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…

It’s all relative. Is 1/8 small? Well, if you’re talking about an extra large pizza, 1/8 might be an entire meal. But certainly it’s less than the whole. Is -100 (a hundred owing) large? If you’re talking about national debt, that’s merely a drop in the bucket. Again, less than the whole amount we’re comparing it to… in fact, probably even less than 1/8!

If we go back to the notion of “unity”, then certainly fractions of 1 are smaller, and multiples of 1 are larger (in magnitude – regardless of positive or negative), but how are we defining “unity”? On a number line? Then consider that the set of numbers between 0 and 1 is actually LARGER than the set of all integers, as proven by earlier infinity discussions…

My calculus students are still having trouble with this distinction. I asked on a test which derivative was greater from a graph. Since both are negative, they need to answer with the one having smaller absolute value. I wish I had given them a hint about this issue. Many will tell me the bigger negative number is the greater. It’s hard, because the way we talk interferes with the way we analyze mathematically.

It’s confusing to compare “less than” with “small” – it is hard!

I would say that it is almost more confusing to try to compare them. Negative numbers typically serve a totally different purpose that “small” numbers.

Plus, “negative” is something of a categorical descriptor, whereas “small” requires a standard to compare to.

And there are certain values that will always be “small”. For example, correlation coefficient for regression models will always be between 1 and -1.

I am a fan of the term “bigger in the positive direction or bigger in the negative direction.”

I tend to carefully use “greater” and “less” to refer to the number-line kind of logic, like in Sue’s example asking “which derivative is greater?” and then getting an opportunity to make the distinction among these terms.

So for limits that really are heading toward negative infinity, I would say “decreasing without bound” or “going arbitrarily far in the negative direction” or things like that, rather than “getting smaller”. I’d reserve “small” for “small in absolute value” so that I can say things like “the error is small” and be sure I don’t mean that I’m ten trillion units too low!

For a limit approaching some finite value, I’d say “the distance from L is getting arbitrarily small”.

But the English vocabulary is always tricky here.