# How to denote a very small number $\epsilon$

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!

• Not an expert, just my opinion: since numbers are relative, calling a number small only makes sense if you are comparing it to something else. So, if your problem looks like $\frac{A}{B+\varepsilon}$, then you might say that $\varepsilon \ll B$, or indeed, if $A,B$ are bounded above and below by constants then $\varepsilon \ll 1$ seems fine to say that $\varepsilon$ is a much smaller contribution than $A$ or $B$.
– Matt
Nov 3 '16 at 13:37
• If you have a particular size in mind, why not just specify it? Like $\epsilon < \frac1N$ for $N=1000$?
– MPW
Nov 3 '16 at 13:37
• Can you explain the difference between "close to zero" and "much smaller than 1"? For instance, which is larger, and why? Nov 3 '16 at 13:39
• Thanks for your comments, I edited the question to incorporate Matt's example. In fact, both $A$ and $B$ are in the range $[0,1]$. I presume $\epsilon \ll B$, and therefore $\epsilon \ll 1$ is adequate after all.
– Lisa
Nov 3 '16 at 14:02
• It might be important that $\epsilon \ll B$ as opposed to $1$. It could be that $B$ itself is very small. If that is important, you should say it. Nov 3 '16 at 14:08

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.