Sujet : Repeating decimals are irrational
De : wyniijj5 (at) *nospam* gmail.com (wij)
Groupes : comp.theoryDate : 26. Mar 2024, 16:45:28
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <e9009d933dc0c3008201ba6cfced892d235192c8.camel@gmail.com>
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Snipet from
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download...
Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
digits may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
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).
Note: If the steps of converting a number x to <fixed_point_number> is not
finite, x is not a ratio of two integers, because the following
statement is always true: ∀x,a∈ℚ, x-a∈ℚ
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