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

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?

> 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?

Is 1/3 a real number?

If 1/3 is a real number, what is its representation according to your definition?

--
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 */

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.

> >          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).

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 */

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>.

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
void Void(void) { Void(); } /* The recursive call of the void */

On 3/26/2024 3:18 PM, Keith Thompson wrote:
> 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.
>

Yes.
https://en.wikipedia.org/wiki/Irrational_number

> 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?
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
> 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.
>
Correct, because that would mean no q can exhaust the conversion in long division, or subtraction.

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

Good luck if you try to conquer irrational

>
> 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?

I just understand the front part of your question.
No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
The semantic of "repeating" already says the number 1/3 cannot be completely
exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
is simple false by semantics.

>
> 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?
>
If your question were about 0.333...*3, simple answer is 0.999... (by infinite series)
But in general, "..." is a place to play magic. Don't use it for controversial proofs

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
>> 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.
>>
> Correct, because that would mean no q can exhaust the conversion in long division, or subtraction.
>
>> That's probably a workable definition.  (I won't get into whether it's a
>> *useful* definition.)
>
> Good luck if you try to conquer irrational
>
>>
>> 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?
>
> I just understand the front part of your question.
> No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
> The semantic of "repeating" already says the number 1/3 cannot be completely
> exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
> is simple false by semantics.

And that's where you're quite simply wrong.

1/3 and 0.333(3) are exactly equal, and rational. "∀x,a∈ℚ, x-a∈ℚ" is
not violated; the difference is 0, which is also rational. You do not
have an internally consistent model of real or rational numbers.

You admit that you don't understand all of my question. I suggest that,
until you've learned enough to understand it, you're not going to be
successful in whatever you're trying to do.

I don't think you understand what "0.333(3)" *means*. Sure, you
understand that it's an unending series of 3s, but to understand its
value, you have to understand *limits*. I don't believe that you do.

You don't know how to represent 1/3 exactly in decimal notation. I do.

>> 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?
>>
> If your question were about 0.333...*3, simple answer is 0.999... (by infinite series)
> But in general, "..." is a place to play magic. Don't use it for controversial proofs

There is no magic. People who understand limits, which can be defined
rigorously, are able to understand how all this stuff works.

There are valid extensions to the real numbers (surreals, hyperreals,
etc.) -- but they were invented by people who understand the reals.

--
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 */

On Tue, 2024-03-26 at 15:01 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
> > > 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.
> > >
> > Correct, because that would mean no q can exhaust the conversion in long division, or subtraction.
> >
> > > That's probably a workable definition.  (I won't get into whether it's a
> > > *useful* definition.)
> >
> > Good luck if you try to conquer irrational
> >
> > >
> > > 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?
> >
> > I just understand the front part of your question.
> > No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
> > The semantic of "repeating" already says the number 1/3 cannot be completely
> > exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
> > is simple false by semantics.
>
> And that's where you're quite simply wrong.
>
> 1/3 and 0.333(3) are exactly equal, and rational.  "∀x,a∈ℚ, x-a∈ℚ" is
> not violated; the difference is 0, which is also rational.  You do not
> have an internally consistent model of real or rational numbers.
>

Then, I would ask you to provide proof, not just assertion that you are right.
My model is not perfect, but it does not matter, I am sure now you cannot really
distinguish good or bad.

> You admit that you don't understand all of my question.  I suggest that,
> until you've learned enough to understand it, you're not going to be
> successful in whatever you're trying to do.
>
> I don't think you understand what "0.333(3)" *means*.  Sure, you
> understand that it's an unending series of 3s, but to understand its
> value, you have to understand *limits*.  I don't believe that you do..
>
> You don't know how to represent 1/3 exactly in decimal notation.  I do.
>

It looked your level of math. is really primitive.

> > > 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?
> > >
> > If your question were about 0.333...*3, simple answer is 0.999... (by infinite series)
> > But in general, "..." is a place to play magic. Don't use it for controversial proofs
>
> There is no magic.  People who understand limits, which can be defined
> rigorously, are able to understand how all this stuff works.
>
> There are valid extensions to the real numbers (surreals, hyperreals,
> etc.) -- but they were invented by people who understand the reals.
>

So far, you only stated copied belief and insult.
Please reply with something worth discussion, I don't have time to exchange words.

