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Say I'm measuring two lengths, $L_1$ and $L_2$, with a measuring stick in cm. For concreteness, let's say that $L_1 = 10 \pm 1$cm and $L_2 = 20 \pm 1$cm. I now want to compute the ratio of those two lengths, $R = \frac{L_1}{L_2}$, and the absolute uncertainty of that ratio, $\Delta R$. From what I know, which I think must be wrong, the absolute uncertainty should be:

$$\Delta R = L_2\Delta L_1 + L_1 \Delta L_2 = 10*1 + 20*1 = 30$$ (with no units)

But that sounds terribly wrong. Furthermore, if I convert the measurements into meters, for example, then the uncertainty is much, much smaller than 30. I'm sure I'm misunderstanding something, so what am I missing?

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If $L_1 = 10 \pm 1$ then, $L_1$ ranges between 9 and 11. It isn't 100% clear from your question what you mean by this, but, nevertheless, by the interpretation you mean, this is the range for $L_1$.

Similarly, $L_2 = 20 \pm 1$ means that $L_2$ ranges between 19 and 21 (again, by whatever interpretation you meant).

So, by this interpretation,

$$ {9 \over 21} \leq {L_1 \over L_2} \leq {11 \over 19}. $$

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