<p>A notation like <span class="math-container">$x=24^{+1}_{-3}$</span> is quite common, it means <span class="math-container">$24-3&lt;x&lt;24+1$</span> with a best estimate of 24, and is a way of indicating uncertainty.</p>
<p>The example you give is less common, by analogy <span class="math-container">$x=24^{\times 2}_{\div3}$</span> means
<span class="math-container">$ 24\div3 &lt; x &lt; 24\times 2$</span> ie <span class="math-container">$x$</span> is between 8 and 48, with a best estimate of 24</p>

CC BY-SA 4.0
2020-09-09T17:06:09.233Z
2020-09-09T19:59:06.247Z
2020-09-09T19:59:06.247Z
James K
<p>Small quibble to the (rightfully) accepted answer by James K that was too long for a comment:</p>
<hr />
<p>To be fair, <span class="math-container">$x=24^{+1}_{-3}$</span> doesn't mean that <span class="math-container">$21 \le x \le 25$</span>, but that <em>with a particular amount of certainty</em> (usually 68%), <span class="math-container">$21 \le x \le 25$</span>.</p>
<p>Correspondingly, <span class="math-container">$x=24^{\times 2}_{\div3}$</span> would mean that, <em>with some certainty</em>, <span class="math-container">$8 \le x \le 48$</span>.</p>
<h3>Symmetric vs. asymmetric uncertainties</h3>
<p>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 <span class="math-container">$n$</span>'th sigma is equal to <span class="math-container">$n\sigma$</span>. That is, if the quoted error represents one standard deviation, then for <span class="math-container">$x=24\pm2$</span> you know that with 99% certainty the result is <span class="math-container">$20 \le x \le 28$</span>.</p>
<p>By analogy, if the errors of this notation are normally distributed in log space, as I would think is the case, the <span class="math-container">$n$</span>'th sigma would be equal to <span class="math-container">$\sigma^n$</span>. That is, if <span class="math-container">$x=4000^{\times}_{\div}4$</span>, then
<span class="math-container">$$
\begin{array}{rcl}
          1000 \le      x\le \phantom{1}16\,000 &amp; (\mathrm{68\% \,\,confidence})\\
\phantom{1}250 \le      x\le \phantom{1}64\,000 &amp; (\mathrm{95\% \,\,confidence})\\
\phantom{10}60 \lesssim x\le           256\,000 &amp; (\mathrm{99\% \,\,confidence})\\
(\mathrm{etc.}) &amp;
\end{array}
$$</span></p>
<h3>Please use logarithms</h3>
<p>Personally, I think this notation is horrible. To avoid confusion, instead of <span class="math-container">$x=4000^{\times}_{\div}4$</span> I'd much rather write <span class="math-container">$\log x = 3.6\pm0.6$</span>. Then
<span class="math-container">$$
\begin{array}{rcl}
3.0 \le \log x \le 4.2 &amp; (\mathrm{68\% \,\,confidence})\\
2.4 \le \log x \le 4.8 &amp; (\mathrm{95\% \,\,confidence})\\
1.8 \le \log x \le 5.4 &amp; (\mathrm{99\% \,\,confidence})\\
(\mathrm{etc.}) &amp;
\end{array}
$$</span>
which is (roughly) the same as above.</p>

CC BY-SA 4.0
2020-09-10T09:05:18.610Z
Thanks, that is wonderfully explained. I am still quite new to this, and if I ever have to write such an equation, I will definitely follow the logarithmic procedure.
lordparthurnaax
CC BY-SA 4.0
2020-09-10T10:49:08.880Z
This is neither small nor a quibble.
uhoh
CC BY-SA 4.0
2020-09-10T08:35:03.080Z
2020-09-11T08:01:15.033Z
2020-09-11T08:01:15.033Z
pela
<p>I have come across a few papers using the notation <span class="math-container">$A^{\times{B}}_{\div{C}}$</span>; where A,B,C are real number. For example, <span class="math-container">$3000^{\times{3}}_{\div{4}}$</span>.</p>
<p>An example can be Eqn (4) in Stern &amp; Laor (2012) &quot;<a href="https://academic.oup.com/mnras/article/426/4/2703/1011509" rel="nofollow noreferrer">Type 1 AGN at low <span class="math-container">$z$</span> – II. The relative strength of narrow lines and the nature of intermediate type AGN</a>&quot;, which reads as follows:</p>
<blockquote>
<p><span class="math-container">$$\frac{L_{\rm bol}}{10^{43}\ \rm erg\ s^{-1}} = 4000^{\times 4}_{\div 4} \left( \frac{L_{\rm [O\ III]}}{10^{43}\ \rm erg\ s^{-1}}\right)^{1.39}$$</span></p>
</blockquote>
<p>What is the generic meaning of this notation?</p>

CC BY-SA 4.0
2020-09-09T16:32:26.267Z
An example can be Eqn (4) in https://academic.oup.com/mnras/article/426/4/2703/1011509.
lordparthurnaax
<radio-astronomy><luminosity><mathematics>
What is meant by the notation $A^{\times{B}}_{\div{C}}$; where A,B,C are real numbers?
lordparthurnaax
# What is meant by the notation $A^{\times{B}}_{\div{C}}$; where A,B,C are real numbers?

(Asked by: username_0 on 2020-09-09)

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](https://academic.oup.com/mnras/article/426/4/2703/1011509)", 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?

**Tags:** radio-astronomy, luminosity, mathematics

> **Comments:**
> **username_0**: An example can be Eqn (4) in https://academic.oup.com/mnras/article/426/4/2703/1011509.  

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## 2 answers

### Answer by username_2 on 2020-09-10. Score: 5 | Accepted answer

Small quibble to the (rightfully) accepted answer by username_1 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
$$
\begin{array}{rcl}
1000 \le x\le \phantom{1}16\,000 & (\mathrm{68\% \,\,confidence})\\
\phantom{1}250 \le x\le \phantom{1}64\,000 & (\mathrm{95\% \,\,confidence})\\
\phantom{10}60 \lesssim x\le 256\,000 & (\mathrm{99\% \,\,confidence})\\
(\mathrm{etc.}) &
\end{array}
$$

### 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.

> **Comments:**
> **username_0**: Thanks, that is wonderfully explained. I am still quite new to this, and if I ever have to write such an equation, I will definitely follow the logarithmic procedure.  
> **username_3**: This is neither small nor a quibble.  

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### Answer by username_1 on 2020-09-09. Score: 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


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