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I have heard that in Canada and the United States of America, when we write a decimal number, then we use a period, like 1.111. When we write large numbers, then we use a comma, like 1,111. Sometimes we use spaces for large numbers, like 1 111. I have also heard that Europe, Southern Africa, Central America, the West Indies, and South America use commas and periods the other way around. So, for Americans, 1.111 is the decimal number, while for Europeans, 1.111 is one thousand one hundred eleven. For Americans, 1,111 is one thousand one hundred eleven, while for Europeans, 1,111 is the decimal number. 1 111 is one thousand one hundred eleven for the entire world. How did the commas and periods develop historically?

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    $\begingroup$ Note that resolution 10 of the 22nd CGPM in 2003 defines that either a comma or a point can be used as decimal marker, but only a space as a separator for groups of three digits. This is exactly to avoid confusion; 1.111 or 1,111 can only mean a decimal number, but it can never mean 1111. See bipm.org/documents/20126/33145759/CGPM22.pdf/… $\endgroup$ Feb 11, 2022 at 21:18
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    $\begingroup$ Note that as edited this is not correct as the UK is in Europe but uses the decimal point not the decimal comma. $\endgroup$
    – mdewey
    May 3 at 12:50

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(I must admit that I'm writing this mostly from memory and I don't have sources and not enough time to look for any, so some details may be wrong.)

Simon Stevin, who developed the notation of decimal fractions in the late 16th century, originally used a circle with numbers inside, so 1.111 would be written as "1⓪1①1②1③" and 365.256 (the length of a year in days) as 365⓪2①5②6③, where ⓪ means the preceding digits are the integer part, ① means 1/10, ② means 1/100 and so on. Stevin published this in the pamphlet "De Thiende" ("The Tenth"), written in Flemish, in 1585 (thanks to @njuffa for providing this detail in the comments).

The positional system for integer numbers had, by that time, already a long history; it was originally developed in India, spread through the Islamic world and was introduced to Europe by Leonardo von Pisa (Fibonacci) in 1202. So Stevin did not need or want to modify the integer part of a number, as this was already well established; Stevin's notation just extends it to include fractional parts by marking it with the circled numbers.

Later people found that the ①, ② etc. are not necessary; you only need the marker ⓪ where the integer part ends and the fractional part begins, and later the ⓪ developed into a dot as it was easier to write, initially centred above the line, but later on the line.

In Germany, Leibniz however introduced a center dot for multiplication (as we still do) instead of the cross in the UK. The reason was that he was working with symbolic algebra and introduced the convention to use x, y, z for unknowns, but the x can easily be confused with the multiplication cross.

But when using a dot for multiplication, having a dot as decimal marker leads again to confusion, so people started using a comma instead.

As France and Germany always had very intense cultural exchange, this became the norm in France too and spread to most other European countries (possibly with Napoleon and the metric system?).

By the way, this was formalised by the 9th Conférence Générale des Poids et Mesures (CGPM) in 1948 and the 22nd CGPM (2003). Resolution 10 of the 22nd CGPM says:

The 22nd General Conference (...)

  • declares that the symbol for the decimal marker shall be either the point on the line or the comma on the line,

  • reaffirms that “Numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups”, as stated in Resolution 7 of the 9th CGPM, 1948.

The second bit of the resolution is important and not well known; you should neither use a comma nor a point to separate the 1000s, just a space. This is exactly in order to avoid confusion, so 1.111 (like 1,111) can only ever mean 1+111/1000, but never 1111.

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    $\begingroup$ Stevin's notation seems to have first appeared in a 36-page pamphlet written in Flemish: Simon Stevin, "De Thiende." Leyden, Christoffel Plantijn 1585. The title translates to "The Tenth". I am unable to find an online scan. Translations into French ("La Disme") and English were also produced. An annotated collection of Stevin's works appeared as Albert Girard, "L'Arithmetique de Simon Stevin de Bruges." Leyden: Elzevier 1625. On page 830 start definitions and examples of the notation $\endgroup$
    – njuffa
    Feb 12, 2022 at 0:46
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    $\begingroup$ @njuffa Yes, plural is better (in the example there was only one digit...). I clarify it. $\endgroup$ Feb 12, 2022 at 9:17
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    $\begingroup$ Good information, but the CGPM seems pointless (no pun intended). Their first point (NPI) was "every current notion is Just Fine" and their second point (NPI) seems to be a compromise to follow no one's existing standard and is thus mostly ignored. (If Excel specifies a style, I imagine that's the actual standard.) $\endgroup$
    – Mark Olson
    May 3 at 18:23
  • $\begingroup$ @MarkOlson: One can change settings in Excel so that it uses commata or points as the decimal marker. (As a German living in Canada, I have made use of that, although I have slowly come around to accepting the point even in private.) $\endgroup$ May 5 at 3:13
  • $\begingroup$ @MarkOlson I'm not sure I understand really what you mean. The CGPM convention for the thousands divider (no symbol or a space) is very common and certainly not "mostly ignored". You have a point that their decision for the decimal marker is not particularly revolutionary, but as this was a question about the history of the notation it is a very important event that should be mentioned. $\endgroup$ May 5 at 6:22

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