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What does the sign '<<' mean, for example angle X << 1?

This was in a brilliant.org problem. I won't post the full problem since that would make me a cheat but the first line of the problem is "A source of ions at point O produces a slightly diverging beam with half-angle of divergence α≪1". My question is what does α≪1 mean?

Thanks.

Martin Sleziak
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abcdefgh
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4 Answers4

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It means (in this case) much smaller than. You can use the first order approximations like $\sin x \approx x$ (unless all the first order terms cancel, then you have to consider second order)

Ross Millikan
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  • Could you please explain it? X is much less than 1, so does that mean that X must be nearly equal to 0 (since in the problem I stated the angle must be positive), so I can use the approximations sinX0 and cosX1? And can you please explain how you got sinX~~X, because as far as I know sin(1)= 1 - 1/3! + 1+5! - 1/7! +... (till infinity)? – abcdefgh Apr 17 '13 at 16:28
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    @abcdefgh Yes, that's correct. Although there are other meanings of the symbol $\ll$, you wouldn't write, say, $-100 \ll 1$. In this context it means $X$ is positive, very close to $0$. Since $\sin X = X - X^3/3! + \cdots$, it is safe to say $\sin X \approx X$ for small $X$. – Erick Wong Apr 17 '13 at 16:34
  • @abcdefgh: The Taylor series for $\sin x=x-\frac {x^3}{3!}+\frac {x^5}{5!}-\ldots$. The zero order approximation (constant term) is then $\sin x \approx 0$, but that won't capture what you want to do. The first (and second) order is $\sin x\approx x$, which is often enough for these problems. For $\cos x$, the zero (and first) order is $\cos x \approx 1$, while the second order is $1-\frac {x^2}2$ – Ross Millikan Apr 17 '13 at 16:36
  • Ok got it. Thank you very much. – abcdefgh Apr 17 '13 at 16:38
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x ≪ y means x is much less than y

http://en.wikipedia.org/wiki/List_of_mathematical_symbols

PS: In programming language it could be a bit operator.

Noturab
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In a numerical context, this is sometimes used to mean "$X$ is much less than 1"

vonbrand
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Writing that $x \ll 1$ means (as others have said) that $x$ is much less than $1$.

That said, the symbol is vague and doesn't actually have a precise meaning. If the symbol had a presice meaning, then we could define a set $$ X = \{x \in \mathbb{R}\mid x \ll 1\}. $$ But this set is not well defined.

Thomas
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