"Mike Oliver" <oliver@math.ucla.edu> wrote in message news://3C06E864.BF44CA7D@math.ucla.edu... > Paul Lutus wrote: > > > Okay. How does this relate to the fact that infinity is not a number? You > > certainly cannot perform numerical operations on infinity. > > I certainly can, in some contexts, and it's a useful thing to do > too.
Any context except a number system as that term is normally defined. See below.
> > For example, consider the one-point compactification of the complex > plane, C u { oo }, where we topologize by saying the open sets are generated > by the open subsets of C, plus F u { oo } where F is any closed subset > of C. We extend the arithmetic operations +,-,*,/ as follows: > > z+oo = oo+z = oo provided z != oo ( oo+oo is undefined) > z-oo = oo-z = oo provided z != oo ( oo-oo is undefined) > z*oo = oo*z = oo provided z != 0 ( 0*oo is undefined) > z/oo = 0 provided z != oo ( oo/oo is undefined) > oo/z = oo provided z != oo > z/0 = oo provided z != 0 ( 0/0 is undefined) > > This is the "extended complex number system" or "complex sphere". > It is the natural structure in which to study "Moebius transformations", > which are functions z |-> (az+b)/(cz+d) for constant complex > values a,b,c,d (a and b nonzero).
Meaningless rhetoric. The "extended complex number system" you describe doesn't meet the definition of "number system," one property of which is the exclusion of infinity.
"No 'infinity' concept exists in the context of any number system, if by number system one means a collection of concepts that have operations like addition and multiplication the way familiar numbers do, operations which obey the usual properties of arithmetic."
