Addition: Multiplication:
Infinity + Finite = Infinity Infinity x Infinity = Infinity
Infinity + Infinity = Infinity Infinity x Finite = Infinity,
but Infinity x 0 is undefined
Infinity + -Infinity can be
absolutely anything finite or not Infinity x -Infinity = -Infinity
-Infinity + Finite = -Infinity -Infinity x Finite = -Infinity,
with the same exception for 0 as before
-Infinity + -Infinity = -Infinity
-Infinity x -Infinity = Infinity
Subtraction:
Same as addition, with u-v treated as u+(-v):
where
-(Infinity) = -Infinity
-(-Infinity) = Infinity
Division:
Same as multiplication, with u/v treated as u x (1/v):
where
1/(-Infinity) = -0
1/(Infinity) = +0
1/(-0) = -Infinity
1/(+0) = Infinity
You'll need to make the distinction between +0 and -0, if you're going to say
anything useful about division with infinity.
These rules are made in such a way that all the properties (+,x,-,/) will
remain true when infinite limits are included. It is possible for a limit
to be infinite without its positive or negative sign being determined. This
limit will represent the unsigned infinity. Its negative is itself and its
reciporical is 0 (without the + or - sign). You'll need to use all three
kinds of infinity. Much of Calculus is devoted to resolving those limits
involving the undefined operations above, like
Infinity - Infinity, Infinity x 0, Infinity/Infinity
There is a theory of infinitesimals based on what is known as Non-Standard
Analysis. Its content is completely equivalent to Calculus. In fact, it is
a reformulation of Calculus that matches very closely the original formulation
of Calculus as a calculation system for infinite and infinitesimal numbers.