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