From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.9 required=5.0 tests=BAYES_00,FREEMAIL_FROM autolearn=unavailable autolearn_force=no version=3.4.4 X-Received: by 2002:ad4:58eb:: with SMTP id di11mr5138667qvb.39.1588356860632; Fri, 01 May 2020 11:14:20 -0700 (PDT) X-Received: by 2002:a4a:a54a:: with SMTP id s10mr4944145oom.73.1588356860278; Fri, 01 May 2020 11:14:20 -0700 (PDT) Path: eternal-september.org!reader01.eternal-september.org!feeder.eternal-september.org!news.gegeweb.eu!gegeweb.org!usenet-fr.net!proxad.net!feeder1-2.proxad.net!209.85.160.216.MISMATCH!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail Newsgroups: comp.lang.ada Date: Fri, 1 May 2020 11:14:19 -0700 (PDT) In-Reply-To: <9f0215ca-2760-47cf-a7cb-50184892e1d0@googlegroups.com> Complaints-To: groups-abuse@google.com Injection-Info: google-groups.googlegroups.com; posting-host=47.185.215.60; posting-account=zwxLlwoAAAChLBU7oraRzNDnqQYkYbpo NNTP-Posting-Host: 47.185.215.60 References: <9f0215ca-2760-47cf-a7cb-50184892e1d0@googlegroups.com> User-Agent: G2/1.0 MIME-Version: 1.0 Message-ID: <1c0449b3-9aa5-4222-88e8-aa6aab0b24b9@googlegroups.com> Subject: Re: What is the history behind Natural'First = 0 ? From: Optikos Injection-Date: Fri, 01 May 2020 18:14:20 +0000 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Xref: reader01.eternal-september.org comp.lang.ada:58545 Date: 2020-05-01T11:14:19-07:00 List-Id: On Thursday, April 30, 2020 at 11:51:09 PM UTC-5, reinert wrote: > I have been wondering about this for years: >=20 > Why Natural'First =3D 0 ? >=20 > There is no consensus about including 0 among the natural numbers. > Since there is a Positive (Positive'First =3D 1), one may expect Natural'= First =3D 0 > Except for this, I find little intuition in "Natural'First =3D 0". >=20 >=20 > Copy form: https://en.wikipedia.org/wiki/Natural_number#History >=20 > Some definitions, including the standard ISO 80000-2,[1][2] begin the nat= ural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3,= =E2=80=A6, whereas others start with 1, corresponding to the positive inte= gers 1, 2, 3, =E2=80=A6,[3][4] while others acknowledge both definitions.[5= ] Texts that exclude zero from the natural numbers sometimes refer to the n= atural numbers together with zero as the whole numbers, but in other writin= gs, that term is used instead for the integers (including negative integers= ).[6] >=20 > Is the key point here: "the standard ISO 80000-2" ? >=20 > reinert The key here is that there has always been a terminology divide between Nor= th America and Europe (with UK & Canada sometimes going a 3rd way along Bri= tish-Empire lines). Generally, in the USA, the set of natural numbers is the set of positive in= tegers, that is denoted =E2=84=95 domestically or =E2=84=95* when interacti= ng with people outside of the USA to show the lack of zero. Generally, in = the USA, the set of whole numbers is the set of positive integers, that is = denoted either =E2=84=95=E2=82=80 or =E2=84=A4=E2=81=BA. Conversely, generally in UK and Europe, the set of natural numbers is the s= et of nonnegative integers, which is denoted =E2=84=A4-=E2=81=BA. (The dis= pute even goes that far: having different double-struck/white Z mnemonic n= otation: =E2=84=A4=E2=81=BA versus =E2=84=A4-=E2=81=BA.) Generally in UK a= nd Europe, the set of counting numbers was formerly the set of positive int= egers, but the further away from the 19th century we get, whole numbers has= at times become synonymous with the UK/European definition of natural numb= ers. https://mathworld.wolfram.com/NaturalNumber.html https://mathworld.wolfram.com/NonnegativeInteger.html https://mathworld.wolfram.com/CountingNumber.html https://en.wikipedia.org/wiki/Natural_number Ichbiah showed his European culture by institutionalizing the European defi= nition as the sole normative definition in Ada. (And don't even get me sta= rted on billion, trillion, and milliard.)