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Let's say I have a measurement $x$ with an uncertainty $\Delta x$. I also have a constant $C$ which has no uncertainty.

I want to find the uncertainty of $y$, which is defined as $C/x$. How do I find the uncertainty $\Delta y$? I know that IF I instead defined $y = C*x$ then $\Delta y = C*\Delta x$, but I'm not sure how it would work for division?

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For small $\Delta x$

$$y+\Delta y=\frac{C}{x+\Delta x} = \frac{C}{x(1+\Delta x/x)} = y(1+\Delta x/x)^{-1}=y(1-\Delta x/x) $$

So $\Delta y = -y\frac{\Delta x}{x}$

If $\Delta x$ is larger, for a specific $y$, a straightforward way is to work out $y$ in two cases, using $x + \Delta x$ and $x - \Delta x$ and see what $\Delta y$ results.

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    $\begingroup$ This is a very good answer. I think you should extend it to the case of multiplication even though it was not requested because in end the uncertainty propagation for multiplication and division ends up looking the same. After all, division is multiplication by the inverse. $\endgroup$
    – garyp
    Commented Nov 7, 2021 at 13:59
  • $\begingroup$ @garyp Aside from the table in the link I provided, there is also the exact result for products: Leo A. Goodman, On the Exact Variance of Products, J. Am. Statis. Assoc., 55(292), 1960, pp. 708-713. There is no exact result for quotients. $\endgroup$
    – Ed V
    Commented Nov 7, 2021 at 14:41

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