On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
> On 3/27/2024 9:05 AM, wij wrote:
> > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
> > > On 27/03/2024 14:32, wij wrote:
> > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
> > > > > Am Wed, 27 Mar 2024 20:12:38 +0800
> > > > > schrieb wij <wyniijj5@gmail.com>:
> > > > >
> > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
> > > > > > > On 26/03/2024 22:43, wij wrote:
> > > > > > > >
> > > > > > > > Just repeat the pattern infinitely, then it is irrational.
> > > > > > >
> > > > > > > Nonsense.
> > > > > > >    
> > > > > > > > As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
> > > > > > > > cannot terminate, 1/7 != 0.(142857)
> > > > > > > >    
> > > > > > >
> > > > > > > Nonsense.
> > > > > > >    
> > > > > >
> > > > > > I am surprise your math. knowledge is so low worse than teenagers.
> > > > >
> > > > > Use the standard trick:
> > > > >
> > > > > x=0.[142857] => 1,000,000*x=142857.[142857]
> > > > >
> > > > > subtract the first equation from the second:
> > > > >
> > > > > 999,999*x=142857 => x=142857/999,999=1/7
> > > > >
> > > >
> > > > To determine whether a number x is rational or not, we can repeatedly subtract
> > > > rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
> > > > steps, then x is rational. Otherwise, x is irrational.
> > > > If x is a repeating decimal, proposition "repeating decimal is rational" is
> > > > simply false by sematics.
> > > >
> > >
> > > Let me just ask you two simple questions:
> > >
> > > Do you think 1/7 is a rational number or an irrational number?
> > >
> > rational
> >
> > > What do you think the decimal expansion of 1/7 is?
> > >
> >
> > When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> > never terminates which means the conversion is never complete.
> >
> >
> >
>
> You can stop iteration as soon as you detect a cycle, or period if you
> will. In 1/7, say it took 6 iterations to hit the period... Sound okay?

Stupid! It is an infinite string. Cycle or period can only be determined for
finite string.

On 3/27/2024 2:39 PM, wij wrote:
> On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
>> On 3/27/2024 9:05 AM, wij wrote:
>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>> On 27/03/2024 14:32, wij wrote:
>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>
>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>
>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>
>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>
>>>>>> Use the standard trick:
>>>>>>
>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>
>>>>>> subtract the first equation from the second:
>>>>>>
>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>
>>>>>
>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>> simply false by sematics.
>>>>>
>>>>
>>>> Let me just ask you two simple questions:
>>>>
>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>
>>> rational
>>>
>>>> What do you think the decimal expansion of 1/7 is?
>>>>
>>>
>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>> never terminates which means the conversion is never complete.
>>>
>>>
>>>
>>
>> You can stop iteration as soon as you detect a cycle, or period if you
>> will. In 1/7, say it took 6 iterations to hit the period... Sound okay?
>
> Stupid! It is an infinite string.

Not sure how to respond to that. A cycle is a finite thingy... ;^)

> Cycle or period can only be determined for
> finite string.
>
>

On 3/27/2024 2:39 PM, wij wrote:
> On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
>> On 3/27/2024 9:05 AM, wij wrote:
>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>> On 27/03/2024 14:32, wij wrote:
>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>
>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>
>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>
>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>
>>>>>> Use the standard trick:
>>>>>>
>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>
>>>>>> subtract the first equation from the second:
>>>>>>
>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>
>>>>>
>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>> simply false by sematics.
>>>>>
>>>>
>>>> Let me just ask you two simple questions:
>>>>
>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>
>>> rational
>>>
>>>> What do you think the decimal expansion of 1/7 is?
>>>>
>>>
>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>> never terminates which means the conversion is never complete.
>>>
>>>
>>>
>>
>> You can stop iteration as soon as you detect a cycle, or period if you
>> will. In 1/7, say it took 6 iterations to hit the period... Sound okay?
>
> Stupid! It is an infinite string. Cycle or period can only be determined for
> finite string.
>
>

For some reason, I think you might be misunderstanding me.

On 3/27/2024 3:31 AM, David Brown wrote:
> On 26/03/2024 21:13, Chris M. Thomasson wrote:
>> On 3/26/2024 7:51 AM, wij wrote:
>> [...]
>>
>> Repeating decimals are rational, say
>>
>> 0.142857 142857 142857
>>
>> That is just 1 / 7 represented in base 10.
>
> Obviously (to everyone except perhaps wij).
>
>>
>> Now, think of using a TRNG to create each digit...
>>
>> That would be, irrational... ;^)
>
> That would not be a defined number.  I am not convinced it is meaningful
> to talk about its properties at all.
>

Say you iterated up to, say 7 digits from a TRNG:

..3274836

We say, okay:

0.3274836 3274836 3274836

Can we say this is rational? How about:

1091612/3333333

? Fair enough or off base, so to speak... ?

On 3/27/2024 5:12 AM, wij wrote:
> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>> On 26/03/2024 22:43, wij wrote:
>>> On Tue, 2024-03-26 at 13:13 -0700, Chris M. Thomasson wrote:
>>>> On 3/26/2024 7:51 AM, wij wrote:
>>>> [...]
>>>>
>>>> Repeating decimals are rational, say
>>>>
>>>> 0.142857 142857 142857
>>>>
>>>> That is just 1 / 7 represented in base 10.
>>>>
>>>> Now, think of using a TRNG to create each digit...
>>>>
>>>> That would be, irrational... ;^)
>>>
>>> Just repeat the pattern infinitely, then it is irrational.
>>
>> Nonsense.
>>
>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>> cannot terminate, 1/7 != 0.(142857)
>>>
>>
>> Nonsense.
>>
>
> I am surprise your math. knowledge is so low worse than teenagers.

Oh wow. What made you say that?

Just a bit interested? Humm...

>
>> Simply stating random things does not make them so.
>>
>> I recommend you stick to C++ in this C++ newsgroup.
>>
>
> I know. You 'occupied' c/c++ forum and think you are speech police.
> For now, this discussion is mainly in comp.theory
> But you have shown your knowledge is so so low, don't go there waste our time.
>
>> As for your maths, you'd do better learning some basics of the
>> mathematics of real numbers and rational numbers, and that being able to
>> find the Unicode characters for some logic symbols does not mean you
>> understand how to write a proof.
>>
>>
>
>

On 27/03/2024 17:05, wij wrote:
> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>> On 27/03/2024 14:32, wij wrote:
>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>
>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>
>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>
>>>>>> Nonsense.
>>>>>>
>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>
>>>>>>
>>>>>> Nonsense.
>>>>>>
>>>>>
>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>
>>>> Use the standard trick:
>>>>
>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>
>>>> subtract the first equation from the second:
>>>>
>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>
>>>
>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>> steps, then x is rational. Otherwise, x is irrational.
>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>> simply false by sematics.
>>>
>>
>> Let me just ask you two simple questions:
>>
>> Do you think 1/7 is a rational number or an irrational number?
>>
> rational
>
>> What do you think the decimal expansion of 1/7 is?
>>
>
> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> never terminates which means the conversion is never complete.
>

It is a repeating decimal. If you try to write it all out, then I agree
you will not finish. That does not mean it is not the decimal expansion
of 1/7 - the list of multiples of (negative) powers of 10 which sum up
to 1/7. You just need a better notation so that you can finish the task
- and 0.(142857), as you wrote, is one such notation.

