How do French mathematicians write ordered pairs, since they already use the comma as a decimal separator?

Question

This is blatantly a terminology question, but it's bugging me.

If a comma is used in French to separate the one's and tenth's place, e.g. $5/4=1,\!25$, how do you write the ordered pair $(5/4,6)$?

Surely $(1,25,6)$ is ambiguous since it could be interpreted as $(5/4,6)$ or $(1,256/10)$? What do they do in school there?

Answer 1

There is no ambiguity. There are no spaces before or after a decimal comma. $$(1{,}25, 6) = (1.25, 6)$$ $$(1, 25{,}6) = (1, 25.6)$$ It is the same for handwritten text. Use correct typography. Only with a longer list of many decimal numbers with thin spaces inside numbers, should semicolons be considered for easier readability.

By the way, various English-speaking countries use a comma for separating thousands, so the "problem" would still be present. $$(10,100, 10) \text{ is a triple}$$ $$(10{,}100, 10) = (10\, 100, 10) \text{ is a pair}$$ $$(10, 10{,}010) = (10, 10\,010) \text{ is a pair}$$

Answer 2

If there might be an ambiguity in notation with a list of decimal values, we use a semicolon, e.g. $\{2{,}3\ ; 5\}$, but this separator is often replaced by a simple comma when there is no ambiguity, e.g. with an interval $[x, 1]$.

Answer 3

We (in Germany) use the comma in banking/accounting/billing and as the official decimal separator in numbers, but the mathematicians and physicists use the dot as soon as they have left high school and are writing in English, so there is no ambiguity. The STVO (Strassenverkehrsordnung) uses commas on signs that indicate maximum height limits for vehicles to pass under bridges or maximum weight (in tons) for lorries.

The Austrian book https://www.oebv.at/flippingbook/9783209068224/77/ uses commas. The German article https://numerik.mi.fu-berlin.de/matheon-G8/comaBuch.pdf uses commas in the text but dots in Tables 5-7. The German article https://www.fim.uni-passau.de/fileadmin/dokumente/fakultaeten/fim/lehrstuhl/sauer/geyer/NumDidaktik.pdf uses dots. The web page https://inf-schule.de/information/darstellunginformation/binaerdarstellungzahlen/exkurs_andere_zahlen/konzept_kommazahlen uses a dot in a graph and otherwise commas. The middle school article https://ddi.uni-wuppertal.de/archiv/madin/material/materialsammlung/mittelstufe/binaer/ab_03_gleitkommazahl.pdf uses commas.

One may note that the decimals after the dot/comma are in German commonly described as "die Stellen hinter dem Komma" and less often "die Stellen hinter dem Dezimalpunkt".

In computations one must follow the coding standards, and the comma is almost always used for separation in lists etc., so the dot is the decimals separator in floating point numbers. [Albeit I had a case where Java stringified with commas because the Locale was set to German..]

Answer 4

We (Finland) also use comma as a decimal separator in common contexts (outside academia, for example in price tags and such). It doesn't really cause confusion for most people.

• My students are free to use the period as a decimal separator. They get used to the alternative at an early enough age given that most pocket calculators use the period.
• Spacing is different. A pro-tip for TeX use: if you write $2{,}3$, it is typeset as $2{,}3$ and you can see the difference in spacing. Recycling your example: $(1{,}25,6,0)$ vs. $(1{,}25, 6{,}0)$. Not very clear (particularly when handwritten). Another possibility is to add thinspaces: $(1{,}25,\,6,\,0)$ vs. $(1{,}25,\, 6{,}0)$.
• I do concede that the difference is a problem, when using some pieces of software like Microsoft Excel. You always need to remember that if you running a Finnish language version (don't remember exactly what the setting is called), then, in its infinite wisdom, MS uses a comma exclusively as a decimal separator, and a semicolon as a list separator. When you search for syntax help within Excel, you need to remember not to just copy/paste snippets because it will be misinterpreted.
• I have occasionally, when needing to print a sequence of decimal numbers inside parens, also used semicolons as separators. Basically to make the meaning clear to me students. As others explained, this is rarely needed in math courses. Engineers and/or physicists may meet this problem more often than math teachers. Anyway $(1{,}25;\, 6;\, 0)$ and $(1{,}25;\, 6{,}0)$ are quite unambiguous.

Answer 5

In Brazil, we also use commas to separate the units and tenths places. I don’t remember a single instance where this ambiguity actually caused confusion.

Outside academia, I suppose people rarely need to write lots of pairs of real numbers in decimal notation.

In academia, during my Bachelor’s in Mathematics, except perhaps in physics, experimental physics, numerical calculus and statistics, we were expected to carry out all calculations with full precision. The exercises were designed so that you could solve them by hand, without using calculators or computers. As a result, we almost always used fractions in analytic geometry, linear algebra, multivariable calculus, and so on. The same practice continued throughout my Master’s and Ph.D. in Mathematics. For better or worse, this is often still true today, even for students from other majors taking Calculus, Linear Algebra, and similar courses.

I believe this is, to some extent, a French influence on our undergraduate and graduate education, but I could be wrong.

Of course, if one adopts the convention of writing at least one decimal place to the right of the units digit, the ambiguity disappears completely, even for triples and $n$-uples:

$$ (1,25,6,0), $$

Integer rarely appeared in experiments/numerical calculus anyway. It is likely we used tables of $x$ and $y$ coordinates separated by vertical lines, instead of pairs of points.

We could just put extra space to separate the numbers from the parentheses and the comma in the middle (and perhaps using slightly enlarged commas)

$$ (\ 1,25 \ \text{,} \ 6 \ ), $$

$$ (\ 1,25 \ \text{,} \ 6,0 \ ), $$

or perhaps we use dots instead. Everybody would understand with no issue.

Answer 6

Here in Austria, we also use the comma: 5/4 = 1,25.

For tuples we use spacing: (1,25, 6,4, 5,3)

Or semicolon: (1,25; 6,4; 5,3)

In schools, we use | (or /): (1,25 | 6,4 | 5,3)

and sometimes sloppy (1,25 / 6,4 / 5,3)

but (1,25 | 6,4 | 5,3) would be the "correct" standard at school.

Answer 7

In true mathematics, I seldom used ordered pairs with raw numeric decimal numbers inside. If I correctly remember, fractions or rational numbers were much more common. But in the rare use cases where it could be required, I just used a different separator, generally a semicolon (;) to avoid any ambiguity.

But for your question, the correct presentation for a French mathematician would certainly be:

$$\left ( \frac 5 4, 6 \right )$$

Because even it $\frac 5 4 = 1.25$ is exact, what about $\frac 4 3$? And what could be the reason for a special processing when the denominator of a rational has only $2$ and $5$ as divisors?

Answer 8

I'm Belgian and have been teaching in an engineering faculty for years. Even though the comma is officially used in Belgium as the decimal separator, I consistently used the point instead. All engineering software packages in our faculty used the decimal point as well, and in LaTeX it was also the default choice. Often, we worked with comma-separated values, which of course is "easier" when using the decimal point too.

P.S. If you still want to work in LaTeX in mathematical mode with a floating comma and do not want spaces after each comma, you can load the package icomma: \usepackage{icomma}