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!

  • 3
    $\begingroup$ 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 $B$. $\endgroup$
    – Matt
    Nov 3 '16 at 13:37
  • $\begingroup$ If you have a particular size in mind, why not just specify it? Like $\epsilon < \frac1N$ for $N=1000$? $\endgroup$
    – MPW
    Nov 3 '16 at 13:37
  • 2
    $\begingroup$ Can you explain the difference between "close to zero" and "much smaller than 1"? For instance, which is larger, and why? $\endgroup$
    – TonyK
    Nov 3 '16 at 13:39
  • $\begingroup$ 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. $\endgroup$
    – Lisa
    Nov 3 '16 at 14:02
  • $\begingroup$ 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. $\endgroup$ Nov 3 '16 at 14:08

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.


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