18
$\begingroup$

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: $\ll$

  • 0
    +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.2011-05-02

7 Answers 7

16

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.

  • 0
    +1. Symbol $\ll$ can, indeed, mean "much smaller than". And $\approx$ can mean "approximately equal". They can also (as Charles said) mean more technical things.2016-09-24
16

"$a\ll b$" 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".

11

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) In slightly more informal terms, it means that the asymptotic growth of $f(x)$ is no faster than that of $g(x)$.

  • 0
    @Zev Chonoles: You can have the check, no big deal to me. I'm just glad you edited your answer to point out the other meaning.2011-05-02
5

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.

3

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.

2

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

  • 0
    @Mar Thank you again for checking. I didn't feel the deleted answers added _anything_ to the conversation (if they add even a little, I leave them). In the future I will use more discretion in both which questions to answer (based on the quality of my given answer) and which answers I delete after a while.2011-08-31
1

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