# Who proposed terminating decimals as a major set and why are them important in France?

After looking at some school sources in French, it is common to provide the various number sets in the following order $$\mathbb{N}\subset \mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$$ where $$\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$$ are the usual natural numbers, integers, rationals, reals and complex, respectively.

However I am puzzled by the $$\mathbb D$$ which represents the décimaux rélatifs (literally "relative decimals", a better term might be terminating decimals). This seems to refer to rationals with a finite sequence of digits in decimal notation (any integer divided by a power of 10, for example 1/3 is not included).

Here are some examples:

What was the need of making the difference between rationals and terminating/relative decimals? Who were the first authors to discuss this difference? Was it a Frenchman?

According to French Wikipedia, nombre décimal, it might be related to French fascination for developing a metric system [unsourced]. But the article seems to mix the history of terminating decimals with the history of of decimal representation.

Ngram seems to indicate that this terminology became popular in the 1970s.

• French Wikipedia has some historical information fr.wikipedia.org/wiki/Nombre_d%C3%A9cimal but I am not sure if it actually answers your question. Oct 24 at 13:05
• For what it's worth, the standard English term for "relative decimals" (a term new to me) is "terminating decimals". Oct 24 at 19:30
• Is it possible that $\mathbb D$ was introduced at a time when there was no well-defined procedure to perform arithmetic on non-terminating decimals, so that $\mathbb D$ distinguishes those numbers which can be added, subtracted, and multiplied in a finite time (though not necessarily divided).
– nwr
Oct 24 at 20:41
• @mdewey interesting aside from pointing to Viète and Stevin, they leave the suggestion that it might be influenced by the metric system, wonder if there is any evidence to this Oct 24 at 23:37
• Since we are talking about school mathematics here, it would be relevant to consider the methods that the school mathematics teaches. Working mainly in the decimal system, surely terminating decimals have a special place there: the methods of adding, subtracting and multiplying by hand are first taught with naturals, then with relative ease extended to the right of the decimal point (comma), giving an algorithm that obviously terminates. With long division you may want to distinguish the situations where your algorithm terminates, and where it does not. Oct 29 at 0:38