In recent years (or perhaps the last decade or so) there has been a rather cyclic phenomenon of a "viral math problem" that somehow stumps people and reveals that they don't know how to calculate it. It seems that every few months the exact same problem (with just perhaps the numbers involved changed) makes the rounds. And it always makes me think: "Sigh, not this again. It's so tiresome."
And the "viral math problem" is a simple arithmetic expression which, however, has been made confusing and obfuscated not only by using unclear operator precedence but, moreover, by abusing the division symbol ÷ instead of using fractional notation. Pretty much invariably the "problem" involves having a division followed by a multiplication, which is what introduces the confusion. A typical version is something like:
12 ÷ 3(2+2) = ?
This "problem" is so tiresome because it deliberately uses the ÷ symbol to keep it all in one line instead of using the actual fractional notation (ie. a horizontal line with the numerator above it and the denominator below it) which would completely disambiguate the expression. And, of course, it deliberately has the division first and the multiplication after that, causing the confusion.
This is deliberately deceptive because, as mentioned, the normal fractional notation would completely disambiguate the expression: If the division of 12 by 3 is supposed to be calculated first and then the result then multiplied by (2+2), then the fraction would have 12 at the top, 3 on the bottom, and the (2+2) would follow the fraction (ie. be at the same level as the horizontal line of the fraction).
Many people would think: "What's the problem? It's quite simple: Follow so-called PEMDAS, where multiplication and division have the same precedence, and operators of the same precedence are evaluated from left to right. In other words, calculate 12 by 3 first, then multiply the result by (2+2)."
Except that it's not that simple. It so happens that "PEMDAS" does not really deal with the "implied multiplication", ie. the symbolless product notation, such as when you write "2x + 3y", which has two implied products.
The fact is that there is no universal consensus on whether the implied product should have a higher precedence than explicit multiplication and division. And the reason for this is that in normal mathematical notation the distinction is unnecessary because you don't get these ambiguous situations, and that's because the ÷ symbol is not usually used to denote division alongside implied multiplication.
In other words, there is no universal consensus on whether "1 ÷ 2x" should be interpreted as "(1÷2)x" or "1 ÷ (2x)". People have actually found published physics and math papers that actually use the latter interpretation, so it's not completely unheard of.
The main problem is that this is deliberately mixing two different notations: Usually the mathematical notation that uses implied multiplication does not use ÷ for division, instead using the fraction notation. And usually the notation that does use ÷ does not use implied multiplication. These are two distinct notations (although not really "standardized" per se, which only adds to the confusion.)
Thus, the only correct answer to "how much is 12 ÷ 3(2+2)?" is: "It depends on your operator precedence agreement when it comes to the ÷ symbol and the implied multiplication." In other words, "tell me the precedence rules you want to use, and then I'll tell you the answer, because it depends on that."
(And, as mentioned, "PEMDAS" is not a valid answer to the question because, ironically, that's ambiguous too. Unless you take it literally and consider ÷ and implied multiplication to be at the same precedence level, and thus to be evaluated from left to right. But you would still want to clarify that that's what's meant.)
Also somewhat ironically, even if instead of implied multiplication we borrowed the actual arithmetic notation for multiplication from the same set as the ÷ symbol, in other words, the expression would be:
12 ÷ 3×(2+2)
that would still be ambiguous because even here there is no 100% consensus.
The entire problem is just disingenuous and specifically designed to confuse and trick people, which is why I really dislike it and am tired of it.
An honest version of the problem would use parentheses to disambiguate. In other words, either:
(12÷3)×(2+2)
or
12 ÷ (3×(2+2))
