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Dedekind gave a construction and explanation of integers and rational in 1858. This was as ordered pairs of natural numbers. I'm not sure if this was the standard view of these objects after this point. Hilbert didn't seem to write much (if anything) about this subject.

If there can be no conclusive "when", are there any textbooks that include general constructions or something like this? Many sources say this is the time period when "negatives" where finally accepted into mathematics, but give no detail about the acceptance.

When did integers start being viewed as a ring structure?

I've gotten side tracked a lot lately with math history, but I think this is what I am most concerned about. This is my last question about this topic, it is the one I originally wanted an answer too I think, but wasn't asking it properly.

I find Dedekind's work to be interesting though.

Thank you.

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    $\begingroup$ The earliest use of the symbol $\mathbb{Z}$ seems to be Bourbaki 1947. $\endgroup$
    – Mauricio
    Commented Mar 1 at 23:30
  • $\begingroup$ @Mauricio , the early use of the symbol "-" seems to be in 15th century, but it doesn't mean anything because it was not commonly accepted. By 1947 field and ring theory was well established, so I'm assuming that integers had already been properly integrated into mathematics, the symbol Z is pretty irrelevant. $\endgroup$
    – Demon
    Commented Mar 2 at 0:33
  • $\begingroup$ Negative numbers are even from before that. But the invention of a symbol $\mathbb{Z}$ is a proof that by that time the concept of positive and negative numbers as one set was well accepted as it needed a symbol and appeared in an important book. $\endgroup$
    – Mauricio
    Commented Mar 2 at 0:48
  • $\begingroup$ For its use in practical science see: hsm.stackexchange.com/questions/9717/… $\endgroup$
    – Mauricio
    Commented Mar 2 at 16:35
  • $\begingroup$ Related: hsm.stackexchange.com/questions/15880/… $\endgroup$
    – Michael E2
    Commented Mar 4 at 13:54

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I am afraid that "fully accepted" is not well defined in this question. Some mathematicians used negative numbers beginning from 2-nd century AD, then they slowly penetrated mathematics, and physics and astronomy. It is safe to say that by 18th century all professional mathematicians routinely used them. And when definitions of such things as fields were given (in early 19 century), rational and real numbers served as trivial examples.

However this does not apply to mathematics textbooks written for non-mathematicians. For example, the authors of 19 century textbooks for mariners explain in detail the decimal system, geometry, trigonometry and logarithms but avoid negative numbers.

High school algebra textbooks of second half of 19th century use negative numbers routinely, about early 19 century I am not sure.

Let me also notice that the instructions to US tax forms avoid negative numbers to this day.

Edit. I consulted an expert in the history of celestial navigation. Here is his reply:

"Celestial navigation textbooks and resources up to the very end of positive development (around 1978, which also happens to be the year the first GPS satellite was launched) continued to behave as if negative numbers either didn't exist or were too complicated for average navigators."

Edit 2. Another group of people which avoids the use of negative numbers is historians. They enumerate years like this ...2 BC, 1 BC, 1 AD, 2 AD..., while astronomers, who really need to calculate with dates, enumerate the same years in the logical order as ...-1,0,1,2,...

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  • $\begingroup$ What about specifically Dedekind's construction and it's continuation in higher mathematics textbooks, or is there no documentation on that. $\endgroup$
    – Demon
    Commented Mar 2 at 22:14
  • $\begingroup$ @Demon: I don't understand our question: what does the Dedekind construction have to do with negative numbers? Dedekind constructs real numbers from rational ones. Both can be positive or negative, of course. $\endgroup$
    – Alexandre Eremenko
    Commented Mar 3 at 2:23
  • $\begingroup$ I would think "19 century" refers to 1800-1899, but that seems to not match some things you say in this. I do not think fields were defined in the early 19th century. $\endgroup$
    – Torsten Schoeneberg
    Commented Mar 3 at 4:36
  • $\begingroup$ @AlexandreEremenko I'm trying to understand what happened to Dedekind's constructions because they seem to be what we still use today $\endgroup$
    – Demon
    Commented Mar 3 at 5:01
  • $\begingroup$ @Torsten Schoeneberg: fields were studied in 19th century, under various names (for example "domains of rationality). Dedekind studies algebraic number fields, etc. $\endgroup$
    – Alexandre Eremenko
    Commented Mar 3 at 13:26

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