What does this symbol "$\gg$" mean

Question

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.

Answer 1

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.

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

Answer 2

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.

Answer 3

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.)

Answer 4

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.

Answer 5

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