# Re: Solidus variations

Date: Tue, 11 Oct 2011 04:26:57 +0200

2011/10/11 "Martin J. Dürst" <duerst_at_it.aoyama.ac.jp>:
> 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).
Received on Mon Oct 10 2011 - 21:30:43 CDT

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