2011/10/11 "Martin J. Dürst" <[email protected]>: > Horizontal bars surely work by using bars of differing length, with shorter > bars having higher priority. Horizontal bars of equal length would be very > weird.
Not so weird. And not exceptionnal, given the implicit top-to-bottom associativity, there's no confusion. Let's not forget cases like "3/2÷4/5": it is most often read with the slash operator having higher priority than the dotted division operator (in the middle), even though this is generally the same operation... You can easily see that "÷" is the linear equivalent of the 2D representation using horizontal bars (the dots are like placeholders for the two numbers or expressions that would fit there), but you can't make differenciation of lengths. In that case, you replace the different associativities of the 2D layout by the division operator variants. Then consider "1÷2÷3" in a linear formula, unambiguously interpreted as "(1÷2)÷3", and convert it back to the 2D layout, there's no need to make distinction of lengths of the horizontal bars... You just keep the same associativity. Weirdness is a matter of choice. There are cases where you still need a form with maximal horizontal compaction to fit a line in complex expressions. Similar considerations are taken with expoentiation, when the using multiple levels of superscripts does not compact enough: the exponentiation operator is generally noted by assuming the right-to-left associativity, so that "2^3^4" means "2^(3^4)"; if you want to avoid excessive vertical layout, aligning the exponents would create a confusion with the products of exponents, so you use a visible operator like "^". This is not definining a new operation, but uses another possible presentation. And let's not forget also that each maths article may redefine all operators and their presentation layout. There's no universal notation. If another notation allows easier reading and shortens the notations, it will be defined and used. That's why we find a lot a variant glyphs for "similar" operations (which are not always equivalent in all contexts, for example the middle dot operator is not always a product, or must note a distinct operation from the cross product, even when one operand is a number "constant", because numbers are not always "constants" but can be used to note a functional operation; a simple formula like "2 x" does not necessarily means the same as "x + x" or "2 · x" or "2 × x" or "2 * x", it may mean "2 ○ 2", where "2" is a function defined in a multiplicative group of functions defined by composition of functions, or may mean the double application of a differenciation operator; more complex interpretations when working with distributions, sets with infinite cardinalities, limits, and so on... because these sets only preserve a few of the properties existing on classical reals, rationals, or integers; operations on cyclic sets, or fields, are even more complex and need specific notations for operations we generally consider equivalent on the simple cases most people assume in usual life).
