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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Tue, 26 Mar 2024 09:45:13 -0700
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wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 08:39 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > 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.
>> 
>> How does a definition that doesn't mention rational numbers imply
>> anything about which numbers are rational?
>> 
> First of all, it is not really my definition (strict meaning of the word 
> 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.

I don't see an answer to my question.

>> >          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).

You didn't actually say what its representation is.  Is it "0." followed
by an infinite sequence of "3"s?

Isn't the representation of 1/3 a "repeating decimal"?  You stated
above that repeating decimals are irrational numbers.  How do you
reconcile that with your (correct) statement that 1/3 is rational?

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */