I've been wondering this ever since I started using internet for maths actively -- reading math books, this website and sites similar to this one. Sometimes I run into some notations that I'm not sure what they mean, or why are they used in a "weird" way. So I guessed that different notations are used in different parts of the world, and since I'm reading mostly american websites and blogs/books written in English, all of the notations are standard notations in America. So why does, for example, Serbia, have different notations for some things?

For example, in Serbia we use $\text{tg}$ and $\text{ctg}$ instead of $\tan$ and $\cot$. We mark derivatives as $f'(x)$, $f''(x)$, $f'''(x)$ and $f^{(n)}(x)$ for larger numbers instead of $\frac{\text{d}}{\text{dx}}f(x)$ (or however it actually works, I've never got the hang of the notation). In a question I posted not so long ago, I realized that $a=\overline{a_na_{n-1}a_{n-2}\dots a_0}$ doesn't represent a number made of stacking digits $a_i$ one to another for other mathematicians around the globe. In Serbia, we use overline to make sure the reader doesn't interpret it as $a=\prod_{i=0}^{n}a_i$, but rather as $a=\sum_{i=0}^{n}{10^{i}\cdot a_i}$. We use it for other purposes as well, such as repeating digits, conjugated complex numbers, of course (which I believe are standard everywhere).

I'm interested in why different notations are used around the world. Why are they not standardized? You could say that the language of mathematics is a language used by everyone around the world, but these are a few counterexamples I get on top of my head.

Please note that I'm not talking about usual marks for certain things such as $A$ for area. It's obviously used in English only because $A$ is the first letter of *a*${}$rea (we use $P$ here for he same reason), although I believe $S$ is internationally accepted.

So where do certain notations come from, and why do (you think) they differ in different countries?

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    $\begingroup$ Why are different languages, idioms, or conventions for anything used around the word? Why aren't they standardized? $\endgroup$ – Zev Chonoles Feb 9 '13 at 21:36
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    $\begingroup$ I always thought that $A$ for area was from the Latin word "area" and only $S$ for "surface" was from English influence ... $\endgroup$ – Hagen von Eitzen Feb 9 '13 at 21:40
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    $\begingroup$ I doubt that Serbian notation for derivatives is as standardized as you indicate. The $f'$, $f''$, $\dots$ notation is also widely used elsewhere, but often the Leibniz notation is very useful. $\endgroup$ – André Nicolas Feb 9 '13 at 21:42
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    $\begingroup$ The notation for derivatives $f'$ versus $\dot f$ versus $\frac d{dx} f$ are due to the different "inventors" (Leibniz/Newton), though today e.g. in physics they are used with different meaning: we use $f'$ for derivative with rtespect to place and $\dot f$ with respect to time. I have encountered $f^{(n)}(x)$ for $n$-fold iteration $\underbrace{f(f(\cdots(f(f}_n(x)))\cdots)$, so someone using that notation would better use $\frac {d^n}{dx^n}$ for higher derivatives ... $\endgroup$ – Hagen von Eitzen Feb 9 '13 at 21:47
  • $\begingroup$ Just a remark, I actually prefer $\operatorname{base}_{10}\left(a_1,a_2,a_3,...,a_n\right)$ rather than writing like a product or writing like the conjugate of a product. Just another remark: The notation \frac{\mbox{d}^n}{\mbox{d}x^n} is Leibniz notation and is much better that Newton notation($\dot f$), in my opnion and Lagrange notation ($f^{(n)}$) because (1) more intuition (2) variable is more clear. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 2 '13 at 10:54

You see, it depends mostly on culture and translation, and also the habits. When you get used to using a notation you cannot give up on it easily. For instance, the notations you gave for derivates are both acceptable. If I am right, one of them is the notation used by Leibniz, and one of them by Newton. They can both be used and whichever comes easy, you use it. However, the upperline notation is a different case. For instance, in my country we use that as you do too, and I guess it is used in every country but it is not so popular you see. The notation itself is usually used (also in my country) in mathematical olympiads. Also if you want to learn more, I suggest you reading this article http://www.stephenwolfram.com/publications/recent/mathml/mathml2.html.

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    $\begingroup$ I think the "unpopularity" of the overline notation comes from the fact that talking about the digit structure of numbers is somewhat frowned upon because it doesn't really touch properties of the numbers (only properties of their representation). $\endgroup$ – Hagen von Eitzen Feb 9 '13 at 21:50
  • $\begingroup$ @HagenvonEitzen I think so, and it seems unimportant to me too. $\endgroup$ – ciceksiz kakarot Feb 9 '13 at 21:54

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