Re: Repeating decimals are irrational

On Wed, 2024-03-27 at 22:38 -0400, Richard Damon wrote:

On 3/27/24 10:18 PM, wij wrote:

On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:

On 3/27/24 10:01 PM, wij wrote:

On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:

On 3/27/24 8:56 PM, wij wrote:

On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:

On 3/26/24 10:45 AM, wij wrote:

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
 
---End of quote
 
 

 
So, if 10 * 0.999... isn't 9.999... what is it?
and if 9 + 0.999... isnt 9.999... what is it?
 
And why aren't the same numbers the same numbers.
 
So, either your "wij-Reals" just fail to have the normal mathematical
operations defined or you have a problem with the proof.
 
Numbers defined with no rules on how to manipulate them are fairly
worthless.

 
The update was available:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
 
Hope, it can solve your doubt.
 

 
But the name "Real" is still very bad.
 
Particularly since you seem to say that any number that can't be
expressed in a finite number of digits in SOME base, is not a number in
your system,

 
I did not say that. ℝ just numbers expressible by <fixed_point_number>.

Near the top of the paper is:
 
 
+-------------+

Real Number |

+-------------+
 
 
 
 

 

since they can not be explicitly defined, OR HAVE MATH DONE
ON THEM, since
 
0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
.999...)
 

 
What are you referring to?
 

 
 
 
      IOW, by repeatedly multiplying 0.999.... with 10, you can only see 9,
      the structure of the rear end of 0.999.... is never seen.
 
 

Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
defined after it. (it isn't even
.999...)"

 
The line above was taken directly from the paper that I downloaded by
clicking on the link.
 
You say, and I quote:
 
0.999.... * 10 = 9. and somthing not defined after it. (it isn't even
.999...)
 
 

I uploaded again: What/where were you referring to?
--------------------------

+-------------+
| Real Number |
+-------------+

n-ary Fixed-Point Number::= Number represented by a string of digits, the
   string may contain a minus sign or a point:

     <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
     <dstr1>::= 0 | <nzd> { 0, <nzd> }
     <dstr2>::= { 0, <nzd> } <nzd>
     <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary

   Two n-ary fixed-point number (same n-ary) x,y are equal iff their
   <fixed_point_number> representation are identical.

Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
   and those that cannot be finitely represented }

   Note: Numbers that is not finitely representable cannot all be explicitly
         defined, this is the property of real number based on discrete symbols
         (like quantum?). E.g.

         A= lim(n->∞) 1-3/10^n = 0.999...
         B= lim(n->∞) 1-2/2^n  = 0.999...
         C= lim(n->∞) 1-1/n    = 0.999...
         ...

         IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
         the structure of the rear end of 0.999... is not seen.

   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: To determine whether a repeating decimal x is rational or not, we can
         repeatedly subtract the repeating pattern p(i) from x.
         If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
         rational. Otherwise, x is irrational, because, if x is rational, the
         last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
         repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
         be pattern p(i). Therefore, repeating decimal is irrational.

 

 

 

So, your system seems more to be just the rationals. and you don't seem
to provide a clear set of axioms of what you allow to be done with these
numbers.