once again, the internet has been violently reminded that the true joy of notational standards is that there are so many to choose from! that's right, social media is once again abuzz over an arithmetic expression using the worst symbol, the obelus, '÷'.
neither the specific expression nor the form of it are new. arguments over "8÷2(2+2)" can be found on forum posts at least as early as 2016, and "48÷2(9+3)" was apparently a major controversy in 2011, having its own Know Your Meme page.
standards
the old rule of PEMDAS (parentheses, exponentiation, multiplication, division, addition, subtraction) is a notational standard for arithmetic expressions, defining (supposedly) a strict ordering of operator precedence. the operators that are earlier in the list get higher precedence when parsing an expression into a tree for evaluation, which one can think of as "getting to be closer to the arguments".
there are of course plenty of other standards for notation, famously Polish Notation (a prefix notation) and Reverse Polish Notation (a postfix notation).
PEMDAS is an infix notation, but (as I hinted before) there is observably dispute over whether this is the true standard, and whether the standard imposes a strict precedence between multiplication and division. multiplication and division, as is commonly known, are closely related operations; it is not entirely unfair to dispute strict precedence between them. there is also another wrinkle, known among people who care about programming languages as "operator associativity".
operator associativity and trees
operator associativity, as la wik tells us, is either rightwards or leftwards. this tells us the tree structure that should be parsed if two operators have the same precedence.
if (÷) and (an implicit) (*) have the same precedence, then associativity of (÷) tells us how the expression is actually structured. we can use parentheses to make it explicit:
- left associative: "(8÷2)*(2+2)", giving us 16
- right associative: "8÷(2*(2+2))", giving us 1
it seems clear to me that preference for one answer vs the other actually comes down to a preference for one associativity over the other. I know that I prefer the obelus to be right associative, and therefore I prefer the second interpretation.
preference, and getting rid of the obelus
the issue, of course, is that there isn't a standard associativity for the obelus. la wik, on the page for the obelus tells us
The ISO 80000-2 standard for mathematical notation recommends only the solidus or fraction bar for division, or the colon for ratios; it says that the obelus "should not be used" for division.
the solidus is (approximately) the "/" symbol. the fraction bar refers to the full-on fractional representation, which the use of "/" is a shorthand for.
while I do not have the standard available to see what the ISO justification for this rule is, the wisdom seems clear to me - the obelus has too much ambiguity, and the alternatives do not.
- full fractional representation makes it extremely clear what other terms the division is to apply to - it effectively parenthesizes the elements above and below the bar.
- the solidus is generally easy to understand as being right associative (as a shorthand in situations where it is hard to provide a full fraction bar).
- a ratio (eg. "5:7") is understood (when recognized) as having an extremely high precedence, with presumably only parentheses exceeding it.
if we were to rewrite the original math then as a full on fraction, the difference between
8
------
2(2+2)
and
8
- * (2+2)
2
is obvious.
written with a fraction bar, "8/2(2+2)" should be more easily accepted as resulting in "1". writing it out as a ratio, "8:2(2+2)" looks silly and can be discarded.
the futility, of course, is in getting grade-school math teachers and math book writers, especially in America, to give up on the obelus. and so we will continue to get trolled by this wretched symbol.