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 15:01 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
[SNIP]
>> > > 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?
>> >
>> > I just understand the front part of your question.
>> > No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
>> > The semantic of "repeating" already says the number 1/3 cannot be completely
>> > exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
>> > is simple false by semantics.
>>
>> And that's where you're quite simply wrong.
>>
>> 1/3 and 0.333(3) are exactly equal, and rational.  "∀x,a∈ℚ, x-a∈ℚ" is
>> not violated; the difference is 0, which is also rational.  You do not
>> have an internally consistent model of real or rational numbers.
>>
>
> Then, I would ask you to provide proof, not just assertion that you are right.
> My model is not perfect, but it does not matter, I am sure now you cannot really
> distinguish good or bad.

Do you understand what a mathematical limit is? Can you explain it?

If you understand limits, you should be able to understand that
0.333..., where the "..." denotes the limit as the number of 3s exceeds
without bound, is exactly equal to 1/3.

[...]

--
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 */

On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 15:01 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
> [SNIP]
> > > > > 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?
> > > >
> > > > I just understand the front part of your question.
> > > > No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
> > > > The semantic of "repeating" already says the number 1/3 cannot be completely
> > > > exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
> > > > is simple false by semantics.
> > >
> > > And that's where you're quite simply wrong.
> > >
> > > 1/3 and 0.333(3) are exactly equal, and rational.  "∀x,a∈ℚ, x-a∈ℚ" is
> > > not violated; the difference is 0, which is also rational.  You do not
> > > have an internally consistent model of real or rational numbers.
> > >
> >
> > Then, I would ask you to provide proof, not just assertion that you are right.
> > My model is not perfect, but it does not matter, I am sure now you cannot really
> > distinguish good or bad.
>
> Do you understand what a mathematical limit is?  Can you explain it?
>
> If you understand limits, you should be able to understand that
> 0.333..., where the "..." denotes the limit as the number of 3s exceeds
> without bound, is exactly equal to 1/3.
>
> [...]
>

I already stated it the file:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
[...]
>> Do you understand what a mathematical limit is?  Can you explain it?
>>
>> If you understand limits, you should be able to understand that
>> 0.333..., where the "..." denotes the limit as the number of 3s exceeds
>> without bound, is exactly equal to 1/3.
>>
>> [...]
>>
>
> I already stated it the file:
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

OK, in that file you wrote:

Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1

What do you mean by that question mark? Do you agree that both limits
in your example (A and B) are equal to 1?

--
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 */

On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> [...]
> > > Do you understand what a mathematical limit is?  Can you explain it?
> > >
> > > If you understand limits, you should be able to understand that
> > > 0.333..., where the "..." denotes the limit as the number of 3s exceeds
> > > without bound, is exactly equal to 1/3.
> > >
> > > [...]
> > >
> >
> > I already stated it the file:
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> OK, in that file you wrote:
>
>     Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
>          B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>
> What do you mean by that question mark?  Do you agree that both limits
> in your example (A and B) are equal to 1?
>

Agree.

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
>> [...]
>> > > Do you understand what a mathematical limit is?  Can you explain it?
>> > >
>> > > If you understand limits, you should be able to understand that
>> > > 0.333..., where the "..." denotes the limit as the number of 3s exceeds
>> > > without bound, is exactly equal to 1/3.
>> > >
>> > > [...]
>> > >
>> >
>> > I already stated it the file:
>> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>
>> OK, in that file you wrote:
>>
>>     Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
>>          B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>>
>> What do you mean by that question mark?  Do you agree that both limits
>> in your example (A and B) are equal to 1?
>>
>
> Agree.

OK. Do you agree that 0.333..., where ... denotes the limit as the
number of 3s increases without bound, is exactly equal to 1/3?

And that that number multipled by 3 is exactly equal to 1?

And that 0.999... is exactly equal to 1?

