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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: wij Newsgroups: comp.theory Subject: Re: Repeating decimals are irrational Date: Wed, 27 Mar 2024 00:15:53 +0800 Organization: A noiseless patient Spider Lines: 65 Message-ID: References: <87wmppuhn8.fsf@nosuchdomain.example.com> MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Date: Tue, 26 Mar 2024 17:15:54 +0100 (CET) Injection-Info: dont-email.me; posting-host="feb37bfb1ae6380338b03ab1c2de35b5"; logging-data="1997415"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+xpt7edRYHBbYDVV//OYw/" User-Agent: Evolution 3.50.2 (3.50.2-1.fc39) Cancel-Lock: sha1:VGqb84+XOSgzbocEDouSy1Cbx9U= In-Reply-To: <87wmppuhn8.fsf@nosuchdomain.example.com> Bytes: 3560 On Tue, 2024-03-26 at 08:39 -0700, Keith Thompson wrote: > wij writes: > > Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/Real= Number-en.txt/download > >=20 > > ... > > Real Nunmber(=E2=84=9D)::=3D {x| x is represented by n-ary , the > > =C2=A0=C2=A0 digits may be infinitely long } > >=20 > > =C2=A0=C2=A0 Note: This definition implies that repeating decimals are = irrational number. >=20 > How does a definition that doesn't mention rational numbers imply > anything about which numbers are rational? >=20 First of all, it is not really my definition (strict meaning of the word= =20 definition). What I showed is a reasonable proof of what the real number re= ally=20 'practically' used world-wide (not the ones in academic theory). is just a representation of real number specified for convenience for math. proofs and discussion of numbers.=20 Your 'rational number' might mean a sub-class defined latter. > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 Let's list a common ma= gic proof in the way as a brief explanation: > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (1) x=3D 0= ..999... > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (2) 10x=3D= 9+x=C2=A0 // 10x=3D 9.999... > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (3) 9x=3D9= =C2=A0=C2=A0=C2=A0=C2=A0=20 > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (4) x=3D1 > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 Ans: There is no axiom= or theorem to prove (1) =3D> (2). > >=20 > > =C2=A0=C2=A0 Note: If the steps of converting a number x to is not > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 finite, x is not a rat= io of two integers, because the following > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 statement is always tr= ue: =E2=88=80x,a=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84=9A > >=20 > > ---End of quote >=20 > Is 1/3 a rational number? >=20 Yes, by definition? > Is 1/3 a real number? >=20 > If 1/3 is a real number, what is its representation according to your def= inition? >=20 Yes, 1/3 is a real number, it's n-nary representaion i= s infinitely long. Infinitely long number is harder to explain by now. I think this part can b= e=20 skip for the moment (no present theory can make this very clear and satisfa= ctory).