Richard says:
> When you use long division to divide 1 by 3, the resultant quotient is
> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> This fact forms the equality:
>
> 1/3= 0.333... + non_zero_remainder

But at the limit of infinite digits, the remainder becomes smaller than
any number and becomes zero, so isn't 'non-zero'.

>
> Is this correct? If so, from the equation above we can deduce
> 1/3 - 0.333...= non_zero_remainder

Except that at in infinite number of digits, the remainder is goes to
zero, as The Reals don't represent infinitesimal values.

>
> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> 1/3 ≠ 0.333...
>
> Is there any flaw?

The fact that while for a FINITE number of digtis, you have a non-zero
result, but as you add more and more digits the value shrinks, and
shrinks fast enough that when we get to the infinite number of digits
(in the limit) it becomes 0.

Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
when you invoke it, you invoke the limit property, and the limit of that
remainder is zero, and of 0.999... is 1.

---
1/3= 0.333... + non_zero_remainder ?

On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> Richard says:
> > When you use long division to divide 1 by 3, the resultant quotient is
> > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > This fact forms the equality:
> >
> > 1/3= 0.333... + non_zero_remainder
>
> But at the limit of infinite digits, the remainder becomes smaller than
> any number and becomes zero, so isn't 'non-zero'.
>
> >
> > Is this correct? If so, from the equation above we can deduce
> > 1/3 - 0.333...= non_zero_remainder
>
> Except that at in infinite number of digits, the remainder is goes to
> zero, as The Reals don't represent infinitesimal values.
>
> >
> > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > 1/3 ≠ 0.333...
> >
> > Is there any flaw?
>
> The fact that while for a FINITE number of digtis, you have a non-zero
> result, but as you add more and more digits the value shrinks, and
> shrinks fast enough that when we get to the infinite number of digits
> (in the limit) it becomes 0.
>
> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> when you invoke it, you invoke the limit property, and the limit of that
> remainder is zero, and of 0.999... is 1.
>
> ---
> 1/3= 0.333... + non_zero_remainder ?

You've narrowed your context a bit too quickly I think.
I recommend you cast your net wider.
Consider closure and operator theory.
You will destroy the rational value in the process.

On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> > Richard says:
> > > When you use long division to divide 1 by 3, the resultant quotient is
> > > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > > This fact forms the equality:
> > >
> > > 1/3= 0.333... + non_zero_remainder
> >
> > But at the limit of infinite digits, the remainder becomes smaller than
> > any number and becomes zero, so isn't 'non-zero'.
> >
> > >
> > > Is this correct? If so, from the equation above we can deduce
> > > 1/3 - 0.333...= non_zero_remainder
> >
> > Except that at in infinite number of digits, the remainder is goes to
> > zero, as The Reals don't represent infinitesimal values.
> >
> > >
> > > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > > 1/3 ≠ 0.333...
> > >
> > > Is there any flaw?
> >
> > The fact that while for a FINITE number of digtis, you have a non-zero
> > result, but as you add more and more digits the value shrinks, and
> > shrinks fast enough that when we get to the infinite number of digits
> > (in the limit) it becomes 0.
> >
> > Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> > when you invoke it, you invoke the limit property, and the limit of that
> > remainder is zero, and of 0.999... is 1.
> >
> > ---
> > 1/3= 0.333... + non_zero_remainder ?
> You've narrowed your context a bit too quickly I think.
> I recommend you cast your net wider.
> Consider closure and operator theory.
> You will destroy the rational value in the process.

Q: 1/3= 0.333... + non_zero_remainder ?

True or false and for what reason (proof)? Idiot.

On 12/18/2021 2:47 AM, wij wrote:
> Richard says:
>> When you use long division to divide 1 by 3, the resultant quotient is
>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
>> This fact forms the equality:
>>
>> 1/3= 0.333... + non_zero_remainder
>
> But at the limit of infinite digits, the remainder becomes smaller than
> any number and becomes zero, so isn't 'non-zero'.

mistake. you have to stop or ignore the three dots ... to conclude that

>
>>
>> Is this correct? If so, from the equation above we can deduce
>> 1/3 - 0.333...= non_zero_remainder
>
> Except that at in infinite number of digits, the remainder is goes to
> zero, as The Reals don't represent infinitesimal values.

mistake.

