On Fri, Feb 11, 2011 at 2:51 PM, Richard Fateman
<fateman at eecs.berkeley.edu>wrote:
> 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> 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".
>
You are probably not aware, but every response you have written has been
very rude. Seek professional help.
Recall that the OP specifically says that he is restricting consideration to
real numbers. So for him, x<0 is not in the domain of the function he or
she wants (or perhaps it's not in the domain of the function he or she
expects).
A real mathematical function must be single valued. To construct a proper
function that we can call "square root", we choose by convention the
"principle square root", the positive number. Hence in many cases we tend
to write \pm\sqrt{16}, where \sqrt{16} is a positive number, but the
negative of that number will work just as well for whatever we are doing.
Also, it's not defined for x<0. And the OP has questions...
I'll agree: the word "function" has slightly different meanings in computer
science, math, maxima, and even physics.
A useful response is to recognize that there can be confusion and
misinterpretation, and help to straighten out the difficulties of
interpretation.
Sage has an alternate CAS, doesn't it? And it has lots of "rings". I
wonder if sqrt has the desired/expected behavior over the ring of reals?
>
> 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
>
>