![]() Additionally, many effects that we observe are non-linear, that is, not too friendly to effectively compute. Physically, then, there is a benchmark, and that is "Does the model work?" We approximate because the real world is a mess of particle interactions. We can control our required levels of tolerance in modelling reality, though. I can see that in many contexts, but I don't find that the notion of control is universally applied, in that we cannot choose the numbers that we measure from reality. I love avid19's answer on this question, as the part on control is very intriguing. You'll rarely if ever see the $\ll$ sign or its counterpart unless the numbers are at least an order of magnitude apart. Are the observed effects so different in order that your simplified model is sufficiently accurate? In this case, sufficient depends on how closely you need to look and how feasible it is to even look deeper. Let me chip in from a physicists vantage. as small as you'd like, provided that you make $x$ small enough. There is some way of making the error in the approximation, argument, etc. But it is precise in the sense that it implies that there is some control. $\ll$ is imprecise in the sense that you don't know how "small" something is. You can make $(1 x)^n$ as close as you'd like to $1 nx$ by making $x$ small enough! I think this is a more rigorous meaning. Can you control the error in a manageable way.įor the example $0 ![]() In most contexts, $\ll$ is used in approximation. What precisely makes something "much less than"? What is "much less than"? In some contexts $1\ll 2$ and in others $ 1\not \ll 2$ but $1 \ll 100000$ and in others, even $1\not\ll 100000$. The entire point is that it's NOT very clear what "much less than" is.
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