And that all the numbers we're discussing are both real and 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 */

On Tue, 2024-03-26 at 17:01 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> > > [...]
> > > > > Do you understand what a mathematical limit is?  Can you explain it?
> > > > >
> > > > > If you understand limits, you should be able to understand that
> > > > > 0.333..., where the "..." denotes the limit as the number of 3s exceeds
> > > > > without bound, is exactly equal to 1/3.
> > > > >
> > > > > [...]
> > > > >
> > > >
> > > > I already stated it the file:
> > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > >
> > > OK, in that file you wrote:
> > >
> > >     Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
> > >          B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
> > >
> > > What do you mean by that question mark?  Do you agree that both limits
> > > in your example (A and B) are equal to 1?
> > >
> >
> > Agree.
>
> OK.  Do you agree that 0.333..., where ... denotes the limit as the
> number of 3s increases without bound, is exactly equal to 1/3?
>

Nop.

> And that that number multipled by 3 is exactly equal to 1?
>
> And that 0.999... is exactly equal to 1?
>
> And that all the numbers we're discussing are both real and rational?
>

Can you make statement really like math?
You still keep asserting your belief, not a proof but strangely formed statement !!!
You cannot read proof !!! Go home and learn more.

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 17:01 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
>> > > wij <wyniijj5@gmail.com> writes:
>> > > > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
>> > > [...]
>> > > > > Do you understand what a mathematical limit is?  Can you explain it?
>> > > > >
>> > > > > If you understand limits, you should be able to understand that
>> > > > > 0.333..., where the "..." denotes the limit as the number of 3s exceeds
>> > > > > without bound, is exactly equal to 1/3.
>> > > > >
>> > > > > [...]
>> > > > >
>> > > >
>> > > > I already stated it the file:
>> > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>> > >
>> > > OK, in that file you wrote:
>> > >
>> > >     Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
>> > >          B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>> > >
>> > > What do you mean by that question mark?  Do you agree that both limits
>> > > in your example (A and B) are equal to 1?
>> > >
>> >
>> > Agree.
>>
>> OK.  Do you agree that 0.333..., where ... denotes the limit as the
>> number of 3s increases without bound, is exactly equal to 1/3?
>>
>
> Nop.

I'll assume that's a typo for "No" or "Nope".

So you're saying that 0.333... is not exactly equal to 1/3.

It seems odd that you agree that 0.999... is exactly equal to 1, but
0.333... is not exactly equal to 1/3.

Is 0.333... a real number? Is 1/3 - 0.333... a real number? What is
the value of 1/3 - 0.333...?

>> And that that number multipled by 3 is exactly equal to 1?
>>
>> And that 0.999... is exactly equal to 1?
>>
>> And that all the numbers we're discussing are both real and rational?
>
> Can you make statement really like math?

Please rephrase that. I don't understand what it means.

> You still keep asserting your belief, not a proof but strangely formed statement !!!
> You cannot read proof !!! Go home and learn more.

I haven't offered a proof because (a) there's no point in doing so if we
don't have enough mutual understanding, and (b) I'm not a mathematician,
so I'm not confident of my ability to construct a rigorous proof. I
might give it a try later, but I see no point in doing so now.

--
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 */

On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 17:01 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
> > > > > wij <wyniijj5@gmail.com> writes:
> > > > > > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> > > > > [...]
> > > > > > > Do you understand what a mathematical limit is?  Can you explain it?
> > > > > > >
> > > > > > > If you understand limits, you should be able to understand that
> > > > > > > 0.333..., where the "..." denotes the limit as the number of 3s exceeds
> > > > > > > without bound, is exactly equal to 1/3.
> > > > > > >
> > > > > > > [...]
> > > > > > >
> > > > > >
> > > > > > I already stated it the file:
> > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > > >
> > > > > OK, in that file you wrote:
> > > > >
> > > > >     Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
> > > > >          B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
> > > > >
> > > > > What do you mean by that question mark?  Do you agree that both limits
> > > > > in your example (A and B) are equal to 1?
> > > > >
> > > >
> > > > Agree.
> > >
> > > OK.  Do you agree that 0.333..., where ... denotes the limit as the
> > > number of 3s increases without bound, is exactly equal to 1/3?
> > >
> >
> > Nop.
>
> I'll assume that's a typo for "No" or "Nope".
>
> So you're saying that 0.333... is not exactly equal to 1/3.
>
> It seems odd that you agree that 0.999... is exactly equal to 1, but
> 0.333... is not exactly equal to 1/3.
>

I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
Read the definition carefully from trustworthy website.