(I have no idea what you think the symbol "≒" might mean.)

But you agree that 0.(142857) is the decimal expansion of 1/7, even
though you could not write it out long-hand, and you agree that 1/7 i
rational. And clearly 0.(142857) is a repeating decimal, since that's
what the notation means.

I can't see how you can still misunderstand this.

On 27/03/2024 22:39, wij wrote:
> On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
>> On 3/27/2024 9:05 AM, wij wrote:
>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>> On 27/03/2024 14:32, wij wrote:
>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>
>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>
>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>
>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>
>>>>>> Use the standard trick:
>>>>>>
>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>
>>>>>> subtract the first equation from the second:
>>>>>>
>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>
>>>>>
>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>> simply false by sematics.
>>>>>
>>>>
>>>> Let me just ask you two simple questions:
>>>>
>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>
>>> rational
>>>
>>>> What do you think the decimal expansion of 1/7 is?
>>>>
>>>
>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>> never terminates which means the conversion is never complete.
>>>
>>>
>>>
>>
>> You can stop iteration as soon as you detect a cycle, or period if you
>> will. In 1/7, say it took 6 iterations to hit the period... Sound okay?
>
> Stupid! It is an infinite string. Cycle or period can only be determined for
> finite string.
>

Nonsense.

You /know/ the cycle for the infinite decimal expansion for 1/7 - it is
the digits "142857", repeated every 6 digits in the decimal expansion.
Again, that's what the notation 0.(142857) - /your/ choice of notation,
so presumably familiar to you - means.

On 27/03/2024 21:17, Chris M. Thomasson wrote:
> On 3/27/2024 3:31 AM, David Brown wrote:
>> On 26/03/2024 21:13, Chris M. Thomasson wrote:
>>> On 3/26/2024 7:51 AM, wij wrote:
>>> [...]
>>>
>>> Repeating decimals are rational, say
>>>
>>> 0.142857 142857 142857
>>>
>>> That is just 1 / 7 represented in base 10.
>>
>> Obviously (to everyone except perhaps wij).
>
> ;^)
>
>
>>> Now, think of using a TRNG to create each digit...
>>>
>>> That would be, irrational... ;^)
>>
>> That would not be a defined number.  I am not convinced it is
>> meaningful to talk about its properties at all.
>>
>
> Well, it would be a "number" at any finite view of it? Or, is that just
> moronic thinking?

It is not a defined number in any way - you are explicitly using a
procedure which does not give any defined or deterministic results. How
could it be a number when no two people could agree to its value?

Now, if you were to take a TRNG and generate a series of digits, writing
each one down as you go along, then at any given time you would have a
finite decimal expansion which would be a number (a rational, if that
matters to you). But that is a particular instance of creating such a
number - the procedure you describe does not define a number nor is it a
number in itself.

> For some reason it makes me think of infinite
> convergents of continued fractions.

I am not sure you are helping yourself or anyone else there. A sequence
of digits obtained randomly does not converge.

> Can we gain a rational that can
> approximate any irrational up to a certain precision, so to speak?

Of course we can. That is a completely separate issue.

If you have a real number "x" (positive for simplicity), and a precision
"e", then we can easily find a rational number q such that |x - q| < e.
We do so by letting "i" be an integer greater than 1/e. Let "j" be the
first integer less than or equal to i * x.

So
j <= (i * x) < (j + 1)

and thus

j / i <= x < (j + 1) / i

q0 = j / i and q1 = (j + 1) / i are both rational numbers, and "x" lies
between them.

q1 - q0 = 1 / i, which is less than "e" since "i" is greater than 1/e.

So since (q1 - q0) < e, and q0 <= x < q1, then we can see that "x" is
"trapped" between two rationals that are both less than "e" from "x".

You can approximate any real number as closely as you like with a series
of rational numbers. That is basically what it means to say that the
real numbers "complete" the rationals - the fill in the gaps that can be
expressed as the limits of bounded sequences of rationals where the
limit is not itself a rational.

On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
> On 27/03/2024 17:05, wij wrote:
> > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
> > > On 27/03/2024 14:32, wij wrote:
> > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
> > > > > Am Wed, 27 Mar 2024 20:12:38 +0800
> > > > > schrieb wij <wyniijj5@gmail.com>:
> > > > >
> > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
> > > > > > > On 26/03/2024 22:43, wij wrote:
> > > > > > > >
> > > > > > > > Just repeat the pattern infinitely, then it is irrational.
> > > > > > >
> > > > > > > Nonsense.
> > > > > > >    
> > > > > > > > As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
> > > > > > > > cannot terminate, 1/7 != 0.(142857)
> > > > > > > >    
> > > > > > >
> > > > > > > Nonsense.
> > > > > > >    
> > > > > >
> > > > > > I am surprise your math. knowledge is so low worse than teenagers.
> > > > >
> > > > > Use the standard trick:
> > > > >
> > > > > x=0.[142857] => 1,000,000*x=142857.[142857]
> > > > >
> > > > > subtract the first equation from the second:
> > > > >
> > > > > 999,999*x=142857 => x=142857/999,999=1/7
> > > > >
> > > >
> > > > To determine whether a number x is rational or not, we can repeatedly subtract
> > > > rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
> > > > steps, then x is rational. Otherwise, x is irrational.
> > > > If x is a repeating decimal, proposition "repeating decimal is rational" is
> > > > simply false by sematics.
> > > >
> > >
> > > Let me just ask you two simple questions:
> > >
> > > Do you think 1/7 is a rational number or an irrational number?
> > >
> > rational
> >
> > > What do you think the decimal expansion of 1/7 is?
> > >
> >
> > When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> > never terminates which means the conversion is never complete.
> >
>
> It is a repeating decimal.  If you try to write it all out, then I agree
> you will not finish.  That does not mean it is not the decimal expansion
> of 1/7 - the list of multiples of (negative) powers of 10 which sum up
> to 1/7.  You just need a better notation so that you can finish the task
> - and 0.(142857), as you wrote, is one such notation.
>
> (I have no idea what you think the symbol "≒" might mean.)
>
> But you agree that 0.(142857) is the decimal expansion of 1/7, even
> though you could not write it out long-hand, and you agree that 1/7 i
> rational.  And clearly 0.(142857) is a repeating decimal, since that's
> what the notation means.
>
> I can't see how you can still misunderstand this.
>

You are restating your assertion without proof, again. I have provided mine..
(If you say that is you proof, I will say it is invalid).

