Re: Repeating decimals are irrational

wij <wyniijj5@gmail.com> writes:

On Tue, 2024-03-26 at 09:45 -0700, Keith Thompson wrote:

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?
 

>
Ah, I see. I should make my statement more clear:
1/3 is representable in 3-nary <fixed_point_number> (e.g. 0.1)
"Infinite long" (for irrational) refers to numbers that is not finitely representable by any
n-ary <fixed_point_number>.

Are you now saying (for the first time, as far as I can tell) that a
number is rational if and only if it has a finite representation in
*any* integer base?  For example, 1/3 has a finite representation in
base 3, and 1/7 has a finite representation in base 7 (both have finite
representations in base 21).  So an infinite decimal representation
doesn't make a number irrational as long as it has a finite
representation in *some* integer base.

That's probably a workable definition.  (I won't get into whether it's a
*useful* definition.)

Still, rational numbers can be represented in decimal, and you already
acknowledged that "the digits may be infinitely long" for real numbers.
So 0.333..., where the sequence of 3s is unending, is a valid
representation of the rational number 1/3, yes?  It's the number itself
that's rational, regardless of which of several valid representations
you choose to express it.

Do you agree so far?

If so, consider the rational number that is the result of dividing 1 by
3, represented in decimal as 0.333..., where the "..." denotes an
unending sequence of 3s.  What is the decimal representation of the
number that is the result of multiplying that number by 3?

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
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