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

On Tue, 26 Mar 2024 22:51:40 +0800
wij <wyniijj5@gmail.com> wrote:

> Snipet from
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> ...
> Real Nunmber(ℝ)::= {x| x is represented by n-ary
> <fixed_point_number>, the digits may be infinitely long }
>
> Note: This definition implies that repeating decimals are
> irrational number. Let's list a common magic proof in the way as a
> brief explanation: (1) x= 0.999...
> (2) 10x= 9+x // 10x= 9.999...
> (3) 9x=9
> (4) x=1
> Ans: There is no axiom or theorem to prove (1) => (2).
>
> Note: If the steps of converting a number x to
> <fixed_point_number> is not finite, x is not a ratio of two integers,
> because the following statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>
> ---End of quote
>

I don't know what you meant to say, but repeating (a.k.a. periodic)
decimals are most certainly rational numbers. I think that proving it
would be rather easy although I didn't try to do it in rigorous
manner. The idea of proof is multiplying repeating decimal with period P
by (10**P-1) will produce finite decimal. Which is obviously rational.

Michael S <already5chosen@yahoo.com> writes:

> On Tue, 26 Mar 2024 22:51:40 +0800
> wij <wyniijj5@gmail.com> wrote:
<nothing relating to C++>

> I don't know what you meant to say,

Indeed! But it was not about C++, that's for sure. He's riding this
hobby horse around some maths groups, but there's no reason to get
comp.lang.c++ involved.

--
Ben.

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... ;^)

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.
As said "∀x,a∈ℚ, x-a∈ℚ", if the subtraction a= 142857/10^(6*i)
cannot terminate, 1/7 != 0.(142857)

26.03.2024 22:13 Chris M. Thomasson kirjutas:
> 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... ;^)

Any number represented by stored digits on Earth has finite number of
digits (because Earth is finite) and therefore is rational, regardless
of how the digits are generated.

If you want to represent irrational numbers you need to use some other
encoding schema, e.g. "sqrt(2)" (8 bytes, voila!).

On 3/26/2024 2:51 PM, Paavo Helde wrote:
> 26.03.2024 22:13 Chris M. Thomasson kirjutas:
>> 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... ;^)
>
> Any number represented by stored digits on Earth has finite number of
> digits (because Earth is finite) and therefore is rational, regardless
> of how the digits are generated.

Even with a TRNG?

>
> If you want to represent irrational numbers you need to use some other
> encoding schema, e.g. "sqrt(2)" (8 bytes, voila!).

pi? ;^D

27.03.2024 04:42 Chris M. Thomasson kirjutas:
> On 3/26/2024 2:51 PM, Paavo Helde wrote:
>> 26.03.2024 22:13 Chris M. Thomasson kirjutas:
>>> 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... ;^)
>>
>> Any number represented by stored digits on Earth has finite number of
>> digits (because Earth is finite) and therefore is rational, regardless
>> of how the digits are generated.
>
> Even with a TRNG?
>

TRNG is defined as a device having access to a "physical entropy
source". For producing an irrational number it ought to produce an
infinite never-repeating sequence of digits.

It's not clear if any physical mechanism would in principle produce a
non-repeating infinite sequence of digits, the fact that the world is
quantized might rather speak against it. Even if it did, the lifetime of
a physical device is finite, so it can only produce a finite sequence of
digits. Even if the lifetime of the hypothetical device would be
infinite, we do not possess the infinite space and time to store the
result. Such infinities are strictly the territory of mathematics, in
real world we can have only finite sequence of digits.

A finite sequence of digits represents a rational number. For example, a
number which I just constructed by throwing my 10-euro-cent TRNG is
0.10010111011 (in binary) which can be expressed as a ratio of two
integers by trivial point shifting:

0.10010111011 (binary) = 10010111011 (binary) / 2^11 = 1211 / 2048

>
>>
>> If you want to represent irrational numbers you need to use some other
>> encoding schema, e.g. "sqrt(2)" (8 bytes, voila!).
>
> pi? ;^D

In mathematics, a notion like pi actually defines an irrational number
by fixing its properties. For some such irrational numbers it is
possible to give an infinite algorithm which produces the sequence of
its digits. The funny thing is that after fixing the number there is no
randomness any more, so e.g. a machine computing subsequent digits of pi
would make a pretty poor RNG ;-)

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.

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.

Simply stating random things does not make them so.

I recommend you stick to C++ in this C++ newsgroup.