On Thu, 2024-03-28 at 18:17 +0100, David Brown wrote:
> On 27/03/2024 22:39, wij wrote:
> > On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
> > > On 3/27/2024 9:05 AM, wij wrote:
> > > > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
> > > > > On 27/03/2024 14:32, wij wrote:
> > > > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
> > > > > > > Am Wed, 27 Mar 2024 20:12:38 +0800
> > > > > > > schrieb wij <wyniijj5@gmail.com>:
> > > > > > >
> > > > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
> > > > > > > > > On 26/03/2024 22:43, wij wrote:
> > > > > > > > > >
> > > > > > > > > > Just repeat the pattern infinitely, then it is irrational.
> > > > > > > > >
> > > > > > > > > Nonsense.
> > > > > > > > >     
> > > > > > > > > > As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
> > > > > > > > > > cannot terminate, 1/7 != 0.(142857)
> > > > > > > > > >     
> > > > > > > > >
> > > > > > > > > Nonsense.
> > > > > > > > >     
> > > > > > > >
> > > > > > > > I am surprise your math. knowledge is so low worse than teenagers.
> > > > > > >
> > > > > > > Use the standard trick:
> > > > > > >
> > > > > > > x=0.[142857] => 1,000,000*x=142857.[142857]
> > > > > > >
> > > > > > > subtract the first equation from the second:
> > > > > > >
> > > > > > > 999,999*x=142857 => x=142857/999,999=1/7
> > > > > > >
> > > > > >
> > > > > > To determine whether a number x is rational or not, we can repeatedly subtract
> > > > > > rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
> > > > > > steps, then x is rational. Otherwise, x is irrational.
> > > > > > If x is a repeating decimal, proposition "repeating decimal is rational" is
> > > > > > simply false by sematics.
> > > > > >
> > > > >
> > > > > Let me just ask you two simple questions:
> > > > >
> > > > > Do you think 1/7 is a rational number or an irrational number?
> > > > >
> > > > rational
> > > >
> > > > > What do you think the decimal expansion of 1/7 is?
> > > > >
> > > >
> > > > When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> > > > never terminates which means the conversion is never complete.
> > > >
> > > >
> > > >
> > >
> > > You can stop iteration as soon as you detect a cycle, or period if you
> > > will. In 1/7, say it took 6 iterations to hit the period... Sound okay?
> >
> > Stupid! It is an infinite string. Cycle or period can only be determined for
> > finite string.
> >
>
> Nonsense.
>
> You /know/ the cycle for the infinite decimal expansion for 1/7 - it is
> the digits "142857", repeated every 6 digits in the decimal expansion.
> Again, that's what the notation 0.(142857) - /your/ choice of notation,
> so presumably familiar to you - means.
>
"0.(142857)" is pre-determined and specified not detected. (unless I misunderstood
what Chris said)

On 3/28/2024 10:47 AM, David Brown wrote:
> On 27/03/2024 21:17, Chris M. Thomasson wrote:
>> On 3/27/2024 3:31 AM, David Brown wrote:
>>> On 26/03/2024 21:13, Chris M. Thomasson wrote:
>>>> On 3/26/2024 7:51 AM, wij wrote:
>>>> [...]
>>>>
>>>> Repeating decimals are rational, say
>>>>
>>>> 0.142857 142857 142857
>>>>
>>>> That is just 1 / 7 represented in base 10.
>>>
>>> Obviously (to everyone except perhaps wij).
>>
>> ;^)
>>
>>
>>>> Now, think of using a TRNG to create each digit...
>>>>
>>>> That would be, irrational... ;^)
>>>
>>> That would not be a defined number.  I am not convinced it is
>>> meaningful to talk about its properties at all.
>>>
>>
>> Well, it would be a "number" at any finite view of it? Or, is that
>> just moronic thinking?
>
> It is not a defined number in any way - you are explicitly using a
> procedure which does not give any defined or deterministic results.  How
> could it be a number when no two people could agree to its value?
>
> Now, if you were to take a TRNG and generate a series of digits, writing
> each one down as you go along, then at any given time you would have a
> finite decimal expansion which would be a number (a rational, if that
> matters to you).  But that is a particular instance of creating such a
> number - the procedure you describe does not define a number nor is it a
> number in itself.

Would it be fair to say it defines a random number that is an actual
number at every finite step... Is that a stupid thought?

>> For some reason it makes me think of infinite convergents of continued
>> fractions.
>
> I am not sure you are helping yourself or anyone else there.

Well, shit happens. Just some fun with numbers? ;^)

> A sequence
> of digits obtained randomly does not converge.

True, but at every "step", it can be a rational? I just think it can be
a fun study, so to speak...

>> Can we gain a rational that can approximate any irrational up to a
>> certain precision, so to speak?
>
> Of course we can.  That is a completely separate issue.

Yeah. Well, okay. I agree.

> If you have a real number "x" (positive for simplicity), and a precision
> "e", then we can easily find a rational number q such that |x - q| < e.
> We do so by letting "i" be an integer greater than 1/e.  Let "j" be the
> first integer less than or equal to i * x.

Using a TRNG to "generate" x, and a precision, say when we stop drawing
digits from the TRNG, would create a rational. Right?

> So
>     j <= (i * x) < (j + 1)
>
> and thus
>
>     j / i <= x < (j + 1) / i
>
> q0 = j / i and q1 = (j + 1) / i are both rational numbers, and "x" lies
> between them.
>
> q1 - q0 = 1 / i, which is less than "e" since "i" is greater than 1/e.
>
> So since (q1 - q0) < e, and q0 <= x < q1, then we can see that "x" is
> "trapped" between two rationals that are both less than "e" from "x".
>
> You can approximate any real number as closely as you like with a series
> of rational numbers.  That is basically what it means to say that the
> real numbers "complete" the rationals - the fill in the gaps that can be
> expressed as the limits of bounded sequences of rationals where the
> limit is not itself a rational.

Are you saying that a number constructed digit-by-digit using a TRNG is
undefined? Its not a number, however it creates many numbers during the
construction process? Fair enough?

Ahhh shit, this is just me having some fun. Sorry.

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.
> 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

What does the arity of the "n-ary <fixed_point_number>" refer to? The
base or the number of digits?

Any rational number can be represented with a single fractional digit
when represented in the base of its denominator.

I.e., 3/7 = 0.3 (base 7)

And as far as I am aware, the rationality of a number does not depend on
its representation...

-Stefan

--
There is no such thing as luck. Luck is nothing but an absence of bad luck.

On Thu, 2024-03-28 at 21:33 +0100, Stefan Große Pawig 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.
> >          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
>
> What does the arity of the "n-ary <fixed_point_number>" refer to? The
> base or the number of digits?
>

Yes, base of the number system.

