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

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?

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.

• +1, but I've never seen this notation and I think it's horrible!
– pela
Sep 9 '20 at 19:11
• @pela - I'm not entirely sold on it myself... I think I prefer notation like $10^{x \pm \sigma}$ instead!
– user24157
Sep 9 '20 at 19:27
• @pela I have never encountered these notations in any books or classes, but these papers treat them as trivial. I wish they would add a concise note making it clear, or use a more intuitive one, like the one antispinwards suggested. Sep 10 '20 at 4:40
• A notation like shown with + and - like that giving errors is common when it is not a symmetric error expressed as +- Sep 10 '20 at 5:18

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

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 $$\begin{array}{rcl} 1000 \le x\le \phantom{1}16\,000 & (\mathrm{68\% \,\,confidence})\\ \phantom{1}250 \le x\le \phantom{1}64\,000 & (\mathrm{95\% \,\,confidence})\\ \phantom{10}60 \lesssim x\le 256\,000 & (\mathrm{99\% \,\,confidence})\\ (\mathrm{etc.}) & \end{array}$$

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.
A notation like $$x=24^{+1}_{-3}$$ is quite common, it means $$24-3 with a best estimate of 24, and is a way of indicating uncertainty.
The example you give is less common, by analogy $$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