<p>Recently, some in the astronomy literature are adopting $\sim$ in the statistics sense to mean "is sampled from" or "has the distribution of" (<a href="http://mathworld.wolfram.com/Tilde.html" rel="nofollow noreferrer">the last definition on mathworld</a>). For example, </p>

<p>$$
x \sim p(x)
$$</p>

<p>would read "$x$ is sampled from the distribution $p(x)$".</p>

CC BY-SA 3.0
2017-03-24T07:03:02.650Z
Please give some examples of where it is used like this in the astronomical, peer-reviewed literature.
ProfRob
CC BY-SA 3.0
2017-03-25T02:58:08.723Z
@RobJeffries Off the top of my head, see Table 1 in [Brewer et al. 2015](https://arxiv.org/abs/1501.06952), or Table 1 in [Foreman-Mackey et al. 2016](https://arxiv.org/abs/1607.08237).
ampw
CC BY-SA 3.0
2017-03-23T23:44:01.853Z
2017-03-23T23:44:01.853Z
ampw
<p>As an (ex) professional astronomer, I can tell you that there are no rules that are followed.  There are very very few exact measurements in Astronomy, <em>x</em> rarely technically <em>equals</em>  anything.  In general, $\sim$ means less accurate than $\approx$ which means less accurate than $=$.</p>

<p>Mathematicians (and Chemists) frequently complain about our loose use of mathematics (and chemistry).</p>

CC BY-SA 3.0
2017-03-24T17:14:23.697Z
2017-03-24T17:14:23.697Z
Jonathan Twite
<p>Wikipedia (where would we be without it) has a <a href="https://en.wikipedia.org/wiki/List_of_mathematical_symbols" rel="nofollow noreferrer">handy list of mathematical symbols</a> which is pretty useful in this context.</p>

<p>Note that in particular :</p>

<ul>
<li>$\sim$ will crop up often as meaning <em><a href="https://en.wikipedia.org/wiki/Asymptotic_analysis" rel="nofollow noreferrer">asymptotically equivalent</a></em> which has a very specific meaning and should <strong>not</strong> be used for as an approximate symbol - they're quite different and some mathematicians will lynch you if you call it approximately.</li>
</ul>

<p>Last time I checked <a href="https://www.youtube.com/watch?v=LYNOGk3ZjFM" rel="nofollow noreferrer">Carl Bender's lectures on mathematical physics</a> were available on YouTube if it's of interest to anyone and these include asymptotic analysis.</p>

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2017-03-22T15:23:25.510Z
This is definitely _not_ how $\sim$ is used in (observational) astronomy.
pela
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2017-03-22T19:55:26.347Z
@pela Yes, I read your answer, and I specifically mentioned this mathematical interpretation because it could lead to ambiguity or downright misunderstanding.  I'd recommend people stick with the more precisely defined symbols (like $\propto$ for proportional to).
StephenG - Help Ukraine
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2017-03-22T21:23:58.447Z
Okay, fair enough. But good luck persuading the astronomical community to do that ;-)
pela
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2017-03-22T10:49:50.007Z
2017-03-22T10:49:50.007Z
StephenG - Help Ukraine
<p>$\simeq$ and $\approx$ both mean "approximately equal to". I don't think $\cong$ is used so often, but if I read it, I would interpret is as the same as the two others.</p>

<p>$\sim$ <em>in principle</em> means "of the order of", i.e. correct to within an order of magnitude. However, astronomers tend to be rather slobby sometimes (we call all elements except H and He for "metals", we call it "gas" even when it's ionized and should be called "plasma", we omit factors of order unity if we can get away with it, we mix up terms like "flux", "flux density", and "intensity", and so on), and thus $\sim$ is very often used to mean "approximately equal to". It is even sometimes used as "proportional to", but (usually) only when there's no risk of misunderstanding (e.g. $L_\mathrm{UV}\!~\sim\!~\mathrm{SFR}$, meaning that the UV luminosity is proportional to the star formation rate, omitting a factor of $10^{27}$).</p>

<p>Journals usually have rather strict standards for style and nomenclature, so they might not accept a $\sim$ for a $\propto$, or even for a $\simeq$. But if you read a conference proceeding or lecture notes or some other non-refereed text (even a paper on arXiv waiting to be refereed), you best be prepared for anything.</p>

CC BY-SA 3.0
2017-03-22T05:57:29.057Z
2017-03-22T05:57:29.057Z
pela
<p>My understanding is $\sim$ generally means "on the order of magnitude of" e.g. $T \sim 10^5$ K</p>

<p>$\approx$ is obviously "approximately equal to" so for example one might write $d \approx 400$ pc rather than $d=4 \times 10^2$ pc</p>

<p>$\simeq$ and $\cong$, are where I am more confused as their usage is rarer and less consistent.</p>

<p>The below IAU resolution, for example, seems to use $\simeq$ for truncation $m_{\mbox{bol}} = -26.83199\ldots \simeq -26.832$, but I have also seen $\simeq$ used as "approximately equal to" in other works.</p>

<p><a href="http://arxiv.org/pdf/1510.06262" rel="noreferrer">http://arxiv.org/pdf/1510.06262</a></p>

<p>How should one be using $\sim$, $\approx$, $\simeq$, and $\cong$?</p>

CC BY-SA 3.0
2017-03-22T04:35:05.597Z
I don't think that there's any set standard for using these approximation signs.
Phiteros
CC BY-SA 3.0
2017-03-22T13:19:31.637Z
Those are just the pen strokes of hand waving astronomers. :)
LocalFluff
CC BY-SA 3.0
2017-03-22T15:01:25.227Z
To point out, you'd likely receive a very different answer if you posted this on the math stack exchange. I think the distinction between these various symbols has more significance in mathematics. Astronomers on the other hand tend to be less strict and will often just use all of these interchangeably, or you'll see different standards being used by different people. There's less uniformity in their meaning in the sciences than there would be in the mathematics groups.
zephyr
<terminology><notation><symbols>
Usage of $\sim$, $\approx$, $\simeq$, and $\cong$ in observational astronomy?
imanorc
# Usage of $\sim$, $\approx$, $\simeq$, and $\cong$ in observational astronomy?