>
>>
>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>> 1/3 ≠ 0.333...
>>
>> Is there any flaw?
>
> The fact that while for a FINITE number of digtis, you have a non-zero
> result, but as you add more and more digits the value shrinks, and
> shrinks fast enough that when we get to the infinite number of digits
> (in the limit) it becomes 0.

almost right, remove "fast enough". there are other ways to word it, it is a limit.

>
> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> when you invoke it, you invoke the limit property, and the limit of that
> remainder is zero, and of 0.999... is 1.

however you can map the real line including infinity onto a circle, or complex plane onto a sphere where infinity is a point on the sphere.

>
> ---
> 1/3= 0.333... + non_zero_remainder ?

true.

On Saturday, December 18, 2021 at 12:47:40 AM UTC-8, wij wrote:
> Richard says:
> > When you use long division to divide 1 by 3, the resultant quotient is
> > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > This fact forms the equality:
> >
> > 1/3= 0.333... + non_zero_remainder
>
> But at the limit of infinite digits, the remainder becomes smaller than
> any number and becomes zero, so isn't 'non-zero'.
>
> >
> > Is this correct? If so, from the equation above we can deduce
> > 1/3 - 0.333...= non_zero_remainder
>
> Except that at in infinite number of digits, the remainder is goes to
> zero, as The Reals don't represent infinitesimal values.
>
> >
> > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > 1/3 ≠ 0.333...
> >
> > Is there any flaw?
>
> The fact that while for a FINITE number of digtis, you have a non-zero
> result, but as you add more and more digits the value shrinks, and
> shrinks fast enough that when we get to the infinite number of digits
> (in the limit) it becomes 0.
>
> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> when you invoke it, you invoke the limit property, and the limit of that
> remainder is zero, and of 0.999... is 1.
>
> ---
> 1/3= 0.333... + non_zero_remainder ?

It is nonzero but closest to it...

Mitchell Raemsch

lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> > On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> > > Richard says:
> > > > When you use long division to divide 1 by 3, the resultant quotient is
> > > > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > > > This fact forms the equality:
> > > >
> > > > 1/3= 0.333... + non_zero_remainder
> > >
> > > But at the limit of infinite digits, the remainder becomes smaller than
> > > any number and becomes zero, so isn't 'non-zero'.
> > >
> > > >
> > > > Is this correct? If so, from the equation above we can deduce
> > > > 1/3 - 0.333...= non_zero_remainder
> > >
> > > Except that at in infinite number of digits, the remainder is goes to
> > > zero, as The Reals don't represent infinitesimal values.
> > >
> > > >
> > > > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > > > 1/3 ≠ 0.333...
> > > >
> > > > Is there any flaw?
> > >
> > > The fact that while for a FINITE number of digtis, you have a non-zero
> > > result, but as you add more and more digits the value shrinks, and
> > > shrinks fast enough that when we get to the infinite number of digits
> > > (in the limit) it becomes 0.
> > >
> > > Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> > > when you invoke it, you invoke the limit property, and the limit of that
> > > remainder is zero, and of 0.999... is 1.
> > >
> > > ---
> > > 1/3= 0.333... + non_zero_remainder ?
> > You've narrowed your context a bit too quickly I think.
> > I recommend you cast your net wider.
> > Consider closure and operator theory.
> > You will destroy the rational value in the process.
> Q: 1/3= 0.333... + non_zero_remainder ?
>
> True or false and for what reason (proof)? Idiot.
1/3=0.333....