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 10:47, Paavo Helde wrote:
> In mathematics, a notion like pi actually defines an irrational number
> by fixing its properties. For some such irrational numbers it is
> possible to give an infinite algorithm which produces the sequence of
> its digits. The funny thing is that after fixing the number there is no
> randomness any more, so e.g. a machine computing subsequent digits of pi
> would make a pretty poor RNG ;-)
>

The digits would actually make quite a good RNG - if you had a practical
way to keep getting the digits. It is not proven, but it is strongly
suspected that pi is normal in all bases - you will get an even
distribution of digits over time, and any finite sequence of digits will
occur equally often as other sequences of the same length. This is the
key characteristic of an unbiased random source. No test of the
sequence of digits could determine that it is not from a random source.

There is an algorithm that lets you find digits of the hexadecimal
expansion of pi without finding all the previous digits, which could be
useful here.

But it won't be very good for security purposes or other such uses of
random numbers!

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.

> 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.
>
>

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

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.

On Wed, 2024-03-27 at 21:32 +0800, 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.
>
>
By the way, this came to me: ε-δ method was used by people to think that we can 
make x-a approach 0 to arbitrary precision, then conclude that the 'limit' is 0,
'therefore', repeating decimal is rational !!!

On 27/03/2024 13:12, 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.
>

It's a long time ago and hard to be sure, but I believe I knew that real
numbers were rational if and only if their decimal expansion was
repeating before I was a teenager. And I don't believe that rational
numbers or decimal expansions have changed their nature since then.

(And as a maths student I wrote an essay on a derivation of the real
numbers from the axioms of Zermelo-Fraenkel set theory, proving each
step along the way, proving the equivalence between constructions from
decimal - or n-ary - representations to constructions based on
completing the rationals, proving the existence of irrational numbers,
and establishing the cardinality of the real numbers. I might be a bit
out of practice, but I know what I am talking about.)

If you hold strange views on a mathematical topic that runs contrary to
the established mainstream, especially something so simple and
non-contentious, you have to be prepared to be treated as a fool and
ridiculed as a flat-earther or a trisector.

But just for your benefit, I had a quick look at the start of your
"RealNumber-en.txt" file - the section on Real Numbers.

1. /You/ don't get to define real numbers. That is a well-established
term in mathematics, and you don't get to replace it with waffle-worded
text.

2. In mathematics, you don't get to say "a definition is not provided"
and then "this definition implies ...". Make /rigorous/ definitions,
and /prove/ their definitions. What you are writing here is not
mathematics - it's a C-grade response to a high-school test question
"What is a real number?".

3. You cannot claim what this definition (such as it is) says about
irrational numbers, when you have not said what rational or irrational
numbers are. Since you are using your own broken definition for reals,
who knows what mistaken ideas you might have about rationals and
irrationals.

4. Arithmetic on repeating decimals is well defined, and if x = 0.999...
then 10.x is 9 + x. This /is/ provable.

5. Please don't try and talk about axioms. You are /very/ far from that
level of rigour.

6. Even if your claim that 0.999... != 1 were true, and even if you had
proven it, it would not have the implications you are claiming.

7. Noting that the rationals are closed under subtraction has not the
slightest bearing on anything that you have been claiming. It's
plausible that it might be involved in a step of the proof of your
claims - if such a proof were possible. But you haven't even made the
vaguest suggestions of a proof - you simply throw out your claims and
expect them to be believed.

(The rest of the document is too jumbled and unclear to critique. I
appreciate that English is not your first language, but you seem to be
able to write it well enough when you try, so I blame your mathematics,
not your language skills.)

I expect you've been told all of this before.

>> 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.

I have not "occupied" anything. This is a newsgroup primarily concerned
with discussions of C++ - that's in the name of the group.

I also recommend you stick to C++ because you apparently have an
interest in and knowledge of C++, and as far as I have noticed, you talk
sensibly about the language.

> For now, this discussion is mainly in comp.theory

You started a new thread in comp.lang.c++. I don't know what there
might or might not be in comp.theory - anything there is irrelevant to
this discussion. (I can't see how your post is remotely on-topic for
computational theory either.)

> But you have shown your knowledge is so so low, don't go there waste our time.

I don't follow comp.theory. But I have seen a few threads over the
years which have "leaked" from there to groups that I do follow, and I
feel confident in guessing that few people there share your ideas about
real numbers.

>
>> 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 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.

I assume that when you say "rational numbers a?", you mean numbers with
finite decimal expansions?

Your method could, I suppose, be used to prove that x is rational - but
not to prove that it is irrational. It is not particularly helpful,
unless you are using it as some way to build up the rationals
inductively from a starting point of "assumed" rationals.

