# What does the notation $8.9875517923(14)$ mean? [duplicate]

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?

It's the uncertainty in the last two digits:

$$8.9875517923(14) = \color{blue}{8.987\,551\,79}\color{red}{23} \pm \color{blue}{0.000\,000\,00}\color{red}{14}.$$

• Interesting notation. What is it called? – l0b0 Jul 5 at 10:43
• Note that $a\pm b$ here really means a random value whose distribution has mean $a$ and standard deviation $b$ (or some multiple thereof depending on the confidence interval used), and is likely to be approximated as Gaussian. – J.G. Jul 5 at 11:53
• @l0b0 That notation doesn't have a specific name. You can find guidance on how to report uncertainty, and the definition of that notation, in this document, section 7, and in particular section 7.2.2. – Massimo Ortolano Jul 5 at 15:13
• @J.G. Unfortunately, I've been taught that means "all our observed values are within the plus and the minus"; I wouldn't be surprised if there were papers out there that used the notation (wrong) like that. – wizzwizz4 Jul 5 at 18:22

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.

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.

• Yep: ISO standard also states that one should state explicitly what the values in the parentheses are representing. – D Duck Jul 6 at 10:49

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/

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