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Madeleine Birchfield (Aug 04 2024 at 02:23):Who was the first mathematician to define the irrational numbers as the set of non-repeating infinite decimals in a published paper?
Joshua Meyers (Aug 04 2024 at 03:39):https://hsm.stackexchange.com/a/2743
Zoltan A. Kocsis (Z.A.K.) (Aug 04 2024 at 03:39):I think this question could benefit from guidelines on what counts as "defining" and "irrational real numbers".
E.g. in the late 1600s Leibniz explicitly characterized ratios of integers as preicsely the periodic expansions in any base, characterized the terminating ones number theoretically, and posed the challenge to obtain approximate decimal expansions for irrational constants. It trivially follows that he knew these would be non-periodic.
But Leibniz didn't define irrational real numbers in any modern sense. He freely used infinite integers and infinitesimal rationals, after all, and presumably we would not count a decimal expansion of length as a terminating decimal expansion nowadays.
If I was asking this question on, say, the HSM Stack Exchange, I would explicitly rule out anything before the publication of Cauchy's Cours d'analyse, or anything after that fails to engage with one of the Caucy-Cantor-Dedekind "triumvirate" reals.
Kevin Carlson (Aug 04 2024 at 20:33):What do you mean by the "triumvirate", Zoltan? Are you thinking of a third construction other than Cauchy sequences and Dedekind cuts? I don't really know how Cantor constructed the reals.
Madeleine Birchfield (Aug 04 2024 at 20:59):It was Cantor who first defined the real numbers as equivalence classes of Cauchy sequences in 1872. Then the constructivists came along and called them "Cauchy reals" because the construction used Cauchy sequences, not because Cauchy defined the real numbers (he was already dead by 15 years).
Kevin Carlson (Aug 04 2024 at 21:06):Sure, thanks, so Cantor gives the Cauchy sequence construction and I think Dedekind is actually correctly credited with the Dedekind construction, which leaves me confused still about the "tri" in the "triumvirate."
David Michael Roberts (Aug 05 2024 at 02:04):The triumvirate might just refer to the three mathematicians.
David Michael Roberts (Aug 05 2024 at 02:07):But I agree that one shouldn't attribute either Cauchy sequences or Cauchy reals to Cauchy. And one should point out that Cauchy wasn't the only person who defined the Cauchy reals as we know them. From Wikipedia
We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to Georg Cantor/Charles Méray, Richard Dedekind/Joseph Bertrand and Karl Weierstrass all occurred within a few years of each other.
David Michael Roberts (Aug 05 2024 at 02:24):Here's a paper reviewing Weierstraß's version, which I don't think I've seen before
Zoltan A. Kocsis (Z.A.K.) (Aug 05 2024 at 05:05):@Kevin Carlson I don't mean 3 "constructions of the reals", I meant the work of 3 mathematicians who made significant efforts to get rid of infinitesimal methods, and clarified that they didn't intend their notion of "reals" to include infinite or infinitesimal quantities. Cauchy's Cours contains a large section where he explains that he's actively trying to get rid of them, but was partly unsuccessful.
Zoltan A. Kocsis (Z.A.K.) (Aug 05 2024 at 05:14):I'm saying that any mention of "infinite decimal expansions" before that time is going to be suspect, since we won't be able to rule out that the author's "infinite expansions" included stuff that we nowadays won't regard as a decimal expansion at all.
Consider e.g. where the indicate an "infinite" number of omitted terms: clearly not a decimal expansion for us. But might have been for Euler. And might have been regarded as "irrational" too, given that it's not periodic.
NB finite decimal and repeating decimal do not have this problem: there was much more consensus on what finite means than on how infinites behave pre-19th century.
So it's much easier to answer "who was the first to publish that rationals correspond exactly to repeating decimals" thant to answer "who was the first to publish that irrationals correspond to non-repeating decimals", even though we regard these observations as equivalent now. Which is why I say this one has to be phrased very carefully, with guidelines for what it means to define irrationals. History is hard.
Kevin Carlson (Aug 05 2024 at 17:36):I like that Weierstrass construction!