How to write graph coordinates in German?

Question

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

Answer 1

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.

EDIT:

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.

Answer 2

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

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And just to prove that this is not outdated notation, here is a screenshot from a 2020 schoolbook I found online:

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

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

Answer 3

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.

Answer 4

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:

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.

Conclusion:

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.

Update:

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:

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

Answer 5

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.

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

Note lines 5 and 6 where real numbers are used.

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