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On Tue, 2024-03-26 at 08:39 -0700, Keith Thompson wrote: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?
definition). What I showed is a reasonable proof of what the real number really
'practically' used world-wide (not the ones in academic theory).
<fixed_point_number> is just a representation of real number specified for
convenience for math. proofs and discussion of numbers.
Your 'rational number' might mean a sub-class defined latter.
>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?
Yes, by definition?
>Is 1/3 a real number?>
If 1/3 is a real number, what is its representation according to your definition?
Yes, 1/3 is a real number, it's n-nary <fixed_point_number> representaion is infinitely long.
Infinitely long number is harder to explain by now. I think this part can be
skip for the moment (no present theory can make this very clear and satisfactory).
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