In English, graphical coordinates are given as (x, y). Since ',' represents the decimal separator in German, is some other punctuation used to prevent confusion?

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    While the question referred by @HubertSchölnast is related, this one is sufficiently different not to be considered as duplicate. – guidot Jan 5 at 9:49
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    I'd like to add that this issue does not apply to the German language in general. In Switzerland, both , and . are commonly used as decimal separator. Thus, in a context where , is used to separate multiple numbers, the simplest solution is to use . as the decimal separator. In my academic career, I've almost never seen anyone (outside of books) use something other than , to separate the elements of a vector when written horizontally. – Feuermurmel Jan 5 at 12:04
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    @Feuermurmel agreed. I personally consequently ignore the 3,14 convention of Germany and Norway, and normally people understand my 3.14 just fine. And although , is definitely the official version, I'm not the only one who prefers .. (Unfortunately it can also flip the other way: one of my Professors (he was from Turkey) used the . as both decimal separator and multiplication sign, and parsing something like from the blackboard is even more annoying than parsing (4,3, 5,9, 3, 67,4).) – leftaroundabout Jan 5 at 23:23
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    Note that , is the thousands separator in English, so the exact same problem exists there, too. – Jörg W Mittag Jan 6 at 9:00

I think you are referring to the normal Cartesian coordinate system. In this case, the coordinates are usually separated by "|" (senkrechter Strich - vertical bar/Pipe) or ";" (Semikolon - semicolon).

In your example, P (x, y) becomes

P (x | y)

P (x; y)

and Q (x, y, z) becomes

Q (x | y | z)

Q (x; y; z)

Note: It can also happen that a comma is used as a separator. (Usually if there is no risk of confusion, for example when listing variables.) In this case the problem you mentioned with confusion as a decimal point can occur.


I checked my old records and in fact all three variants are used. The (x | y) notation was usually used in school. The other two were used at the university. (x, y) for integer values. (x; y) for floating point numbers. But I think at least one professor used the (x | y) option, at least in the introductory lessons.

  • Comments are not for extended discussion; this conversation has been moved to chat. – jonathan.scholbach Jan 4 at 23:23
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    I have never seen (x | y) in over a decade of mathematical education in Switzerland. In my experience, tuples were consistently written as (a, b), or written as vectors (a b). I guess it is possible that (x | y) is common in Germany, but one should probably not generalize this to all German speaking countries. – meriton Jan 6 at 20:09
  • @meriton I think you should write this as an answer to this question. Ideally with a link to a swiss textbook or other source. – mtwde Jan 6 at 21:39

Here is a picture from my schoolbook from 2006 using the vertical line as a separator to avoid a clash with decimal separators:

enter image description here

Nein, er hat nicht recht. Der untere Graph gehört zu einer anderen Funktion. Er schneidet zwar ebenso wie der obere Graph die y-Achse beim Wert 2,5. Jedoch verläuft der untere Graph im weiteren [sic] durch den Punkt (2,5 | 5), wohingegen der obere Graph im weiteren [sic] durch den Punkt (2,5 | 7,5) verläuft. Daher gehören sie zu unterschiedlichen Funktionen.

Anmerkung: Zeichnet man einen größeren Ausschnitt des mittleren Graphen, so erkennt man, das auch dieser durch A (0 | 2,5) und B (2,5 | 7,5) verläuft. Er gehört demnach zur selben Funktion wie der obere Graph.

And just to prove that this is not outdated notation, here is a screenshot from a 2020 schoolbook I found online:

enter image description here

[...] Ermittle, ob der Punkt Z (11 | 0,1) auf dem Graphen Gg liegt.

In handwriting, we would also use regular slashes, as in (2,5 / 7,5).

  • @DavidFoerster: Done. – Mophotla Jan 6 at 22:40

Yes and no.

Taking a look at wikipedia, you will find the same sign: a comma. Thus these examples live without any real numbers, this is sufficient and does not help in your question.

In education this is especially common for explantations as long as I write with variables (x,y) or I'm in a system with integers only (good for beginners): (2,3)

Other given options are the vertical bar: (2|3). (uncommon to me, never seen as coordinates before)

In school I used for non-integers a semicolon (2,3;3,4) to avoid (2,3,3,4) - as this implicates a four-dimensional integer value. Adding spaces (2,3, 3,4) might help in digital writing and still leaves room for misinterpretation. Handwriting has the additional flaw that a space comes with different lengthes thus hard to detect. And I used a comma for any formulas with variables, because that was quicker.

  • Concerning adding spaces does not help at all: Spaces are used as a delimiter, i.e. (2,3 1,3 5,7) – infinitezero Jan 4 at 12:53
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    "Adding spaces (2,3 , 3,4) does not help at all" Err, what? Of course, the space should only be after the comma that separates the coordinates, but how does it not help? – Carsten S Jan 4 at 13:57
  • That said, I think your last paragraph is important. It would be interesting to look at school maths books because they will be more likely to deviate from common mathematics practice to avoid confusions and also likely need concrete decimal fractions more often. – Carsten S Jan 4 at 14:11
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    @CarstenS: Space or no space is hard to see in handwriting. Thus, at least in handwriting, spaces do not help. There are other ways to clarify the intended meaning - e.g. this article uses small and large commas (but I guess it's up to you whether you consider that a typographically better solution than just using another more visibly different symbol altogether). – O. R. Mapper Jan 4 at 14:36
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    @O.R.Mapper: Das ist doch (:hüstel:)! Wer aufmerksam genug ist, einen senkrechten Strich statt eines Kommas zu setzen, um die Werte lesbar zu gestalten, der sollte auch in der Lage sein einen Leerraum zu lassen, der groß genug ist, Missverständnisse zu vermeiden. – user unknown Jan 5 at 22:17

This is a very interesting question because for all abstract purposes the standard notation is to use a comma. As long as no concrete numbers are involved and one deals with variables, one definitely says that a point in the plane has (Cartesian) coordinates (x,y). Of course one could use any other separator instead of a comma, but that would be highly unusual and does almost never occur in the professional literature (mathematics, physics, engineering sciences, ...).

