From: Philippe Verdy <verdy_p_at_wanadoo.fr>

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

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

*
This archive was generated by hypermail 2.2.0
: Mon Oct 10 2011 - 21:30:44 CDT
*