Does “arbitrarily small” mean very close to zero or very negative?

Question

In mathematical writing, does “arbitrarily small” mean very close to zero (like $0.000001$) or very negative (like $-1000000$)?

Are there better phrases to distinguish these two cases?

Answer 1

Traditionally, it's the first one: "arbitrarily small" means "small in magnitude", close to zero.

I see the logic behind the second interpretation, but numbers like $-10^9$ are normally thought of as large negative numbers, so using "small" to describe them is bound to be confusing.

Also on English.SE: What is an adjective for a very large negative number?

Answer 2

Usually, "Arbitrarily Close" is used to reference a proximity or distance between two points. If two points are arbitrarily close, the difference is such that they are all but the same point; they are so close the difference is insignificant - except that they are different.