> Any rational number can be represented with a single fractional digit
> when represented in the base of its denominator.
>
> I.e., 3/7 = 0.3 (base 7)
>
> And as far as I am aware, the rationality of a number does not depend on
> its representation...
>

Exactly.

You quoted a slightly outdated text. Click the link above to see the most updated text.

> -Stefan
>

On 28/03/2024 19:25, wij wrote:
> On Thu, 2024-03-28 at 18:17 +0100, David Brown wrote:
>> On 27/03/2024 22:39, wij wrote:
>>> On Wed, 2024-03-27 at 13:40 -0700, Chris M. Thomasson wrote:
>>>> On 3/27/2024 9:05 AM, wij wrote:
>>>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>>>> On 27/03/2024 14:32, wij wrote:
>>>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>>>
>>>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>>>
>>>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>>>
>>>>>>>>>> Nonsense.
>>>>>>>>>>
>>>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Nonsense.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>>>
>>>>>>>> Use the standard trick:
>>>>>>>>
>>>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>>>
>>>>>>>> subtract the first equation from the second:
>>>>>>>>
>>>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>>>
>>>>>>>
>>>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>>>> simply false by sematics.
>>>>>>>
>>>>>>
>>>>>> Let me just ask you two simple questions:
>>>>>>
>>>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>>>
>>>>> rational
>>>>>
>>>>>> What do you think the decimal expansion of 1/7 is?
>>>>>>
>>>>>
>>>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>>>> never terminates which means the conversion is never complete.
>>>>>
>>>>>
>>>>>
>>>>
>>>> You can stop iteration as soon as you detect a cycle, or period if you
>>>> will. In 1/7, say it took 6 iterations to hit the period... Sound okay?
>>>
>>> Stupid! It is an infinite string. Cycle or period can only be determined for
>>> finite string.
>>>
>>
>> Nonsense.
>>
>> You /know/ the cycle for the infinite decimal expansion for 1/7 - it is
>> the digits "142857", repeated every 6 digits in the decimal expansion.
>> Again, that's what the notation 0.(142857) - /your/ choice of notation,
>> so presumably familiar to you - means.
>>
> "0.(142857)" is pre-determined and specified not detected. (unless I misunderstood
> what Chris said)
>

You are so confused that I would not be surprised if you misunderstood
Chris. And Chris' posts score much higher on enthusiasm than on clarity.

The fact that 1/7 has the decimal expansion 0.(142857) - that is, an
unending repetition of the digits 142857 - is simple to calculate and
easy to prove correct. It is "pre-determined" in the sense that it is
the unique decimal expansion for 1/7. But I don't understand what you
mean by "specified and not detected" - it is not something that Chris
invented out of thin air.

On 28/03/2024 19:23, wij wrote:
> On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
>> On 27/03/2024 17:05, wij wrote:
>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>> On 27/03/2024 14:32, wij wrote:
>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>
>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>
>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>
>>>>>>>>
>>>>>>>> Nonsense.
>>>>>>>>
>>>>>>>
>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>
>>>>>> Use the standard trick:
>>>>>>
>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>
>>>>>> subtract the first equation from the second:
>>>>>>
>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>
>>>>>
>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>> simply false by sematics.
>>>>>
>>>>
>>>> Let me just ask you two simple questions:
>>>>
>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>
>>> rational
>>>
>>>> What do you think the decimal expansion of 1/7 is?
>>>>
>>>
>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>> never terminates which means the conversion is never complete.
>>>
>>
>> It is a repeating decimal.  If you try to write it all out, then I agree
>> you will not finish.  That does not mean it is not the decimal expansion
>> of 1/7 - the list of multiples of (negative) powers of 10 which sum up
>> to 1/7.  You just need a better notation so that you can finish the task
>> - and 0.(142857), as you wrote, is one such notation.
>>
>> (I have no idea what you think the symbol "≒" might mean.)
>>
>> But you agree that 0.(142857) is the decimal expansion of 1/7, even
>> though you could not write it out long-hand, and you agree that 1/7 i
>> rational.  And clearly 0.(142857) is a repeating decimal, since that's
>> what the notation means.
>>
>> I can't see how you can still misunderstand this.
>>
>
> You are restating your assertion without proof, again. I have provided mine.
> (If you say that is you proof, I will say it is invalid).
>
>

There is no point in giving you a rigorous proof that 0.(142857) is the
decimal expansion of 1/7, if that is what you are contesting. To be
fully rigorous, it requires an understanding of the definition of the
real numbers, sequence limits, and the meaning and validity of
operations on infinite sequences. You have demonstrated that you don't
understand any of that. You have learned a few of the terms, but failed
to understand the concepts. Oh, and it also requires understanding what
a proof is, which again is clearly outside your expertise.

Ralf gave a proof earlier - it is still in the quoted material above.
That is as good as we can get at your level of mathematical
understanding. To be more rigorous, we would need to demonstrate that
the manipulation (multiplication by a finite integer, and subtraction of
sequences) of infinite decimal expansions is valid. That is all
standard stuff, known to mathematics students the world over, but you
are not nearly ready.

You are going to have to go back-track a long way in what you think you
know about mathematics. Somewhere along the line in your education,
you've got things badly wrong. And instead of stopping up and trying to
figure out why everyone else is saying something different from you, or
asking your teachers for help, you have battered on with your mistakes,
leading you to sillier and steadily less logical conclusions.

I think mathematics is a great hobby. It's a shame to see someone spend
their time and effort on doing it so badly.

On 28/03/2024 20:41, Chris M. Thomasson wrote:
> On 3/28/2024 10:47 AM, David Brown wrote:
>> On 27/03/2024 21:17, Chris M. Thomasson wrote:
>>> On 3/27/2024 3:31 AM, David Brown wrote:
>>>> On 26/03/2024 21:13, Chris M. Thomasson wrote:
>>>>> On 3/26/2024 7:51 AM, wij wrote:
>>>>> [...]
>>>>>
>>>>> Repeating decimals are rational, say
>>>>>
>>>>> 0.142857 142857 142857
>>>>>
>>>>> That is just 1 / 7 represented in base 10.
>>>>
>>>> Obviously (to everyone except perhaps wij).
>>>
>>> ;^)
>>>
>>>
>>>>> Now, think of using a TRNG to create each digit...
>>>>>
>>>>> That would be, irrational... ;^)
>>>>
>>>> That would not be a defined number.  I am not convinced it is
>>>> meaningful to talk about its properties at all.
>>>>
>>>
>>> Well, it would be a "number" at any finite view of it? Or, is that
>>> just moronic thinking?
>>
>> It is not a defined number in any way - you are explicitly using a
>> procedure which does not give any defined or deterministic results.
>> How could it be a number when no two people could agree to its value?
>>
>> Now, if you were to take a TRNG and generate a series of digits,
>> writing each one down as you go along, then at any given time you
>> would have a finite decimal expansion which would be a number (a
>> rational, if that matters to you).  But that is a particular instance
>> of creating such a number - the procedure you describe does not define
>> a number nor is it a number in itself.
>
> Would it be fair to say it defines a random number that is an actual
> number at every finite step... Is that a stupid thought?
>

It is not defining a number at all. It is defining a /procedure/ for
generating different numbers.

