How to divide by a number with error 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!
 A: 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 $$\delta f = 5.74 \times 10^{-5} \times \frac{1}{101} \approx 5.7 \times 10^{-7}$$

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

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