# What does $\ll$ mean?

I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically.

http://en.wikipedia.org/wiki/German_tank_problem

EDIT: It looks like this can use TeX commands. So I think this is the symbol: $\ll$

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 +1 for bringing up the German tank problem. Reminds me of a variant a prof once posed to us: when you arrive in a town, and you see a tram with number 14, how can you estimate the number of trams riding in that town. – Raskolnikov May 2 '11 at 16:45

In the occurrence of "$\ll$" you are asking about, it means "much less than". If you look at the fourth entry here, this is the first meaning listed for $\ll$.

As Charles has correctly pointed out, this symbol is also used in advanced mathematics to describe a certain relationship in the growth of two functions. That is the second meaning listed.

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That works for me! – ssiddi38 May 2 '11 at 5:09
@ssiddi38: glad to help! – Zev Chonoles May 2 '11 at 5:41
Downvoter: Why the downvote? – Zev Chonoles May 2 '11 at 5:41

"$a\ll b$" can also mean "$a$ at least as smaller than $b$ as it is needed for my arguments to be true".

It is in that sense that one sometimes writes, for example, "let $x$ be such that $0< x\ll 1$" to mean "let $x$ be a positive number as small as needed for the following to hold".

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It does not mean "much less than". It is the Vinogradov symbol, similar to the Hardy-Landau-etc. Big O notation.

$$f(x)\ll g(x)$$ means that there exists some N and k > 0 such that, for all x > N, $f(x)<k\cdot g(x).$ In slightly more informal terms, it means that the asymptotic growth of f(x) is no faster than that of g(x).

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@Charles: That is one use of it, yes, but I don't believe it is the meaning in the occurrence that the OP is asking about. However, I will edit my answer to be less categorical in saying that $\ll$ means "much less than". – Zev Chonoles May 2 '11 at 5:44
Actually, it is. The precise version would replace $\approx$ with = and multiply by a factor of (1 + o(1)). – Charles May 2 '11 at 5:47
@Charles: Unfortunately I am inexperienced with both statistics and asymptotic analysis. Could you describe in more detail the way in which the $k$ and $N$ which occur in the OP's link are functions that are asymptotically related like this? – Zev Chonoles May 2 '11 at 6:05
@Moron: Yes, I mentioned Vinogradov and Big O. – Charles May 2 '11 at 12:40
@Charles: Yes, I was only agreeing with your answer (you have my +1 already). – Aryabhata May 2 '11 at 15:52

Another way to think about the much less than $\ll$ is in the spirit of approx $\approx$. When you write $a \ll b$ you say that errors of size $a$ don't matter in assessing a quantity of size $b$. So in the article you reference, saying $k \ll N$ allows replacement of $N-k$ by $N$ to simplify the expression, or $N-k \approx N$. For practical purposes, such as the one in the article, an answer within $10\%$ is plenty good enough.

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This is the perfect example of the overloaded symbol. In measure theory, we use $\nu << \mu$ if the measure $\nu$ is absolutely continuous with respect to $\mu$, i.e., for any measurable $E$, we have $\mu(E) = 0\Rightarrow \nu(E) =0.$ Most mathematical symbols require a context to be interpreted unambiguously.

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It means significantly smaller than, if I'm not mistaken.

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Perhaps not its original intention, but we (my collaborators and former advisor) use $X \gg Y$ to mean that $X \geq c Y$ for a sufficiently large constant $c$. Precisely, we usually use it when we write things like:

$$f(x) = g(x) + O(h(x)) \quad \Longrightarrow \quad f(x) = g(x) (1 + o(1))$$

when $g(x) \gg h(x)$.

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