Thee proofs are endless

On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
>>>> Richard says:
>>>>> When you use long division to divide 1 by 3, the resultant quotient is
>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
>>>>> This fact forms the equality:
>>>>>
>>>>> 1/3= 0.333... + non_zero_remainder
>>>>
>>>> But at the limit of infinite digits, the remainder becomes smaller than
>>>> any number and becomes zero, so isn't 'non-zero'.
>>>>
>>>>>
>>>>> Is this correct? If so, from the equation above we can deduce
>>>>> 1/3 - 0.333...= non_zero_remainder
>>>>
>>>> Except that at in infinite number of digits, the remainder is goes to
>>>> zero, as The Reals don't represent infinitesimal values.
>>>>
>>>>>
>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>>>>> 1/3 ≠ 0.333...
>>>>>
>>>>> Is there any flaw?
>>>>
>>>> The fact that while for a FINITE number of digtis, you have a non-zero
>>>> result, but as you add more and more digits the value shrinks, and
>>>> shrinks fast enough that when we get to the infinite number of digits
>>>> (in the limit) it becomes 0.
>>>>
>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
>>>> when you invoke it, you invoke the limit property, and the limit of that
>>>> remainder is zero, and of 0.999... is 1.
>>>>
>>>> ---
>>>> 1/3= 0.333... + non_zero_remainder ?
>>> You've narrowed your context a bit too quickly I think.
>>> I recommend you cast your net wider.
>>> Consider closure and operator theory.
>>> You will destroy the rational value in the process.
>> Q: 1/3= 0.333... + non_zero_remainder ?
>>
>> True or false and for what reason (proof)? Idiot.
> 1/3=0.333....
>
> Thee proofs are endless

1/3 can be represented as .333... in base 10. Therefore:

..333... * 3 = 1, as 1/3*3=1

On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
> > lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
> >> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> >>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> >>>> Richard says:
> >>>>> When you use long division to divide 1 by 3, the resultant quotient is
> >>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> >>>>> This fact forms the equality:
> >>>>>
> >>>>> 1/3= 0.333... + non_zero_remainder
> >>>>
> >>>> But at the limit of infinite digits, the remainder becomes smaller than
> >>>> any number and becomes zero, so isn't 'non-zero'.
> >>>>
> >>>>>
> >>>>> Is this correct? If so, from the equation above we can deduce
> >>>>> 1/3 - 0.333...= non_zero_remainder
> >>>>
> >>>> Except that at in infinite number of digits, the remainder is goes to
> >>>> zero, as The Reals don't represent infinitesimal values.
> >>>>
> >>>>>
> >>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> >>>>> 1/3 ≠ 0.333...
> >>>>>
> >>>>> Is there any flaw?
> >>>>
> >>>> The fact that while for a FINITE number of digtis, you have a non-zero
> >>>> result, but as you add more and more digits the value shrinks, and
> >>>> shrinks fast enough that when we get to the infinite number of digits
> >>>> (in the limit) it becomes 0.
> >>>>
> >>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> >>>> when you invoke it, you invoke the limit property, and the limit of that
> >>>> remainder is zero, and of 0.999... is 1.
> >>>>
> >>>> ---
> >>>> 1/3= 0.333... + non_zero_remainder ?
> >>> You've narrowed your context a bit too quickly I think.
> >>> I recommend you cast your net wider.
> >>> Consider closure and operator theory.
> >>> You will destroy the rational value in the process.
> >> Q: 1/3= 0.333... + non_zero_remainder ?
> >>
> >> True or false and for what reason (proof)? Idiot.
> > 1/3=0.333....
> >
> > Thee proofs are endless
> 1/3 can be represented as .333... in base 10. Therefore:
>
> .333... * 3 = 1, as 1/3*3=1

1/3= 0.333... + non_zero_remainder
0.333...= 1/3 - non_zero_remainder

--- Pythagoreans' Code ---
Infinitely approaching means equal. Number too small equals zero.
Any real number can be approached by ratio number.