>
>>> For some reason it makes me think of infinite convergents of
>>> continued fractions.
>>
>> I am not sure you are helping yourself or anyone else there.
>
> Well, shit happens. Just some fun with numbers? ;^)
>

Fun is always good. And there's plenty of fun to be had with continued
fractions - but pick sequences with a pattern, not random numbers.

>
>> A sequence of digits obtained randomly does not converge.
>
> True, but at every "step", it can be a rational? I just think it can be
> a fun study, so to speak...
>

Certainly random sequences can be interesting to study, though I prefer
defined sequences. And certainly every finite sequence of digits is a
rational. But your sequences of randomly generated digits don't
converge. (What would they converge to?)

>
>>> Can we gain a rational that can approximate any irrational up to a
>>> certain precision, so to speak?
>>
>> Of course we can.  That is a completely separate issue.
>
> Yeah. Well, okay. I agree.
>
>
>> If you have a real number "x" (positive for simplicity), and a
>> precision "e", then we can easily find a rational number q such that
>> |x - q| < e. We do so by letting "i" be an integer greater than 1/e.
>> Let "j" be the first integer less than or equal to i * x.
>
> Using a TRNG to "generate" x, and a precision, say when we stop drawing
> digits from the TRNG, would create a rational. Right?
>

Yes, but it is a different rational each time. You haven't defined a
number. If you had defined a number, then you could calculate it to a
given precision, and /I/ could calculate it to the same precision, and
we'd each have the same result. That doesn't happen with random generators.

>
>> So
>>      j <= (i * x) < (j + 1)
>>
>> and thus
>>
>>      j / i <= x < (j + 1) / i
>>
>> q0 = j / i and q1 = (j + 1) / i are both rational numbers, and "x"
>> lies between them.
>>
>> q1 - q0 = 1 / i, which is less than "e" since "i" is greater than 1/e.
>>
>> So since (q1 - q0) < e, and q0 <= x < q1, then we can see that "x" is
>> "trapped" between two rationals that are both less than "e" from "x".
>>
>> You can approximate any real number as closely as you like with a
>> series of rational numbers.  That is basically what it means to say
>> that the real numbers "complete" the rationals - the fill in the gaps
>> that can be expressed as the limits of bounded sequences of rationals
>> where the limit is not itself a rational.
>
> Are you saying that a number constructed digit-by-digit using a TRNG is
> undefined? Its not a number, however it creates many numbers during the
> construction process? Fair enough?
>

Any given finite sequence of digits taken from a TRNG will give you a
rational number. But "take 20 digits from a TRNG" does not define a
number - it defines a procedure for generating numbers. Do you see the
difference?

>
> Ahhh shit, this is just me having some fun. Sorry.

That's no problem. As long as you are trying to learn, and trying to
have fun, it's fine by me.

On Fri, 2024-03-29 at 11:53 +0100, David Brown wrote:
> On 28/03/2024 19:23, wij wrote:
> > On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
> > > On 27/03/2024 17:05, wij wrote:
> > > > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
> > > > > On 27/03/2024 14:32, wij wrote:
> > > > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
> > > > > > > Am Wed, 27 Mar 2024 20:12:38 +0800
> > > > > > > schrieb wij <wyniijj5@gmail.com>:
> > > > > > >
> > > > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
> > > > > > > > > On 26/03/2024 22:43, wij wrote:
> > > > > > > > > >
> > > > > > > > > > Just repeat the pattern infinitely, then it is irrational.
> > > > > > > > >
> > > > > > > > > Nonsense.
> > > > > > > > >     
> > > > > > > > > > As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
> > > > > > > > > > cannot terminate, 1/7 != 0.(142857)
> > > > > > > > > >     
> > > > > > > > >
> > > > > > > > > Nonsense.
> > > > > > > > >     
> > > > > > > >
> > > > > > > > I am surprise your math. knowledge is so low worse than teenagers.
> > > > > > >
> > > > > > > Use the standard trick:
> > > > > > >
> > > > > > > x=0.[142857] => 1,000,000*x=142857.[142857]
> > > > > > >
> > > > > > > subtract the first equation from the second:
> > > > > > >
> > > > > > > 999,999*x=142857 => x=142857/999,999=1/7
> > > > > > >
> > > > > >
> > > > > > To determine whether a number x is rational or not, we can repeatedly subtract
> > > > > > rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
> > > > > > steps, then x is rational. Otherwise, x is irrational.
> > > > > > If x is a repeating decimal, proposition "repeating decimal is rational" is
> > > > > > simply false by sematics.
> > > > > >
> > > > >
> > > > > Let me just ask you two simple questions:
> > > > >
> > > > > Do you think 1/7 is a rational number or an irrational number?
> > > > >
> > > > rational
> > > >
> > > > > What do you think the decimal expansion of 1/7 is?
> > > > >
> > > >
> > > > When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> > > > never terminates which means the conversion is never complete.
> > > >
> > >
> > > It is a repeating decimal.  If you try to write it all out, then I agree
> > > you will not finish.  That does not mean it is not the decimal expansion
> > > of 1/7 - the list of multiples of (negative) powers of 10 which sum up
> > > to 1/7.  You just need a better notation so that you can finish the task
> > > - and 0.(142857), as you wrote, is one such notation.
> > >
> > > (I have no idea what you think the symbol "≒" might mean.)
> > >
> > > But you agree that 0.(142857) is the decimal expansion of 1/7, even
> > > though you could not write it out long-hand, and you agree that 1/7 i
> > > rational.  And clearly 0.(142857) is a repeating decimal, since that's
> > > what the notation means.
> > >
> > > I can't see how you can still misunderstand this.
> > >
> >
> > You are restating your assertion without proof, again. I have provided mine.
> > (If you say that is you proof, I will say it is invalid).
> >
> >
>
> There is no point in giving you a rigorous proof that 0.(142857) is the
> decimal expansion of 1/7, if that is what you are contesting.  To be
> fully rigorous, it requires an understanding of the definition of the
> real numbers, sequence limits, and the meaning and validity of
> operations on infinite sequences.  You have demonstrated that you don't
> understand any of that.  You have learned a few of the terms, but failed
> to understand the concepts.  Oh, and it also requires understanding what
> a proof is, which again is clearly outside your expertise.
>
> Ralf gave a proof earlier - it is still in the quoted material above.
> That is as good as we can get at your level of mathematical
> understanding.  To be more rigorous, we would need to demonstrate that
> the manipulation (multiplication by a finite integer, and subtraction of
> sequences) of infinite decimal expansions is valid.  That is all
> standard stuff, known to mathematics students the world over, but you
> are not nearly ready.
>
> You are going to have to go back-track a long way in what you think you
> know about mathematics.  Somewhere along the line in your education,
> you've got things badly wrong.  And instead of stopping up and trying to
> figure out why everyone else is saying something different from you, or
> asking your teachers for help, you have battered on with your mistakes,
> leading you to sillier and steadily less logical conclusions.
>
> I think mathematics is a great hobby.  It's a shame to see someone spend
> their time and effort on doing it so badly.
>

Have you ever wondered why you cannot prove something you hold true for granted
for so long?

