How to divide by a number with error

Question

How do I divide a constant number by a number with error? For example...

$$\frac{1}{(101 \pm 0.0058)} = 0.0099 \space\pm\space ???$$

Please help!

Answer 1

Let's define $$f(x) = \frac{1}{x}$$ so we want to estimate the uncertainty in $f$, written $\\\delta f$, when $x = 101$ and the uncertainty in $x$ is $\delta x = 0.0058$.

The fractional uncertainty in $x$ is defined as $$\frac{\delta x}{|x|}$$ so in our case the fractional uncertainty in $x$ is $0.0058 / 101 \approx 5.74 \times 10^{-5}$.

The rule for calculating the fractional uncertainty of a quotient is that the fractional uncertainty is the sum of the fractional uncertainties of the quantities involved. (Reference: An Introduction to Error Analysis, Second Edition by John R. Taylor.) In this case we only have one quantity, $x$, so the fractional uncertainty in $f$ is $$\frac{\delta f}{|f|} =\frac{\delta x}{|x|} \approx 5.74 \times 10^{-5}$$

To get the uncertainty in $f$, use $$\delta f = \frac{\delta f}{|f|} \times |f|$$ so $$\delta f = 5.74 \times 10^{-5} \times \frac{1}{101} \approx 5.7 \times 10^{-7}$$

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Alternatively, we could use some basic facts from calculus. $$f(x+h) \approx f(x) + h f'(x)$$ so $\delta f \approx |h f'(x)|$ and $$f'(x) = - \frac{1}{x^2}$$ so $$\delta f \approx 0.0058 \times \frac{1}{101^2} \approx 5.7 \times 10^{-7}$$ which is the same answer we got using fractional uncertainties.

Answer 2

$$\frac{1}{101 \pm 0.0058} = 0.0099 \space\pm\space ???$$

For small (by absolute value) $x\ll1$ we have:

$$\frac1{1+x}\approx1-x$$

Thus we want to have the denominator in the form "1+small value":

$$\begin{align} \frac{1}{101 \pm 0.0058} &= \frac1{101}·\frac{1}{1 \pm 0.0058/101}\\ &\approx \frac1{101}·(1\mp 0.0058/101)\\ &\approx 0.0099 \mp 0.0058·0.0099^2 \end{align}$$