To put it another way, I am afraid I completely disagree with both of you.
-----Original Message-----
From: Rich Hennessy
Sent: Friday, February 11, 2011 7:23 PM
To: maxima at math.utexas.edu ; Michel Talon
Subject: Re: [Maxima] sqrt(x)*sqrt(x)
I am a computer programmer, not a mathematician. I know one simple fact.
In school I am taking a programming course right now so let's put it in this
light.
x=sqrt(-16);
The computer will not let you put in two values for x. I x is a floating
point register then it just will not fit. You can have structures in
computer science that can hold a complex number so I can handle the idea
that x=sqrt(-16) yields an x with two parts. x.realpart, x.imagpart. This
is sort of like a record. In practical terms you can't have two physical
values in the same place. It is a law of physics that complex numbers
cannot be the result of any measurement. Even in quantum mechanics you have
a complex wave form, but only when you take the conjugate(psi)*psi do you
get something that can be called a possible result of a measurement, because
the answer must be real. Maybe in some abstract sense mathematicians feel
that complex numbers are just as real as real numbers. I don't know much
about that. I do know that you can't put a square peg in round hole. I
also know that all of this is moot when you have things like this.
assume(x>0)$
x:-16$
sqrt(x);
[-4*%i, 4*%i]
At least be consistent. Maxima should spit out an error when it see's that
x is -16 in line 2. But what happens when you try this, nothing. Maxima
ignores it's inputs and does one thing. There is no way you can choose
among two different results when you allow contradictions to squeak through
without any complaints.
Math is supposed to be based on axioms. This is not math, if it was the
above might be okay for some people. This is computer science though. In
computer science you want facts, not arbitrary rules and arguments that can
be tailored to fit just about any proposed hypothesis. I never call myself
a mathematician because of this. I am a computer science major. If I get
my way I would be a computer scientist. I believe in science, which is
based on facts. Math is just conjecture that sometimes coincides with
reality, sometime not.
So you have to pick one or the other, there is no third possibility. I
challenge you to find a third possibility that can actually work and make
sense too.
Rich
-----Original Message-----
From: Michel Talon
Sent: Friday, February 11, 2011 6:41 PM
To: maxima at math.utexas.edu
Subject: Re: [Maxima] sqrt(x)*sqrt(x)
Rich Hennessy wrote:
> I don?t know why I jump in to say anything on this, but a function can
> have
> only one value. If it has more than one it is not a function, by
> definition
> of mathematical functions. So sqrt(16)=4, not 4 and ?4 too.
>
> So sqrt(-16) = 4*%i
>
> I take the positive imaginary axis as the preferred answer for completely
> arbitrary reasons. You know what positive imaginary means right? The
> part
> of the imaginary axis ABOVE the real axis. I say it thus because I like
> it better.
>
Rich, i must say i agree completely with prof Fateman discussion on this
point and thus disagree with you. The notion of a function having just one
value on a given domain to a given range is certainly the notion commonly
used in elementary courses, but so-called multivalued functions have been
used continuously by most great mathematicians since at least 19^th century
up to now, particularly in algebraic geometry. For example sqrt(-16) is
4*%i if you continue analytically from sqrt(16)=4 through the upper half
plane, but -4*%i if you continue through the lower half plane. It jumps
abruptly on the (x<0) axis. Since in a CAS when you consider a square root,
it is generally the square root of some complicated expression of x it is
basically impossible to enforce the rules that students learn in elementary
courses. To reconcile single valuedness and the obvious multivaluedness of
the sqrt, one considers "Riemann surfaces" where basically you duplicate
the x plane into two planes touching at 0 and infinity. So a point in this
surface is a couple (x ,sqrt(x)), and (x, -sqrt(x)) is another point.
Now you can define a single valued sqrt on *that surface* which is simply
the projection (x,sqrt(x)) -> sqrt(x). This is a bona fide function in the
Bourbaki sense, or in the elementary school sense. The other projection
(x,sqrt(x)) -> x is called a branched covering of the complex plane. Whether
this sort of ratiocination is able to clarify your objection, i don't know,
but basically Fateman's argument that a degree N equation has N roots is
all that is really to understand. By analytic continuation in parameters of
the polynomial these roots exchange (this is related to Galois theory) so
there is no way for a CAS to really distinguish them. In particular, as
Fateman says, this rule of elementary school sqrt(x^2)=abs(x) which is true
for x real, is false when you go to complex x. I would be much happier
with sqrt(x^2)=x which is always true, provided you understand that sqrt has
two values and you have expressed one of them.
--
Michel Talon
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