How do operations apply to infinity? Can it be treated like a number when we do operations?
like:
X + infinity
X - infinity
(X)(infinity)
(X)/(infinity)
infinity[sup]X[/sup]
(infinity)!
Will the answer to all of these questions be infinity?
What about
(Infinity)/(infinity)
Does that equal one?
Also, I’ve heard that the slope of a function like X = 4 is infinite. Is this true? Does this mean that dividing by zero gives an infinite ratio?
If this is the case, does (infinity)(0) = something other than 0?
Division by zero is undefined.
Infinity divided by infinity is undefined.
Infinity times zero is undefined.
The answers come from examining limits.
You can often treat infinity like a variable, but not always. In your examples, all approach infinity except x- (inf), which approaches -(inf), and x/(inf), which approaches zero.
There’s no such thing as “infinity,” of course, so the situation you’re actually dealing with in algebra is some function whose value you wish to estimate as x approaches infinity; y = 1/x approaches zero as x approaches infinity. An expression like y = e[sup]x[/sup]/10[sup]x[/sup] has to be calculated in terms of x before you can determine its asymptote(s)
You can extend the real numbers by adding a new element, infinity. Problem is, arithmetic doesn’t work as nicely in the extended real number system (in particular, the old rule a < a + b for b > 0 fails if a is infinity and b is not). See Rudin’s Principles of Mathematical Analysis. Better yet, do a search on transfinite arithmetic, but be prepared for some pretty heady stuff.
Most of the time, what Nametag said is the right idea.