What does the notation $8.9875517923(14)$ mean? The number $8.9875517923(14)$ appears in Coulomb's constant.  I have read that it has something to do with the uncertainty of the accuracy of the number but answers have been unclear.
Can somebody define the meaning of such notation?
 A: Since a key part of the answer is only stated in a comment (see comment from J.G. above), I will restate this here again.
If we write something like $\hat m = 1.234\, 5(67)$ we usually mean that
\begin{align}
\textrm{the }\bf{average \;value}\textrm{ of our measurement is }
\bar m &= 1.23\color{red}{4\, 5}\textrm{, and that}\\
\textrm{the }\bf{standard \;deviation}\textrm{ of the measured values is }
\hat\sigma_m &= 0.00\color{red}{6\,7}. 
\end{align}
If we assume that the error of our measurement is a normally distributed random variable, this implies that we are
approx. 68% confident that the "true value" (=mean value of the population) lies within the interval $\bar m \pm \hat\sigma_m$, and
approx. 95% confident that the "true value" lies within the interval $\bar m \pm 2\hat\sigma_m$ -- the confidence statements are only valid if the sample size is "large enough" so that Student's $t_\nu$-distribution is approx. equal to the normal distribution.
According to ISO/IEC GUIDE 98-3:2008 the parentheses format is recommended, while using $\hat m = 1.234\, 5 \pm 0.006\,7$ should be avoided for historical reasons. However, the ISO standard also states that one should state explicitly what the values in the parentheses are representing. It also helps if you state explicitly which "type/component of uncertainty" you are referring to, e.g.
accuracy, repeatability, reproducibility etc.
A: Go to the BIPM website which describes the notation and the language used. The key document is Guide to the Expression of Uncertainty in Measurement. https://www.bipm.org/en/publications/guides/
https://www.iso.org/sites/JCGM/GUM/JCGM100/C045315e-html/C045315e.html?csnumber=50461

Could also have a look at the NIST site, which clearly explains the notation as well.
https://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html
A: It's the uncertainty in the last two digits:
$$8.9875517923(14) = \color{blue}{8.987\,551\,79}\color{red}{23} \pm \color{blue}{0.000\,000\,00}\color{red}{14}. $$
A: The number $8.9875517923(14)$ can also be expressed as $8.987\,551\,7923\pm0.000\,000\,0014$. The parentheses give you the precision on the last digits of the number. As you can see, this is a much more compact way of writing that expression than the usual $a\pm b$ format.
