Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: comp.theory Subject: Re: Real Number & Restoring Interpretation of Calculus Date: Sun, 24 Mar 2024 18:13:19 +0200 Organization: - Lines: 19 Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Info: dont-email.me; posting-host="22625e93c44a53efab0764cca38c1c63"; logging-data="494602"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX188AaoOhIqhn6FpGECXZ3NJ" User-Agent: Unison/2.2 Cancel-Lock: sha1:HDxbyfMdI4Qr9TaqjXi6pJrf1g8= Bytes: 1449 On 2024-03-24 14:59:52 +0000, wij said: > Note: This definition implies that repeating decimals are irrational number. This contradicts the definition that an irrationa number is not rational. > Let's list a common magic proof in the way as a brief explanation: > (1) x= 0.999... > (2) 10x= 9+x // 10x= 9.999... > (3) 9x=9 > (4) x=1 > Ans: There is no axiom or theorem to prove (1) => (2). A theory of numbers should have enough axioms to determine what the sum or product of any two numbers is. -- Mikko