## Isn’t negative infinity small?

Studying limits again, this time a pair was finding a limit that they said got very small, but then wrote negative infinity instead of zero. When I acknowledged the difficulty of this question, one student mocked me “It’s not like I asked you a touchy subject!”  I laughed and explained that negative infinity has large absolute value, so it is a large negative number.  His partner protested “but negative numbers are small!” so I countered with “If you owe someone 100 dollars is that a lot?” They both concurred that it is, so maybe my explanation was valid.  The original student wanted a non-economic example though, and he proposed toothpicks (this kid cracks me up all the time).  Then I was stuck talking about the non-existence of negative toothpicks, unless we return to the concept of owing someone.  The entire conversation was amusing and I think I was moderately convincing this time.

How do you differentiate between small numbers and negative numbers?  We didn’t even include “less than” in our discussion…

A student asked me “Is infinity a number?” while I was introducing limits.  My immediate response was “infinity is a concept” and I started talking about $\infty$ vs. $2\infty$.  I didn’t really know where I was going with that though, so I let the students debate for a bit and finally related the whole conversation back to the problem at hand.  The discussion was interesting, but other than acknowledging the difficulty of comprehending infinity, I didn’t contribute much (which is usually good, but I certainly didn’t provide clarity).