If you cannot provide a proof, what you said above only make you more a sinner.

On 29/03/2024 16:14, wij wrote:
> On Fri, 2024-03-29 at 11:53 +0100, David Brown wrote:
>> On 28/03/2024 19:23, wij wrote:
>>> On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
>>>> On 27/03/2024 17:05, wij wrote:
>>>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>>>> On 27/03/2024 14:32, wij wrote:
>>>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>>>
>>>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>>>
>>>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>>>
>>>>>>>>>> Nonsense.
>>>>>>>>>>
>>>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Nonsense.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>>>
>>>>>>>> Use the standard trick:
>>>>>>>>
>>>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>>>
>>>>>>>> subtract the first equation from the second:
>>>>>>>>
>>>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>>>
>>>>>>>
>>>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>>>> simply false by sematics.
>>>>>>>
>>>>>>
>>>>>> Let me just ask you two simple questions:
>>>>>>
>>>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>>>
>>>>> rational
>>>>>
>>>>>> What do you think the decimal expansion of 1/7 is?
>>>>>>
>>>>>
>>>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>>>> never terminates which means the conversion is never complete.
>>>>>
>>>>
>>>> It is a repeating decimal.  If you try to write it all out, then I agree
>>>> you will not finish.  That does not mean it is not the decimal expansion
>>>> of 1/7 - the list of multiples of (negative) powers of 10 which sum up
>>>> to 1/7.  You just need a better notation so that you can finish the task
>>>> - and 0.(142857), as you wrote, is one such notation.
>>>>
>>>> (I have no idea what you think the symbol "≒" might mean.)
>>>>
>>>> But you agree that 0.(142857) is the decimal expansion of 1/7, even
>>>> though you could not write it out long-hand, and you agree that 1/7 i
>>>> rational.  And clearly 0.(142857) is a repeating decimal, since that's
>>>> what the notation means.
>>>>
>>>> I can't see how you can still misunderstand this.
>>>>
>>>
>>> You are restating your assertion without proof, again. I have provided mine.
>>> (If you say that is you proof, I will say it is invalid).
>>>
>>>
>>
>> There is no point in giving you a rigorous proof that 0.(142857) is the
>> decimal expansion of 1/7, if that is what you are contesting.  To be
>> fully rigorous, it requires an understanding of the definition of the
>> real numbers, sequence limits, and the meaning and validity of
>> operations on infinite sequences.  You have demonstrated that you don't
>> understand any of that.  You have learned a few of the terms, but failed
>> to understand the concepts.  Oh, and it also requires understanding what
>> a proof is, which again is clearly outside your expertise.
>>
>> Ralf gave a proof earlier - it is still in the quoted material above.
>> That is as good as we can get at your level of mathematical
>> understanding.  To be more rigorous, we would need to demonstrate that
>> the manipulation (multiplication by a finite integer, and subtraction of
>> sequences) of infinite decimal expansions is valid.  That is all
>> standard stuff, known to mathematics students the world over, but you
>> are not nearly ready.
>>
>> You are going to have to go back-track a long way in what you think you
>> know about mathematics.  Somewhere along the line in your education,
>> you've got things badly wrong.  And instead of stopping up and trying to
>> figure out why everyone else is saying something different from you, or
>> asking your teachers for help, you have battered on with your mistakes,
>> leading you to sillier and steadily less logical conclusions.
>>
>> I think mathematics is a great hobby.  It's a shame to see someone spend
>> their time and effort on doing it so badly.
>>
>
> Have you ever wondered why you cannot prove something you hold true for granted
> for so long?

Yes, regularly. Sometimes I will then try to find a proof, or look up
and learn about the proofs. Sometimes I will have to accept that
proving the particular thing is beyond my mathematical skills, or my
time and energy, or my interest, and I will defer to accepting that
others have proven it.

>
> If you cannot provide a proof, what you said above only make you more a sinner.
>

In this particular case, I most certainly /can/ provide a proof. But I
can't provide a proof that /you/ would understand. And since writing a
proof would be a fair effort, off-topic, and clearly a waste of time
since you are impervious to mathematical reasoning, I will not bother.
You can look up such proofs online - I'm sure there are countless
Youtube videos that will explain it to anyone who is actually interested
in learning and not merely trying to claim the whole world is wrong
except them.