On 12/20/2021 9:56 AM, wij wrote:
> On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
>> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
>>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
>>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
>>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
>>>>>> Richard says:
>>>>>>> When you use long division to divide 1 by 3, the resultant quotient is
>>>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
>>>>>>> This fact forms the equality:
>>>>>>>
>>>>>>> 1/3= 0.333... + non_zero_remainder
>>>>>>
>>>>>> But at the limit of infinite digits, the remainder becomes smaller than
>>>>>> any number and becomes zero, so isn't 'non-zero'.
>>>>>>
>>>>>>>
>>>>>>> Is this correct? If so, from the equation above we can deduce
>>>>>>> 1/3 - 0.333...= non_zero_remainder
>>>>>>
>>>>>> Except that at in infinite number of digits, the remainder is goes to
>>>>>> zero, as The Reals don't represent infinitesimal values.
>>>>>>
>>>>>>>
>>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>>>>>>> 1/3 ≠ 0.333...
>>>>>>>
>>>>>>> Is there any flaw?
>>>>>>
>>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
>>>>>> result, but as you add more and more digits the value shrinks, and
>>>>>> shrinks fast enough that when we get to the infinite number of digits
>>>>>> (in the limit) it becomes 0.
>>>>>>
>>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
>>>>>> when you invoke it, you invoke the limit property, and the limit of that
>>>>>> remainder is zero, and of 0.999... is 1.
>>>>>>
>>>>>> ---
>>>>>> 1/3= 0.333... + non_zero_remainder ?
>>>>> You've narrowed your context a bit too quickly I think.
>>>>> I recommend you cast your net wider.
>>>>> Consider closure and operator theory.
>>>>> You will destroy the rational value in the process.
>>>> Q: 1/3= 0.333... + non_zero_remainder ?
>>>>
>>>> True or false and for what reason (proof)? Idiot.
>>> 1/3=0.333....
>>>
>>> Thee proofs are endless
>> 1/3 can be represented as .333... in base 10. Therefore:
>>
>> .333... * 3 = 1, as 1/3*3=1
>
> 1/3= 0.333... + non_zero_remainder
> 0.333...= 1/3 - non_zero_remainder
>
> --- Pythagoreans' Code ---
> Infinitely approaching means equal. Number too small equals zero.
> Any real number can be approached by ratio number.

the representation of 0.333... *is not approaching 1/3*, it is merely an abbreviation of
0.33333333333333333333333333333333333333333333333333333333333333333333
333333333333333333333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333333333...

On Tuesday, 21 December 2021 at 00:04:49 UTC+8, Serg io wrote:
> On 12/20/2021 9:56 AM, wij wrote:
> > On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
> >> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
> >>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
> >>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> >>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> >>>>>> Richard says:
> >>>>>>> When you use long division to divide 1 by 3, the resultant quotient is
> >>>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> >>>>>>> This fact forms the equality:
> >>>>>>>
> >>>>>>> 1/3= 0.333... + non_zero_remainder
> >>>>>>
> >>>>>> But at the limit of infinite digits, the remainder becomes smaller than
> >>>>>> any number and becomes zero, so isn't 'non-zero'.
> >>>>>>
> >>>>>>>
> >>>>>>> Is this correct? If so, from the equation above we can deduce
> >>>>>>> 1/3 - 0.333...= non_zero_remainder
> >>>>>>
> >>>>>> Except that at in infinite number of digits, the remainder is goes to
> >>>>>> zero, as The Reals don't represent infinitesimal values.
> >>>>>>
> >>>>>>>
> >>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> >>>>>>> 1/3 ≠ 0.333...
> >>>>>>>
> >>>>>>> Is there any flaw?
> >>>>>>
> >>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
> >>>>>> result, but as you add more and more digits the value shrinks, and
> >>>>>> shrinks fast enough that when we get to the infinite number of digits
> >>>>>> (in the limit) it becomes 0.
> >>>>>>
> >>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> >>>>>> when you invoke it, you invoke the limit property, and the limit of that
> >>>>>> remainder is zero, and of 0.999... is 1.
> >>>>>>
> >>>>>> ---
> >>>>>> 1/3= 0.333... + non_zero_remainder ?
> >>>>> You've narrowed your context a bit too quickly I think.
> >>>>> I recommend you cast your net wider.
> >>>>> Consider closure and operator theory.
> >>>>> You will destroy the rational value in the process.
> >>>> Q: 1/3= 0.333... + non_zero_remainder ?
> >>>>
> >>>> True or false and for what reason (proof)? Idiot.
> >>> 1/3=0.333....
> >>>
> >>> Thee proofs are endless
> >> 1/3 can be represented as .333... in base 10. Therefore:
> >>
> >> .333... * 3 = 1, as 1/3*3=1
> >
> > 1/3= 0.333... + non_zero_remainder
> > 0.333...= 1/3 - non_zero_remainder
> >
> > --- Pythagoreans' Code ---
> > Infinitely approaching means equal. Number too small equals zero.
> > Any real number can be approached by ratio number.
> the representation of 0.333... *is not approaching 1/3*, it is merely an abbreviation of
> 0.33333333333333333333333333333333333333333333333333333333333333333333
> 333333333333333333333333333333333333333333333333333333333333333333333
> 33333333333333333333333333333333333333333333333333333333333...

