Path: csiph.com!weretis.net!feeder6.news.weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: wij Newsgroups: comp.lang.c++ Subject: Re: Repeating decimals are irrational Date: Fri, 29 Mar 2024 02:23:46 +0800 Organization: A noiseless patient Spider Lines: 85 Message-ID: References: <5b363d215e2c4dab1d496efee2655409dcecf655.camel@gmail.com> <459e2c88fcadc581ef016624894880c24270652f.camel@gmail.com> <930fdb5b775d5512e471c82bf434648c56fd1009.camel@gmail.com> <435b9fe3357bbc4dd5a25f8f9e75637e270dd55b.camel@gmail.com> MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Date: Thu, 28 Mar 2024 18:23:47 +0100 (CET) Injection-Info: dont-email.me; posting-host="b1a05f6d6181cc9398f199026fad047a"; logging-data="3875388"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19ctrRVafaWZmeGeiZLDHIi" User-Agent: Evolution 3.50.2 (3.50.2-1.fc39) Cancel-Lock: sha1:HQjGvXBG7WVPQJLP2FC2KepeUDI= In-Reply-To: Xref: csiph.com comp.lang.c++:118610 On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote: > On 27/03/2024 17:05, wij wrote: > > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote: > > > On 27/03/2024 14:32, wij wrote: > > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote: > > > > > Am Wed, 27 Mar 2024 20:12:38 +0800 > > > > > schrieb wij : > > > > >=20 > > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote: > > > > > > > On 26/03/2024 22:43, wij wrote: > > > > > > > >=20 > > > > > > > > Just repeat the pattern infinitely, then it is irrational. > > > > > > >=20 > > > > > > > Nonsense. > > > > > > > =C2=A0=C2=A0=C2=A0=20 > > > > > > > > As said "=E2=88=80x,a=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84= =9A", if the subtraction a=3D 142857/10^(6*i) > > > > > > > > cannot terminate, 1/7 !=3D 0.(142857) > > > > > > > > =C2=A0=C2=A0=C2=A0=20 > > > > > > >=20 > > > > > > > Nonsense. > > > > > > > =C2=A0=C2=A0=C2=A0=20 > > > > > >=20 > > > > > > I am surprise your math. knowledge is so low worse than teenage= rs. > > > > >=20 > > > > > Use the standard trick: > > > > >=20 > > > > > x=3D0.[142857] =3D> 1,000,000*x=3D142857.[142857] > > > > >=20 > > > > > subtract the first equation from the second: > > > > >=20 > > > > > 999,999*x=3D142857 =3D> x=3D142857/999,999=3D1/7 > > > > >=20 > > > >=20 > > > > To determine whether a number x is rational or not, we can repeated= ly subtract > > > > rational numbers a? from x. If x-a1-a2-a3-...=3D0 can be verified i= n finite > > > > steps, then x is rational. Otherwise, x is irrational. > > > > If x is a repeating decimal, proposition "repeating decimal is rati= onal" is > > > > simply false by sematics. > > > >=20 > > >=20 > > > Let me just ask you two simple questions: > > >=20 > > > Do you think 1/7 is a rational number or an irrational number? > > >=20 > > rational > >=20 > > > What do you think the decimal expansion of 1/7 is? > > >=20 > >=20 > > When converting 1/7 to decimal, the result =E2=89=92 0.(142857), the pr= ocedure > > never terminates which means the conversion is never complete. > >=20 >=20 > It is a repeating decimal.=C2=A0 If you try to write it all out, then I a= gree=20 > you will not finish.=C2=A0 That does not mean it is not the decimal expan= sion=20 > of 1/7 - the list of multiples of (negative) powers of 10 which sum up= =20 > to 1/7.=C2=A0 You just need a better notation so that you can finish the = task=20 > - and 0.(142857), as you wrote, is one such notation. >=20 > (I have no idea what you think the symbol "=E2=89=92" might mean.) >=20 > But you agree that 0.(142857) is the decimal expansion of 1/7, even=20 > though you could not write it out long-hand, and you agree that 1/7 i=20 > rational.=C2=A0 And clearly 0.(142857) is a repeating decimal, since that= 's=20 > what the notation means. >=20 > I can't see how you can still misunderstand this. >=20 You are restating your assertion without proof, again. I have provided mine= . (If you say that is you proof, I will say it is invalid).