[Math] 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

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.