Be noted that this is still nothing to do with what we are discussing.
If you understand "circular reasoning", limit cannot define the number it is using.

> Is 0.333... a real number?  Is 1/3 - 0.333... a real number?  What is
> the value of 1/3 - 0.333...?
>
> > > And that that number multipled by 3 is exactly equal to 1?
> > >
> > > And that 0.999... is exactly equal to 1?
> > >
> > > And that all the numbers we're discussing are both real and rational?
> >
> > Can you make statement really like math?
>
> Please rephrase that.  I don't understand what it means.
>
> > You still keep asserting your belief, not a proof but strangely formed statement !!!
> > You cannot read proof !!! Go home and learn more.
>
> I haven't offered a proof because (a) there's no point in doing so if we
> don't have enough mutual understanding, and (b) I'm not a mathematician,
> so I'm not confident of my ability to construct a rigorous proof.  I
> might give it a try later, but I see no point in doing so now.

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
[...]
>> So you're saying that 0.333... is not exactly equal to 1/3.
>>
>> It seems odd that you agree that 0.999... is exactly equal to 1, but
>> 0.333... is not exactly equal to 1/3.
>>
>
> I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
> Read the definition carefully from trustworthy website.

So you're distinguishing between "the limit of 0.999..." and "0.999...".

I see no difference between them. To me, the "..." notation *means* the
limit. Can you explain what difference you see?

When I write "0.999...", I mean the limit as the number of 9s increases
without bound. That limit, I think we both agree, is equal to 1. And
perhaps we also both agree that the limit of 0.333... as the number of
3s increases without bound is equal to 1/3.

Are you saying that:

- 0.999... is something other than the limit as the number of 9s
increases without bound?
- 0.999... is a real number?
- 0.999... is less than 1?

If so, what is the real value of 1 - 0.999...?

[...]

--
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 */

On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
> [...]
> > > So you're saying that 0.333... is not exactly equal to 1/3.
> > >
> > > It seems odd that you agree that 0.999... is exactly equal to 1, but
> > > 0.333... is not exactly equal to 1/3.
> > >
> >
> > I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
> > Read the definition carefully from trustworthy website.
>
> So you're distinguishing between "the limit of 0.999..." and "0.999...".
>
> I see no difference between them.  To me, the "..." notation *means* the
> limit.  Can you explain what difference you see?
>
> When I write "0.999...", I mean the limit as the number of 9s increases
> without bound.  That limit, I think we both agree, is equal to 1.  And
> perhaps we also both agree that the limit of 0.333... as the number of
> 3s increases without bound is equal to 1/3.
>
> Are you saying that:
>
> - 0.999... is something other than the limit as the number of 9s
>   increases without bound?

Don't know what you are asking for

> - 0.999... is a real number?

Yes

> - 0.999... is less than 1?
>
Yes

> If so, what is the real value of 1 - 0.999...?

Infinitesimal

>
> [...]
>

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
>> [...]
>> > > So you're saying that 0.333... is not exactly equal to 1/3.
>> > >
>> > > It seems odd that you agree that 0.999... is exactly equal to 1, but
>> > > 0.333... is not exactly equal to 1/3.
>> > >
>> >
>> > I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
>> > Read the definition carefully from trustworthy website.
>>
>> So you're distinguishing between "the limit of 0.999..." and "0.999...".
>>
>> I see no difference between them.  To me, the "..." notation *means* the
>> limit.  Can you explain what difference you see?
>>
>> When I write "0.999...", I mean the limit as the number of 9s increases
>> without bound.  That limit, I think we both agree, is equal to 1.  And
>> perhaps we also both agree that the limit of 0.333... as the number of
>> 3s increases without bound is equal to 1/3.
>>
>> Are you saying that:
>>
>> - 0.999... is something other than the limit as the number of 9s
>>   increases without bound?
>
> Don't know what you are asking for

I don't know how to explain it more clearly.

