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I was reading a paper and came to a symbol as follows: "$\gg$" (e.g. $x\gg 5$).

What does that mean? Is it larger than or has more information to mention?

Thanks.

user1729
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5 Answers5

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Often it means "much greater than", and you can interpret by setting $\frac{5}{x}\approx0$. This is done by computing the Taylor expansion in powers of $\frac{1}{x}$ (i.e. "around infinity") and dropping higher order terms. If you had $x\ll5$, then you would instead compute the Taylor expansion in $x$, and drop the higher order terms.

As an example consider, in the context of special relativity, the formula for the total energy: $$E_\mathrm{tot}=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ We want to show that it approaches the formula for the energy in classical mechanics in the limit of small velocity, i.e. $v\ll c$. We can do this by computing the Taylor expansion of $E_\mathrm{tot}$ in $v$ around $v=0$. Doing this we obtain: $$E_\mathrm{tot}=mc^2+\frac{1}{2}mv^2+O(v^3)$$ which is what we expected.

Daniel Robert-Nicoud
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  • (since there is not PM-system I ask here) Can you tell me if this method is called anything specific, that is to do an expansion in (in this case) v, where v is "bounded". I should try to translate into another case, namely http://math.stackexchange.com/questions/1046774/approximate-solutions-for-quintic-equation?noredirect=1#comment2138422_1046774. But for example I am not sure if I should expand around 0 (but I think I should), thank you. – user147163 Dec 07 '14 at 19:00
  • @user147163 I don't think it has a specific name. Try to look up Taylor's theorem. – Daniel Robert-Nicoud Dec 08 '14 at 16:47
  • Well thanks, but it's not really what I'm looking for (I don't think). If you have experience in doing such things expansions in limits, would you then take a look at my question refered to above? I believe I have a case with two variables, where I want to approximately find solutions to the one variable, while the other variable is a limit. Could you maybe confirm? – user147163 Dec 08 '14 at 21:57
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As people have said: it depends on the context.

In the context of analysis (especially analysis of differential equations and such), the statement "$x\gg y$" often means that there exists some implicit large constant $C$ such that $x > Cy$. It is a convenient shorthand for cleaning up the statements of theorems. Compare

Assuming ... then for every $\epsilon \ll \eta \ll \delta \ll 1$ the following statements are true

with

Assuming ... then there exists constants $C, D, E > 1$ such that for every $\epsilon, \delta, \eta >0$ satisfying $C\epsilon < \eta$, $D\eta < \delta$, and $E\delta < 1$ the following statements are true

(These type of quantifier gymnastics are very familiar to the experts, but fraught with perils for the new-comers to the field.)

Willie Wong
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  • As an aside: since there is an implicit constant, one will almost never see a statement of the form $x\gg 5$, since it does not give any more information than $x \gg 1$. – Willie Wong Jan 17 '14 at 13:39
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    I think that this (your last comment) is not true, because saying $x\gg K$ instead of $x\gg1$ gives an idea of what is the relevant scale of the situation (this is particularly important in physics). – Daniel Robert-Nicoud Jan 17 '14 at 13:42
  • @DanielRobert-Nicoud: it is a sad day for physics if 5 is considered to be "much bigger than 1" :-) (Note that I do not disagree with you if one also has $K \gg 1$.) – Willie Wong Jan 17 '14 at 14:38
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It is also sometimes used in the following way: "Blah holds for $x\gg 0$" meaning that for all sufficiently large $x$, blah holds (so we suppress how large $x$ might need to be). Of course, given that the example was $x\gg 5$ this is unlikely to be the case here.

Tobias Kildetoft
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Usually we use $\gg$ (LaTeX code \gg) and $\ll$ (LaTeX code \ll) to represent "much greater than" and "much less than".

There is no often no explicit bound on how much greater or less than the comparison is, but it will usually be somewhat obvious given the context.

N.B: This wikipedia article may be useful in future when trying to determine what various mathematical symbols mean.

Thomas Russell
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It can also indicate absolute continuity of measures. This is seen in lots of books on measure and integral.

ncmathsadist
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