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wij <wyniijj5@gmail.com> writes:First of all, it is not really my definition (strict meaning of the wordSnipet 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.
How does a definition that doesn't mention rational numbers imply
anything about which numbers are 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).
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
---End of quote
Is 1/3 a rational number?
Is 1/3 a real number?
If 1/3 is a real number, what is its representation according to your definition?
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