I'm adding a small number $\epsilon$ to a denominator for numerical stability. Is it correct to introduce it as $\epsilon \ll 1$? In fact, it should be close to zero, not just (much) smaller than 1. What's the best way to describe a small number mathematically?
Specifically, the term in question is $\frac{A}{B + \epsilon}$, where both $A$ and $B$ are in the range $[0,1]$.
Thank you!
On
There is really no "best" way, I think. I would rather say a "precise" or "unambiguous" way. Phrases like "very small", "small enough" in a mathematical statement are usually (shall I say "always"?) informal and should be understood in context.
The notation "$\ll$" has no precise meaning when one interprets it as "much less than", as it is discussed in this question.
When adding a "small" positive number $\epsilon$ to some quantity, one should/might have some criterion in mind that $\epsilon$ should be "small enough" so that some properties are satisfied. If one really wants to be precise, than one might want to state explicitly those properties out.
Does the number need to be larger than 0? If so, specify that.
Is it important that the number is not to large? Give an upper bound on it.
Are you using a specific number? Say what number you use and give a justification for the size.
Basically, say what properties the number must have, explain those and justify those to the extent necessary in the context.