with some notation that's popular in high-school. This isn't an embarrassment for working mathematicians, because working mathematicians never use the division symbol. All our divisions are expressed using fractions, almost always with a horizontal bar so it's unambiguous where the numerator and denominator are.

We still insist that multiplication happens before addition, and I'd personally be sad to see the internet being confused over that convention, but it still wouldn't be too big a deal. I'd be happy to let majority vote determine which notational conventions we should use going forward.

But if the internet was being driven crazy by the problem expressed in full-prose, that would be a serious problem. I would not let a majority vote determine who was right in that case, and would be very worried about the state of our education systems.