What is meant by the notation $A^{\times{B}}_{\div{C}}$; where A,B,C are real numbers?

Question

I have come across a few papers using the notation $A^{\times{B}}_{\div{C}}$; where A,B,C are real number. For example, $3000^{\times{3}}_{\div{4}}$.

An example can be Eqn (4) in Stern & Laor (2012) "Type 1 AGN at low $z$ – II. The relative strength of narrow lines and the nature of intermediate type AGN", which reads as follows:

$$\frac{L_{\rm bol}}{10^{43}\ \rm erg\ s^{-1}} = 4000^{\times 4}_{\div 4} \left( \frac{L_{\rm [O\ III]}}{10^{43}\ \rm erg\ s^{-1}}\right)^{1.39}$$

What is the generic meaning of this notation?

Answer 1

A confirmation of the explanation is given slightly further on where it talks about how the uncertainty is derived:

In equation (4) we use the 0.6 dex scatter of $\log L_{\rm UV}$ around the $L_{\rm UV}$ versus $L_{\rm [O\ III]}$ relation, as an estimate for the uncertainty in deriving $L_{\rm bol}$ from $L_{\rm [O\ III]}$.

The term "dex" indicates a base-10 logarithm of a unit, which translates into a factor of $10^{0.6} \approx 4$. So translating the error expressed in the logarithm of the quantity into an error expressed on the quantity itself, this becomes "multiply-or-divide by 4", which here is rendered $4000^{\times 4}_{\div 4}$.

Pela's excellent answer explicitly goes into detail about what this implies for the probability distribution. I usually prefer notation like $10^{x \pm \sigma}$ which sticks to more familiar notation and explicitly implies something vaguely log-normal rather than normal, although perhaps it gets a bit cramped when asymmetric error bars come into play.

Answer 2

Small quibble to the (rightfully) accepted answer by James K that was too long for a comment:

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To be fair, $x=24^{+1}_{-3}$ doesn't mean that $21 \le x \le 25$, but that with a particular amount of certainty (usually 68%), $21 \le x \le 25$.

Correspondingly, $x=24^{\times 2}_{\div3}$ would mean that, with some certainty, $8 \le x \le 48$.

Symmetric vs. asymmetric uncertainties

With non-Gaussian, asymmetric errors, given only the two values for the lower and upper error there's no way to know the corresponding 95% interval, 99% interval, and so on. You would have to know the full PDF for that. But if the errors are Gaussian, the $n$'th sigma is equal to $n\sigma$. That is, if the quoted error represents one standard deviation, then for $x=24\pm2$ you know that with 99% certainty the result is $20 \le x \le 28$.

By analogy, if the errors of this notation are normally distributed in log space, as I would think is the case, the $n$'th sigma would be equal to $\sigma^n$. That is, if $x=4000^{\times}_{\div}4$, then $$ \begin{array}{rcl} 1000 \le x\le \phantom{1}16\,000 & (\mathrm{68\% \,\,confidence})\\ \phantom{1}250 \le x\le \phantom{1}64\,000 & (\mathrm{95\% \,\,confidence})\\ \phantom{10}60 \lesssim x\le 256\,000 & (\mathrm{99\% \,\,confidence})\\ (\mathrm{etc.}) & \end{array} $$

Please use logarithms

Personally, I think this notation is horrible. To avoid confusion, instead of $x=4000^{\times}_{\div}4$ I'd much rather write $\log x = 3.6\pm0.6$. Then $$ \begin{array}{rcl} 3.0 \le \log x \le 4.2 & (\mathrm{68\% \,\,confidence})\\ 2.4 \le \log x \le 4.8 & (\mathrm{95\% \,\,confidence})\\ 1.8 \le \log x \le 5.4 & (\mathrm{99\% \,\,confidence})\\ (\mathrm{etc.}) & \end{array} $$ which is (roughly) the same as above.

Answer 3

A notation like $x=24^{+1}_{-3}$ is quite common, it means $24-3<x<24+1$ with a best estimate of 24, and is a way of indicating uncertainty.

The example you give is less common, by analogy $x=24^{\times 2}_{\div3}$ means $ 24\div3 < x < 24\times 2$ ie $x$ is between 8 and 48, with a best estimate of 24