We've both been using notations like "0.999...". I've been using it to
mean exactly the limit as the number of 9s increases without bound.
That particular limit is exactly equal to 1.

You apparently mean something other than that limit when you write
"0.999...". I'm asking you what you mean by "0.999...", and in
particular how that differs from the described limit.

>> - 0.999... is a real number?
>
> Yes

Agreed.

>> - 0.999... is less than 1?
>>
> Yes

Incorrect, assuming we're talking about real numbers as that term is
universally used in mathematics.

>> If so, what is the real value of 1 - 0.999...?
>
> Infinitesimal

"Infinitesimal" is not a real number. No real number is infinitesimal.

If 0.999... and 1 are both real numbers, and they're not equal, then
their difference is a non-zero real number.

--
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 */

On 3/26/2024 8:01 PM, wij wrote:
> On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>>> On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
>> [...]
>>>> So you're saying that 0.333... is not exactly equal to 1/3.
>>>>
>>>> It seems odd that you agree that 0.999... is exactly equal to 1, but
>>>> 0.333... is not exactly equal to 1/3.
>>>>
>>>
>>> I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
>>> Read the definition carefully from trustworthy website.
>>
>> So you're distinguishing between "the limit of 0.999..." and "0.999...".
>>
>> I see no difference between them.  To me, the "..." notation *means* the
>> limit.  Can you explain what difference you see?
>>
>> When I write "0.999...", I mean the limit as the number of 9s increases
>> without bound.  That limit, I think we both agree, is equal to 1.  And
>> perhaps we also both agree that the limit of 0.333... as the number of
>> 3s increases without bound is equal to 1/3.
>>
>> Are you saying that:
>>
>> - 0.999... is something other than the limit as the number of 9s
>>   increases without bound?
>
> Don't know what you are asking for
>
>> - 0.999... is a real number?
>
> Yes
>
>> - 0.999... is less than 1?
>>
> Yes
>
>> If so, what is the real value of 1 - 0.999...?
>
> Infinitesimal
>

Wow, I think that I agree with you on this.

I came up with my own infinitesimal number system that also agrees:
The line segment specified by [0.0, 1.0] is one geometric point
longer than [0.0, 1.0)

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Tue, 2024-03-26 at 18:11 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
> > > [...]
> > > > > So you're saying that 0.333... is not exactly equal to 1/3.
> > > > >
> > > > > It seems odd that you agree that 0.999... is exactly equal to 1, but
> > > > > 0.333... is not exactly equal to 1/3.
> > > > >
> > > >
> > > > I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
> > > > Read the definition carefully from trustworthy website.
> > >
> > > So you're distinguishing between "the limit of 0.999..." and "0.999....".
> > >
> > > I see no difference between them.  To me, the "..." notation *means* the
> > > limit.  Can you explain what difference you see?
> > >
> > > When I write "0.999...", I mean the limit as the number of 9s increases
> > > without bound.  That limit, I think we both agree, is equal to 1..  And
> > > perhaps we also both agree that the limit of 0.333... as the number of
> > > 3s increases without bound is equal to 1/3.
> > >
> > > Are you saying that:
> > >
> > > - 0.999... is something other than the limit as the number of 9s
> > >   increases without bound?
> >
> > Don't know what you are asking for
>
> I don't know how to explain it more clearly.
>
> We've both been using notations like "0.999...".  I've been using it to
> mean exactly the limit as the number of 9s increases without bound.
> That particular limit is exactly equal to 1.
>
Prove it. I don't think you understand what you say even though we both agree on this part.

> You apparently mean something other than that limit when you write
> "0.999...".  I'm asking you what you mean by "0.999...", and in
> particular how that differs from the described limit.
>

If you cannot tell the difference, what can I say, and what can you expect?
Go home and learn more.

> > > - 0.999... is a real number?
> >
> > Yes
>
> Agreed.

Prove it.

>
> > > - 0.999... is less than 1?
> > >
> > Yes
>
> Incorrect, assuming we're talking about real numbers as that term is
> universally used in mathematics.
>

Prove it.