On Fri, 2024-03-29 at 16:48 +0100, David Brown wrote:
> On 29/03/2024 16:14, wij wrote:
> > On Fri, 2024-03-29 at 11:53 +0100, David Brown wrote:
> > > On 28/03/2024 19:23, wij wrote:
> > > > On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
> > > > > On 27/03/2024 17:05, wij wrote:
> > > > > > On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
> > > > > > > On 27/03/2024 14:32, wij wrote:
> > > > > > > > On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
> > > > > > > > > Am Wed, 27 Mar 2024 20:12:38 +0800
> > > > > > > > > schrieb wij <wyniijj5@gmail.com>:
> > > > > > > > >
> > > > > > > > > > On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
> > > > > > > > > > > On 26/03/2024 22:43, wij wrote:
> > > > > > > > > > > >
> > > > > > > > > > > > Just repeat the pattern infinitely, then it is irrational.
> > > > > > > > > > >
> > > > > > > > > > > Nonsense.
> > > > > > > > > > >      
> > > > > > > > > > > > As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
> > > > > > > > > > > > cannot terminate, 1/7 != 0.(142857)
> > > > > > > > > > > >      
> > > > > > > > > > >
> > > > > > > > > > > Nonsense.
> > > > > > > > > > >      
> > > > > > > > > >
> > > > > > > > > > I am surprise your math. knowledge is so low worse than teenagers.
> > > > > > > > >
> > > > > > > > > Use the standard trick:
> > > > > > > > >
> > > > > > > > > x=0.[142857] => 1,000,000*x=142857.[142857]
> > > > > > > > >
> > > > > > > > > subtract the first equation from the second:
> > > > > > > > >
> > > > > > > > > 999,999*x=142857 => x=142857/999,999=1/7
> > > > > > > > >
> > > > > > > >
> > > > > > > > To determine whether a number x is rational or not, we can repeatedly subtract
> > > > > > > > rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
> > > > > > > > steps, then x is rational. Otherwise, x is irrational.
> > > > > > > > If x is a repeating decimal, proposition "repeating decimal is rational" is
> > > > > > > > simply false by sematics.
> > > > > > > >
> > > > > > >
> > > > > > > Let me just ask you two simple questions:
> > > > > > >
> > > > > > > Do you think 1/7 is a rational number or an irrational number?
> > > > > > >
> > > > > > rational
> > > > > >
> > > > > > > What do you think the decimal expansion of 1/7 is?
> > > > > > >
> > > > > >
> > > > > > When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
> > > > > > never terminates which means the conversion is never complete.
> > > > > >
> > > > >
> > > > > It is a repeating decimal.  If you try to write it all out, then I agree
> > > > > you will not finish.  That does not mean it is not the decimal expansion
> > > > > of 1/7 - the list of multiples of (negative) powers of 10 which sum up
> > > > > to 1/7.  You just need a better notation so that you can finish the task
> > > > > - and 0.(142857), as you wrote, is one such notation.
> > > > >
> > > > > (I have no idea what you think the symbol "≒" might mean.)
> > > > >
> > > > > But you agree that 0.(142857) is the decimal expansion of 1/7, even
> > > > > though you could not write it out long-hand, and you agree that 1/7 i
> > > > > rational.  And clearly 0.(142857) is a repeating decimal, since that's
> > > > > what the notation means.
> > > > >
> > > > > I can't see how you can still misunderstand this.
> > > > >
> > > >
> > > > You are restating your assertion without proof, again. I have provided mine.
> > > > (If you say that is you proof, I will say it is invalid).
> > > >
> > > >
> > >
> > > There is no point in giving you a rigorous proof that 0.(142857) is the
> > > decimal expansion of 1/7, if that is what you are contesting.  To be
> > > fully rigorous, it requires an understanding of the definition of the
> > > real numbers, sequence limits, and the meaning and validity of
> > > operations on infinite sequences.  You have demonstrated that you don't
> > > understand any of that.  You have learned a few of the terms, but failed
> > > to understand the concepts.  Oh, and it also requires understanding what
> > > a proof is, which again is clearly outside your expertise.
> > >
> > > Ralf gave a proof earlier - it is still in the quoted material above.
> > > That is as good as we can get at your level of mathematical
> > > understanding.  To be more rigorous, we would need to demonstrate that
> > > the manipulation (multiplication by a finite integer, and subtraction of
> > > sequences) of infinite decimal expansions is valid.  That is all
> > > standard stuff, known to mathematics students the world over, but you
> > > are not nearly ready.
> > >
> > > You are going to have to go back-track a long way in what you think you
> > > know about mathematics.  Somewhere along the line in your education,
> > > you've got things badly wrong.  And instead of stopping up and trying to
> > > figure out why everyone else is saying something different from you, or
> > > asking your teachers for help, you have battered on with your mistakes,
> > > leading you to sillier and steadily less logical conclusions.
> > >
> > > I think mathematics is a great hobby.  It's a shame to see someone spend
> > > their time and effort on doing it so badly.
> > >
> >
> > Have you ever wondered why you cannot prove something you hold true for granted
> > for so long?
>
> Yes, regularly.  Sometimes I will then try to find a proof, or look up
> and learn about the proofs.  Sometimes I will have to accept that
> proving the particular thing is beyond my mathematical skills, or my
> time and energy, or my interest, and I will defer to accepting that
> others have proven it.
>
> >
> > If you cannot provide a proof, what you said above only make you more a sinner.
> >
>
> In this particular case, I most certainly /can/ provide a proof.  But I
> can't provide a proof that /you/ would understand.  And since writing a
> proof would be a fair effort, off-topic, and clearly a waste of time
> since you are impervious to mathematical reasoning, I will not bother.
> You can look up such proofs online - I'm sure there are countless
> Youtube videos that will explain it to anyone who is actually interested
> in learning and not merely trying to claim the whole world is wrong
> except them.
>

Not the whole world, you can see some on the internet claiming "0.999...!=1",
although the proof is also invalid. And, in every generation, every kid 
(developed IQ) in school will keep wondering why 1/3=0.333... 'will stop' and
why the the number very close to the left of 1 is not 0.999.... !

Have you wondered if 1/3=0.333..., the conversion algorithm is theoretically flawed?

David Brown <david.brown@hesbynett.no> writes:
[...]
> I think mathematics is a great hobby. It's a shame to see someone
> spend their time and effort on doing it so badly.

It's also a shame to see someone engaging here in a discussion that has
nothing to do with C++. David, if you must feed this particular troll,
I suggest doing so in comp.theory.

*You don't have to reply to everything.*

--
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 3/29/2024 5:03 AM, David Brown wrote:
> On 28/03/2024 20:41, Chris M. Thomasson wrote:
[...]
>> Are you saying that a number constructed digit-by-digit using a TRNG
>> is undefined? Its not a number, however it creates many numbers during
>> the construction process? Fair enough?
>>
>
> Any given finite sequence of digits taken from a TRNG will give you a
> rational number.  But "take 20 digits from a TRNG" does not define a
> number - it defines a procedure for generating numbers.  Do you see the
> difference?

Indeed I do.

>> Ahhh shit, this is just me having some fun. Sorry.
>
> That's no problem.  As long as you are trying to learn, and trying to
> have fun, it's fine by me.

:^)

On 3/29/2024 8:14 AM, wij wrote:
[...]
> Have you ever wondered why you cannot prove something you hold true for granted
> for so long?
>
> If you cannot provide a proof, what you said above only make you more a sinner.

Oh common. You know better. Well, at least I think you do.

On 3/29/2024 9:16 AM, wij wrote:
[...]
> Have you wondered if 1/3=0.333..., the conversion algorithm is theoretically flawed?

Yawn.

