Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.lang.c++ > #118554 > unrolled thread

Repeating decimals are irrational

Back to article view | Back to comp.lang.c++

Contents

  Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-26 22:51 +0800
    Re: Repeating decimals are irrational Michael S <already5chosen@yahoo.com> - 2024-03-26 17:11 +0200
      Re: Repeating decimals are irrational Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-03-26 16:13 +0000
      Re: Repeating decimals are irrational Marcel Mueller <news.5.maazl@spamgourmet.org> - 2024-03-27 21:34 +0100
    Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-26 13:13 -0700
      Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-27 05:43 +0800
        Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 12:50 +0100
          Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-27 20:12 +0800
            Re: Repeating decimals are irrational Ralf Goertz <me@myprovider.invalid> - 2024-03-27 13:57 +0100
              Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-27 21:32 +0800
                Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-27 21:49 +0800
                Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 16:01 +0100
                  Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 13:34 -0700
                Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 16:02 +0100
                  Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-28 00:05 +0800
                    Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 13:40 -0700
                      Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-28 05:39 +0800
                        Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 15:10 -0700
                        Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 15:14 -0700
                        Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-28 18:17 +0100
                          Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-29 02:25 +0800
                            Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-29 11:36 +0100
                    Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-28 18:16 +0100
                      Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-29 02:23 +0800
                        Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-29 11:53 +0100
                          Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-29 23:14 +0800
                            Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-29 16:48 +0100
                              Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-30 00:16 +0800
                                Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-29 15:43 -0700
                                Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-30 15:44 +0100
                            Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-29 15:40 -0700
                          Re: Repeating decimals are irrational Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-03-29 11:35 -0700
                            Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-30 15:49 +0100
                              Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-30 23:14 +0800
                                Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-30 19:26 +0100
                                  Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-31 03:30 +0800
                Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 13:29 -0700
            Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 15:51 +0100
            Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 21:52 -0700
      Re: Repeating decimals are irrational Paavo Helde <eesnimi@osa.pri.ee> - 2024-03-26 23:51 +0200
        Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-26 19:42 -0700
          Re: Repeating decimals are irrational Paavo Helde <eesnimi@osa.pri.ee> - 2024-03-27 11:47 +0200
            Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 13:10 +0100
              Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 13:45 -0700
            Re: Repeating decimals are irrational Tim Rentsch <tr.17687@z991.linuxsc.com> - 2024-04-25 16:33 -0700
              Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-04-26 12:46 +0200
      Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-27 11:31 +0100
        Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 13:17 -0700
          Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-28 18:47 +0100
            Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-28 12:41 -0700
              Re: Repeating decimals are irrational David Brown <david.brown@hesbynett.no> - 2024-03-29 13:03 +0100
                Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-29 15:33 -0700
        Re: Repeating decimals are irrational "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-03-27 19:20 -0700
    Re: Repeating decimals are irrational usenet@stegropa.de (Stefan Große Pawig) - 2024-03-28 21:33 +0100
      Re: Repeating decimals are irrational wij <wyniijj5@gmail.com> - 2024-03-29 05:06 +0800

Page 1 of 3  [1] 2 3  Next page →

#118554 — Repeating decimals are irrational

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

[toc] | [next] | [standalone]

#118555

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.

[toc] | [prev] | [next] | [standalone]

#118556

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.

[toc] | [prev] | [next] | [standalone]

#118585

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

[toc] | [prev] | [next] | [standalone]

#118558

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... ;^)

[toc] | [prev] | [next] | [standalone]

#118563

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)

[toc] | [prev] | [next] | [standalone]

#118569

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.

[toc] | [prev] | [next] | [standalone]

#118571

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.
> 
> 

[toc] | [prev] | [next] | [standalone]

#118572

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

[toc] | [prev] | [next] | [standalone]

#118573

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.

[toc] | [prev] | [next] | [standalone]

#118574

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 !!!

[toc] | [prev] | [next] | [standalone]

#118576

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.

[toc] | [prev] | [next] | [standalone]

#118584

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

[toc] | [prev] | [next] | [standalone]

#118577

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?

[toc] | [prev] | [next] | [standalone]

#118580

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.

[toc] | [prev] | [next] | [standalone]

#118586

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?

[toc] | [prev] | [next] | [standalone]

#118589

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.

[toc] | [prev] | [next] | [standalone]

#118590

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.
> 
> 

[toc] | [prev] | [next] | [standalone]

#118591

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.

[toc] | [prev] | [next] | [standalone]

#118608

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.

[toc] | [prev] | [next] | [standalone]

Page 1 of 3  [1] 2 3  Next page →

Back to top | Article view | comp.lang.c++

csiph-web