> > > If so, what is the real value of 1 - 0.999...?
> >
> > Infinitesimal
>
> "Infinitesimal" is not a real number.  No real number is infinitesimal.
>
> If 0.999... and 1 are both real numbers, and they're not equal, then
> their difference is a non-zero real number.
>

Prove it.

On Tue, 2024-03-26 at 20:20 -0500, olcott wrote:
> On 3/26/2024 8:01 PM, wij wrote:
> > On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
> > > [...]
> > > > > So you're saying that 0.333... is not exactly equal to 1/3.
> > > > >
> > > > > It seems odd that you agree that 0.999... is exactly equal to 1, but
> > > > > 0.333... is not exactly equal to 1/3.
> > > > >
> > > >
> > > > I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
> > > > Read the definition carefully from trustworthy website.
> > >
> > > So you're distinguishing between "the limit of 0.999..." and "0.999....".
> > >
> > > I see no difference between them.  To me, the "..." notation *means* the
> > > limit.  Can you explain what difference you see?
> > >
> > > When I write "0.999...", I mean the limit as the number of 9s increases
> > > without bound.  That limit, I think we both agree, is equal to 1..  And
> > > perhaps we also both agree that the limit of 0.333... as the number of
> > > 3s increases without bound is equal to 1/3.
> > >
> > > Are you saying that:
> > >
> > > - 0.999... is something other than the limit as the number of 9s
> > >    increases without bound?
> >
> > Don't know what you are asking for
> >
> > > - 0.999... is a real number?
> >
> > Yes
> >
> > > - 0.999... is less than 1?
> > >
> > Yes
> >
> > > If so, what is the real value of 1 - 0.999...?
> >
> > Infinitesimal
> >
>
> Wow, I think that I agree with you on this.
>
> I came up with my own infinitesimal number system that also agrees:
> The line segment specified by [0.0, 1.0] is one geometric point
> longer than [0.0, 1.0)
>

Yes, you are such a genius !!!

wij <wyniijj5@gmail.com> writes:
> On Tue, 2024-03-26 at 18:11 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Tue, 2024-03-26 at 17:53 -0700, Keith Thompson wrote:
>> > > wij <wyniijj5@gmail.com> writes:
>> > > > On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
>> > > [...]
>> > > > > So you're saying that 0.333... is not exactly equal to 1/3.
>> > > > >
>> > > > > It seems odd that you agree that 0.999... is exactly equal to 1, but
>> > > > > 0.333... is not exactly equal to 1/3.
>> > > > >
>> > > >
>> > > > I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you asked)
>> > > > Read the definition carefully from trustworthy website.
>> > >
>> > > So you're distinguishing between "the limit of 0.999..." and "0.999...".
>> > >
>> > > I see no difference between them.  To me, the "..." notation *means* the
>> > > limit.  Can you explain what difference you see?
>> > >
>> > > When I write "0.999...", I mean the limit as the number of 9s increases
>> > > without bound.  That limit, I think we both agree, is equal to 1.  And
>> > > perhaps we also both agree that the limit of 0.333... as the number of
>> > > 3s increases without bound is equal to 1/3.
>> > >
>> > > Are you saying that:
>> > >
>> > > - 0.999... is something other than the limit as the number of 9s
>> > >   increases without bound?
>> >
>> > Don't know what you are asking for
>>
>> I don't know how to explain it more clearly.
>>
>> We've both been using notations like "0.999...".  I've been using it to
>> mean exactly the limit as the number of 9s increases without bound.
>> That particular limit is exactly equal to 1.
>>
> Prove it. I don't think you understand what you say even though we both agree on this part.

I am not ready to offer a rigorous proof until and unless we agree on
some key concepts. I might not do so even then, for reasons I explained
earlier.

>> You apparently mean something other than that limit when you write
>> "0.999...".  I'm asking you what you mean by "0.999...", and in
>> particular how that differs from the described limit.
>
> If you cannot tell the difference, what can I say, and what can you expect?
> Go home and learn more.

I'm asking what the difference is *to you*. What do *YOU* mean by
0.999...? The only possible way I can learn that is for you to tell me.
If you're unwilling to do so, we can end this discussion.

[snip]

--
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 */