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On Wed, 2024-03-27 at 22:38 -0400, Richard Damon wrote:^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^On 3/27/24 10:18 PM, wij wrote:I uploaded again: What/where were you referring to?On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:>On 3/27/24 10:01 PM, wij wrote:Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig notOn Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:Near the top of the paper is: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
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...
Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
digits may be infinitely long }
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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).
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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
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---End of quote
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So, if 10 * 0.999... isn't 9.999... what is it?
and if 9 + 0.999... isnt 9.999... what is it?
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And why aren't the same numbers the same numbers.
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So, either your "wij-Reals" just fail to have the normal mathematical
operations defined or you have a problem with the proof.
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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
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Hope, it can solve your doubt.
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But the name "Real" is still very bad.
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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>.
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+-------------+Real Number |+-------------+
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>>>since they can not be explicitly defined, OR HAVE MATH DONE>
ON THEM, since
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0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
.999...)
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What are you referring to?
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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.
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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.
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You say, and I quote:
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0.999.... * 10 = 9. and somthing not defined after it. (it isn't even
.999...)
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--------------------------
+-------------+
| 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.
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