On 2/11/2011 10:26 AM, Gary Pajer wrote:
>
>
> On Fri, Feb 11, 2011 at 10:10 AM, Richard Fateman
> <fateman at eecs.berkeley.edu <mailto:fateman at eecs.berkeley.edu>> wrote:
>
> ...
> It seems hopeless to point out that x>0 does not mean that
> sqrt(x)>0, mathematically.
> There are 2 square roots. For example sqrt(16) is {-4, 4}, even
> though 16>0.
>
>
> Perhaps that is true maxima-tically.
No, there are 2 square roots of 16, mathematically. Maxima chooses
one of them, 4,
and thus it is not true maxima-tically that there are 2 roots.
> And perhaps 16 has two real roots.
There is nothing "perhaps" about it, unless you do not believe in
negative numbers.
S^2-16=0 defines the value(s) for S, corresponding to the square root
of 16.
By some obscure theorem, the so-called "fundamental theorem of
algebra" there are 2 roots.
> But as a mathematical *function* sqrt(16) = +4.
>
Nope. You mean "as a program written by one or more people, the
conjunction of
circumstances leads the Maxima system to return 4 when you type
sqrt(16)." Your
statement probably misuses at least one technical term, and maybe three.
"mathematical" "function" and maybe "sqrt".
The computer programming term "function" in common usage does not
correspond really to
the mathematical concept "function" except superficially.
Why am I bothering to comment on this?
Because failing to make such distinctions leads to
subtle but very bad consequences, even though it may seem harmless
enough for the
moment.
RJF