2011-02-28

unary operators not always taking precedence over binary

2.7: news.adda/math/unary not always taking precedence:
. negation has the same precedence as
multiplication and division;
because, negation means mult by -1.
So -a^b should be -1*a^b = -(a^b).
details:
. programmers were accustomed to the C language,
in which unary operators such as negation
have higher precedence than any binary operator;
(and there was no exponent operator in C
to cause them to think twice about the matter).
so, when programmers use an exponent operator,
they may have wished to remain consistent with C;
however, for centuries,
the polynomial -x^2
has meant -1*x^2 = -(x^2)
not (-x)^2 = x^2 .

. look at the HP48G User Guide/order of operations:
priority#1:
Prefix functions (such as sin, ln, ...)
and Postfix functions (such as ! (factorial)).
--[. many could say negation is a prefix -();
2.16: nevertheless,
notice the way math has superscripted powers
(rather than using an operator);
as if it was an extension of the symbol's name
like the way subscripts actually are,
and thus intuitively having higher precedence
than any operation applied to the name .]
priority#2: Power (^) and square root.
priority#3: Negation (-), multiplication, and division.
--[. here is the 2nd place -() fits;
but, only because of its equvalence to -1*();
many think it's obvious that the negative
is part of the number's value .]
priority#4: Addition and subtraction.

. clarity should take precedence over correctness;
so, the system needs to ask new users
-- at least those who use the form (-x^n):
"( how would you eval -2^2 ?
{ 4, -4 } ??
. -1*2^2 is definitely = -1(2^2) = -4 .
whereas (-2)^2 = (-2)(-2) = 4 . )
. furthermore, when exporting adda`binary,
or allowing copies to text
always write it unambiguously { -(2^2), (-2)^2 }.