In practice this notation may lead into a mess - expressions like (2,3,4,187) are very unpleasant and may lead to misunderstanding: Does it mean x = 2,3 and y = 4,187 or does (2,3,4,187) have four integer components?

The notation (x|y) could be useful and is in fact often used in school mathematics, especially in geometry. But I have never seen it outside school, hence I would not recommend to use it.

Sometimes the coordinates are written in a vertical arrangement (i.e. in the form of "column vectors"). Here is an example with three coordinates:

enter image description here

You can see the difference between between the abstract mathematical level where (φ,θ) denote two coordinates on the earth's surface (given by the degrees of longitude and latitude) and the concrete level of decimal numbers.


There is no cure-all, in practical applications involving decimal numbers I suggest to use a semicolon as a separator and to explain explicitly its purpose.

By the way, a similar notational problem can also occur if a point is used as decimal separator because in that case often a comma is used for digit grouping. An example is (2,419.6, 41,964,301). Of course, the use of digit grouping can be avoided, but then clarity may suffer.


In a comment Hagen von Eitzen says that with x = 2,3 and y = 4,187, one should write (2,3, 4,187) with a space character after the comma-separator and not as I did above (2,3,4,187). This is of course better, but I think it may still be somewhat ambiguous for the reader. Let us present some options:

enter image description here

This is why I suggest to use a semicolon as a separator, and one should insert a space after it.

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    Actually, with x = 2,3 and y = 4,187, on should write (2,3, 4,187) and not (2,3,4,187) – Hagen von Eitzen Jan 4 at 20:40
  • KÖnnte es sein, dass die concrete numbers eher integer (numbers) sind? Concrete numbers klingt eher nach diskreten oder abzählbare Zahlen. en.wikipedia.org/wiki/Concrete_number – Shegit Brahm Jan 4 at 22:19
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    @HagenvonEitzen You are right, but I wouldn't really regard that as a particularly pleasant notation. – Paul Frost Jan 4 at 23:12
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    @ShegitBrahm Vielleicht ist der Begriff "concrete number" unglücklich gewählt, ich meinte es im Sinne konkreter Zahlenwerte. Das brauchen keine ganzen Zahlen zu sein, sondern können auch Dezimalzahlen mit Nachkommastellen sein. – Paul Frost Jan 4 at 23:17
  • @PaulFrost: mein Verständnis vom verlinkten Artikel reicht nur soweit, dass ich mangels Maßeinheiten der Meinung bin, dass der englische Ausdruck "concrete numbers" hier ein Falscher Freund ist. Ich kann weder genug Mathe noch englisch, um was besseres als "as long as no numbers are involved" vorzuschlagen. Von mir daher die Frage, ob hier eine - für englische Mathematiker - falsche Übersetzung bei rum kommt. – Shegit Brahm Jan 5 at 17:13

The standard notation for this in German, English, Russian and many other languages is to use a comma. See as one of many possible examples the image taken from a page in the textbook Algebraische Geometrie by Markus Brodmann.

the image taken from a page in the textbook Algebraische Geometrie by Markus Brodmann.

In order to adress Shegit Brahm´s comment I have added two more images of the same book.

page 5

Note lines 5 and 6 where real numbers are used.

charts for projective space

Look at the formula after Schließlich wird durch ....

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    I agree, that (as far as I know/remember) this is the standard notation. Your example is not a good example, though, since it contains variables and not floating point numbers who have a comma themselves. Hence the relevant scenario the OP is interested in, is not met here. Could you improve your post in order to fix this? – jonathan.scholbach Jan 4 at 19:43
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    @jonathan.scholbach For one, nobody would switch from f(x,y) when using variables to f(31,4|17) when using numbers. Second, observe that there is some space after each of the listing commas (while there would be no space after a decimal comma). Finally, actually using floating point numbers (in particular as one of several arguments of a function) in a math text is perhaps quite rare. You'd see $f(\sqrt 2,\pi)$ way more often than $f(1{,}41,3{14})$ – Hagen von Eitzen Jan 4 at 20:46
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    @jonathan.scholbach Well, you can just plug in numbers and then with good typesetting the meaning will be clear. – Martin Peters Jan 5 at 7:06
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    @ShegitBrahm The variables $z_i$ can be real numbers. One would ask that they are in a field, and real numbers form a field. – Martin Peters Jan 5 at 7:14
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    @MartinPeters: es ging mir nicht darum, ein weiteres Beispiel mit Variablen zu sehen. Sondern eins mit echten gebrochenen Zahlen. Bei Variablen reicht das pofelige Komma vollständig aus. Danke für den Versuch. – Shegit Brahm Jan 5 at 12:17

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