# Q & A: infinity

Q:
If infinity covers all numbers then it must cover both positive and negative numbers This would mean that it is both infinitely large AND infinitely small at the SAME time. How can this be? Thanks
- James
U.S
A:
This is really a math question, so it evokes my old math-major traits- mainly pickiness.

Math questions really require some precision before they can be answered.

Here, I'm not quite sure what you mean by 'infinity' as a noun.
I definitely have no idea what you mean by the verb 'covers'.

So perhaps it would be possible to rephrase the question using terms whose meaning can be explained. In the process, you might discover the answer yourself. Meanwhile, I'm clueless.

Mike W.

I'm having a bit of trouble with "large" and "small" as well, but only in a trivial sense -- I think of large, negative numbers as large, but less than negative numbers of less absolute value.

"Infinity" is usually just shorthand for a process of taking a limit with a finite number as an input, and observing how something changes as that input number gets larger and larger. Sometimes the limit of the same function as the input parameter gets more and more negative is very different. Consider the function f(x)=Arctan(x). The limit of f(x) as x goes to infinity is pi/2, while the limit of f(x) as x goes to negative infinity is -pi/2, so in this restricted application these are different. If x can be complex, then f(x) has no limit as x goes to infinity; Arctan(x) has an essential singularity at infinity in the complex plane, so it doesn't take much to make this argument break down.

Tom

(published on 10/22/2007)