(Asked by: username_0 on 2017-03-22)

My understanding is $\sim$ generally means "on the order of magnitude of" e.g. $T \sim 10^5$ K

$\approx$ is obviously "approximately equal to" so for example one might write $d \approx 400$ pc rather than $d=4 \times 10^2$ pc

$\simeq$ and $\cong$, are where I am more confused as their usage is rarer and less consistent.

The below IAU resolution, for example, seems to use $\simeq$ for truncation $m\_{\mbox{bol}} = -26.83199\ldots \simeq -26.832$, but I have also seen $\simeq$ used as "approximately equal to" in other works.

<http://arxiv.org/pdf/1510.06262>

How should one be using $\sim$, $\approx$, $\simeq$, and $\cong$?

**Tags:** terminology, notation, symbols
> **Comments:**
> **username_1**: I don't think that there's any set standard for using these approximation signs.  
> **username_2**: Those are just the pen strokes of hand waving astronomers. :)  
> **username_3**: To point out, you'd likely receive a very different answer if you posted this on the math stack exchange. I think the distinction between these various symbols has more significance in mathematics. Astronomers on the other hand tend to be less strict and will often just use all of these interchangeably, or you'll see different standards being used by different people. There's less uniformity in their meaning in the sciences than there would be in the mathematics groups.  

---

## 4 answers

### Answer by username_8 on 2017-03-22. Score: 13 | Accepted answer

$\simeq$ and $\approx$ both mean "approximately equal to". I don't think $\cong$ is used so often, but if I read it, I would interpret is as the same as the two others.

$\sim$ *in principle* means "of the order of", i.e. correct to within an order of magnitude. However, astronomers tend to be rather slobby sometimes (we call all elements except H and He for "metals", we call it "gas" even when it's ionized and should be called "plasma", we omit factors of order unity if we can get away with it, we mix up terms like "flux", "flux density", and "intensity", and so on), and thus $\sim$ is very often used to mean "approximately equal to". It is even sometimes used as "proportional to", but (usually) only when there's no risk of misunderstanding (e.g. $L\_\mathrm{UV}\!~\sim\!~\mathrm{SFR}$, meaning that the UV luminosity is proportional to the star formation rate, omitting a factor of $10^{27}$).

Journals usually have rather strict standards for style and nomenclature, so they might not accept a $\sim$ for a $\propto$, or even for a $\simeq$. But if you read a conference proceeding or lecture notes or some other non-refereed text (even a paper on arXiv waiting to be refereed), you best be prepared for anything.


---

### Answer by username_4 on 2017-03-23. Score: 3

Recently, some in the astronomy literature are adopting $\sim$ in the statistics sense to mean "is sampled from" or "has the distribution of" ([the last definition on mathworld](http://mathworld.wolfram.com/Tilde.html)). For example,

$$
x \sim p(x)
$$

would read "$x$ is sampled from the distribution $p(x)$".

> **Comments:**
> **username_5**: Please give some examples of where it is used like this in the astronomical, peer-reviewed literature.  
> **username_4**: @RobJeffries Off the top of my head, see Table 1 in [Brewer et al. 2015](https://arxiv.org/abs/1501.06952), or Table 1 in [Foreman-Mackey et al. 2016](https://arxiv.org/abs/1607.08237).  

---

### Answer by username_6 on 2017-03-24. Score: 1

As an (ex) professional astronomer, I can tell you that there are no rules that are followed. There are very very few exact measurements in Astronomy, *x* rarely technically *equals* anything. In general, $\sim$ means less accurate than $\approx$ which means less accurate than $=$.

Mathematicians (and Chemists) frequently complain about our loose use of mathematics (and chemistry).


---

### Answer by username_7 on 2017-03-22. Score: 1

Wikipedia (where would we be without it) has a [handy list of mathematical symbols](https://en.wikipedia.org/wiki/List_of_mathematical_symbols) which is pretty useful in this context.

Note that in particular :

* $\sim$ will crop up often as meaning *[asymptotically equivalent](https://en.wikipedia.org/wiki/Asymptotic_analysis)* which has a very specific meaning and should **not** be used for as an approximate symbol - they're quite different and some mathematicians will lynch you if you call it approximately.

Last time I checked [Carl Bender's lectures on mathematical physics](https://www.youtube.com/watch?v=LYNOGk3ZjFM) were available on YouTube if it's of interest to anyone and these include asymptotic analysis.

> **Comments:**
> **username_8**: This is definitely _not_ how $\sim$ is used in (observational) astronomy.  
> **username_7**: @pela Yes, I read your answer, and I specifically mentioned this mathematical interpretation because it could lead to ambiguity or downright misunderstanding.  I'd recommend people stick with the more precisely defined symbols (like $\propto$ for proportional to).  
> **username_8**: Okay, fair enough. But good luck persuading the astronomical community to do that ;-)  

---