On 29/03/2024 17:16, wij wrote:
> On Fri, 2024-03-29 at 16:48 +0100, David Brown wrote:
>> On 29/03/2024 16:14, wij wrote:
>>> On Fri, 2024-03-29 at 11:53 +0100, David Brown wrote:
>>>> On 28/03/2024 19:23, wij wrote:
>>>>> On Thu, 2024-03-28 at 18:16 +0100, David Brown wrote:
>>>>>> On 27/03/2024 17:05, wij wrote:
>>>>>>> On Wed, 2024-03-27 at 16:02 +0100, David Brown wrote:
>>>>>>>> On 27/03/2024 14:32, wij wrote:
>>>>>>>>> On Wed, 2024-03-27 at 13:57 +0100, Ralf Goertz wrote:
>>>>>>>>>> Am Wed, 27 Mar 2024 20:12:38 +0800
>>>>>>>>>> schrieb wij <wyniijj5@gmail.com>:
>>>>>>>>>>
>>>>>>>>>>> On Wed, 2024-03-27 at 12:50 +0100, David Brown wrote:
>>>>>>>>>>>> On 26/03/2024 22:43, wij wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>> Just repeat the pattern infinitely, then it is irrational.
>>>>>>>>>>>>
>>>>>>>>>>>> Nonsense.
>>>>>>>>>>>>
>>>>>>>>>>>>> As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
>>>>>>>>>>>>> cannot terminate, 1/7 != 0.(142857)
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Nonsense.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> I am surprise your math. knowledge is so low worse than teenagers.
>>>>>>>>>>
>>>>>>>>>> Use the standard trick:
>>>>>>>>>>
>>>>>>>>>> x=0.[142857] => 1,000,000*x=142857.[142857]
>>>>>>>>>>
>>>>>>>>>> subtract the first equation from the second:
>>>>>>>>>>
>>>>>>>>>> 999,999*x=142857 => x=142857/999,999=1/7
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> To determine whether a number x is rational or not, we can repeatedly subtract
>>>>>>>>> rational numbers a? from x. If x-a1-a2-a3-...=0 can be verified in finite
>>>>>>>>> steps, then x is rational. Otherwise, x is irrational.
>>>>>>>>> If x is a repeating decimal, proposition "repeating decimal is rational" is
>>>>>>>>> simply false by sematics.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Let me just ask you two simple questions:
>>>>>>>>
>>>>>>>> Do you think 1/7 is a rational number or an irrational number?
>>>>>>>>
>>>>>>> rational
>>>>>>>
>>>>>>>> What do you think the decimal expansion of 1/7 is?
>>>>>>>>
>>>>>>>
>>>>>>> When converting 1/7 to decimal, the result ≒ 0.(142857), the procedure
>>>>>>> never terminates which means the conversion is never complete.
>>>>>>>
>>>>>>
>>>>>> It is a repeating decimal.  If you try to write it all out, then I agree
>>>>>> you will not finish.  That does not mean it is not the decimal expansion
>>>>>> of 1/7 - the list of multiples of (negative) powers of 10 which sum up
>>>>>> to 1/7.  You just need a better notation so that you can finish the task
>>>>>> - and 0.(142857), as you wrote, is one such notation.
>>>>>>
>>>>>> (I have no idea what you think the symbol "≒" might mean.)
>>>>>>
>>>>>> But you agree that 0.(142857) is the decimal expansion of 1/7, even
>>>>>> though you could not write it out long-hand, and you agree that 1/7 i
>>>>>> rational.  And clearly 0.(142857) is a repeating decimal, since that's
>>>>>> what the notation means.
>>>>>>
>>>>>> I can't see how you can still misunderstand this.
>>>>>>
>>>>>
>>>>> You are restating your assertion without proof, again. I have provided mine.
>>>>> (If you say that is you proof, I will say it is invalid).
>>>>>
>>>>>
>>>>
>>>> There is no point in giving you a rigorous proof that 0.(142857) is the
>>>> decimal expansion of 1/7, if that is what you are contesting.  To be
>>>> fully rigorous, it requires an understanding of the definition of the
>>>> real numbers, sequence limits, and the meaning and validity of
>>>> operations on infinite sequences.  You have demonstrated that you don't
>>>> understand any of that.  You have learned a few of the terms, but failed
>>>> to understand the concepts.  Oh, and it also requires understanding what
>>>> a proof is, which again is clearly outside your expertise.
>>>>
>>>> Ralf gave a proof earlier - it is still in the quoted material above.
>>>> That is as good as we can get at your level of mathematical
>>>> understanding.  To be more rigorous, we would need to demonstrate that
>>>> the manipulation (multiplication by a finite integer, and subtraction of
>>>> sequences) of infinite decimal expansions is valid.  That is all
>>>> standard stuff, known to mathematics students the world over, but you
>>>> are not nearly ready.
>>>>
>>>> You are going to have to go back-track a long way in what you think you
>>>> know about mathematics.  Somewhere along the line in your education,
>>>> you've got things badly wrong.  And instead of stopping up and trying to
>>>> figure out why everyone else is saying something different from you, or
>>>> asking your teachers for help, you have battered on with your mistakes,
>>>> leading you to sillier and steadily less logical conclusions.
>>>>
>>>> I think mathematics is a great hobby.  It's a shame to see someone spend
>>>> their time and effort on doing it so badly.
>>>>
>>>
>>> Have you ever wondered why you cannot prove something you hold true for granted
>>> for so long?
>>
>> Yes, regularly.  Sometimes I will then try to find a proof, or look up
>> and learn about the proofs.  Sometimes I will have to accept that
>> proving the particular thing is beyond my mathematical skills, or my
>> time and energy, or my interest, and I will defer to accepting that
>> others have proven it.
>>
>>>
>>> If you cannot provide a proof, what you said above only make you more a sinner.
>>>
>>
>> In this particular case, I most certainly /can/ provide a proof.  But I
>> can't provide a proof that /you/ would understand.  And since writing a
>> proof would be a fair effort, off-topic, and clearly a waste of time
>> since you are impervious to mathematical reasoning, I will not bother.
>> You can look up such proofs online - I'm sure there are countless
>> Youtube videos that will explain it to anyone who is actually interested
>> in learning and not merely trying to claim the whole world is wrong
>> except them.
>>
>
> Not the whole world, you can see some on the internet claiming "0.999...!=1",
> although the proof is also invalid.

Of course there are no valid proofs that 0.999... != 1, since it
0.999... is equal to 1.

But there are folks on the internet claiming the earth is flat, birds
are not real, and every other bit of nonsense you could imagine.

> And, in every generation, every kid
> (developed IQ) in school will keep wondering why 1/3=0.333... 'will stop'

I've never known anyone to wonder that - any kid who learns about this
learns that it does /not/ stop. That's what the three dots mean. But
maybe you didn't write quite what you meant to write here.

> and
> why the the number very close to the left of 1 is not 0.999.... !

Certainly people wonder about things like this. They wonder if 0.999...
really is the same as 1, and how could they prove it. (It /is/ the
same, and the proof is easy.) They wonder if there is a number "just to
the left of 1", and what it might be. (There is no such number.)

On 29/03/2024 19:35, Keith Thompson wrote:
> David Brown <david.brown@hesbynett.no> writes:
> [...]
>> I think mathematics is a great hobby. It's a shame to see someone
>> spend their time and effort on doing it so badly.
>
> It's also a shame to see someone engaging here in a discussion that has
> nothing to do with C++. David, if you must feed this particular troll,
> I suggest doing so in comp.theory.
>
> *You don't have to reply to everything.*
>

It is Easter, and Usenet traffic is low. No, I don't have to reply to
everything (and I don't - I have replied to very few of wij's broken
maths threads), and this thread will soon die away. I am trying to get
some idea of why wij thinks the way he does, and perhaps even help him
think differently (though that's quite optimistic).