> If x-a1-a2-a3-...=0 can be verified in finite
> steps, then x is rational.

Correct.

> Otherwise, x is irrational.

Incorrect.

All you have proven is that you have not picked appropriate rationals in
the sequence, or that x is a number with a non-finite decimal expansion.
You haven't demonstrated that it is irrational.

Your method here doesn't give you anything new. It boils down to saying
that if we assume that all rationals have finite decimal expansions, we
can prove that numbers without finite decimal expansions are not
rational - and that's a simple tautology. The assumption is, of course,
wrong.

> If x is a repeating decimal, proposition "repeating decimal is rational" is
> simply false by sematics.
>

Incorrect.

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?

What do you think the decimal expansion of 1/7 is?

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.

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? For some reason it makes me think of infinite
convergents of continued fractions. Can we gain a rational that can
approximate any irrational up to a certain precision, so to speak?

On 3/27/2024 6:32 AM, 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,[...]

Check for a period...

On 3/27/2024 8:01 AM, 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.
>
> I assume that when you say "rational numbers a?", you mean numbers with
> finite decimal expansions?
>
> Your method could, I suppose, be used to prove that x is rational - but
> not to prove that it is irrational.  It is not particularly helpful,
> unless you are using it as some way to build up the rationals
> inductively from a starting point of "assumed" rationals.
>
>> If x-a1-a2-a3-...=0 can be verified in finite
>> steps, then x is rational.
>
> Correct.
>
>> Otherwise, x is irrational.
>
> Incorrect.
>
> All you have proven is that you have not picked appropriate rationals in
> the sequence, or that x is a number with a non-finite decimal expansion.
>  You haven't demonstrated that it is irrational.
>
> Your method here doesn't give you anything new.  It boils down to saying
> that if we assume that all rationals have finite decimal expansions, we
> can prove that numbers without finite decimal expansions are not
> rational - and that's a simple tautology.  The assumption is, of course,
> wrong.
>
>
>> If x is a repeating decimal, proposition "repeating decimal is
>> rational" is
>> simply false by sematics.
>>
>
> Incorrect.
>
>
>

Think of doing long division, as soon as you hit a period (aka the same
number), you can stop. Fair enough?

base 10:

1 / 3 = .(3), .3..., whatever

We will notice during long division that a period has been encountered,
there we can stop iteration, and say .3 repeating. Fair enough? Its fun
to record how many steps to took to hit a period. Fun... Akin to escape
time fractals wrt how many iterations it took for a number to escape a
given limit, so to speak.

Make any sense? Or stupid? ;^o

Am 26.03.24 um 16:11 schrieb Michael S:
> I don't know what you meant to say, but repeating (a.k.a. periodic)
> decimals are most certainly rational numbers.

Exactly.

> I think that proving it
> would be rather easy although I didn't try to do it in rigorous
> manner.

They are always rational numbers because any repeated sequence is just
equivalent to a denominator withe base^length - 1, e.g.
0,(142857) = 142857 / 999999

> The idea of proof is multiplying repeating decimal with period P
> by (10**P-1) will produce finite decimal. Which is obviously rational.

Indeed.

Marcel

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?

On 3/27/2024 5:10 AM, David Brown wrote:
> On 27/03/2024 10:47, Paavo Helde wrote:
>> In mathematics, a notion like pi actually defines an irrational number
>> by fixing its properties. For some such irrational numbers it is
>> possible to give an infinite algorithm which produces the sequence of
>> its digits. The funny thing is that after fixing the number there is
>> no randomness any more, so e.g. a machine computing subsequent digits
>> of pi would make a pretty poor RNG ;-)
>>
>
> The digits would actually make quite a good RNG - if you had a practical
> way to keep getting the digits.

Fwiw, here is an example of a highly non-practical way to reap base 2
symbols to construct a number:

https://groups.google.com/g/comp.lang.c++/c/7u_rLgQe86k/m/fYU9SnuAFQAJ

;^D

> It is not proven, but it is strongly
> suspected that pi is normal in all bases - you will get an even
> distribution of digits over time, and any finite sequence of digits will
> occur equally often as other sequences of the same length.  This is the
> key characteristic of an unbiased random source.  No test of the
> sequence of digits could determine that it is not from a random source.
>
> There is an algorithm that lets you find digits of the hexadecimal
> expansion of pi without finding all the previous digits, which could be
> useful here.
>
> But it won't be very good for security purposes or other such uses of
> random numbers!
>
>