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: $\ll$

Solution 1:

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.

Solution 2:

"$a\ll b$" 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".

Solution 3:

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 $g(x)$.