The point is the representation of 0.333... is indeterminant. As is written, I assume you mean
Σ(n=1,∞) 3/10^n, you can try to add the sum infinitely, the result will never be exact 1/3.

On Sunday, December 19, 2021 at 9:55:01 PM UTC-8, zelos...@gmail.com wrote:
> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
> > On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> > > On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> > > > Richard says:
> > > > > When you use long division to divide 1 by 3, the resultant quotient is
> > > > > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > > > > This fact forms the equality:
> > > > >
> > > > > 1/3= 0.333... + non_zero_remainder
> > > >
> > > > But at the limit of infinite digits, the remainder becomes smaller than
> > > > any number and becomes zero, so isn't 'non-zero'.
> > > >
> > > > >
> > > > > Is this correct? If so, from the equation above we can deduce
> > > > > 1/3 - 0.333...= non_zero_remainder
> > > >
> > > > Except that at in infinite number of digits, the remainder is goes to
> > > > zero, as The Reals don't represent infinitesimal values.
> > > >
> > > > >
> > > > > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > > > > 1/3 ≠ 0.333...
> > > > >
> > > > > Is there any flaw?
> > > >
> > > > The fact that while for a FINITE number of digtis, you have a non-zero
> > > > result, but as you add more and more digits the value shrinks, and
> > > > shrinks fast enough that when we get to the infinite number of digits
> > > > (in the limit) it becomes 0.
> > > >
> > > > Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> > > > when you invoke it, you invoke the limit property, and the limit of that
> > > > remainder is zero, and of 0.999... is 1.
> > > >
> > > > ---
> > > > 1/3= 0.333... + non_zero_remainder ?
> > > You've narrowed your context a bit too quickly I think.
> > > I recommend you cast your net wider.
> > > Consider closure and operator theory.
> > > You will destroy the rational value in the process.
> > Q: 1/3= 0.333... + non_zero_remainder ?
> >
> > True or false and for what reason (proof)? Idiot.
> 1/3=0.333....
>
> Thee proofs are endless

You don't have a real proof.
You are just brainwashed about zero.

Mitchell Raemsch

On 12/20/2021 10:16 AM, wij wrote:
> On Tuesday, 21 December 2021 at 00:04:49 UTC+8, Serg io wrote:
>> On 12/20/2021 9:56 AM, wij wrote:
>>> On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
>>>> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
>>>>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
>>>>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
>>>>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
>>>>>>>> Richard says:
>>>>>>>>> When you use long division to divide 1 by 3, the resultant quotient is
>>>>>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
>>>>>>>>> This fact forms the equality:
>>>>>>>>>
>>>>>>>>> 1/3= 0.333... + non_zero_remainder
>>>>>>>>
>>>>>>>> But at the limit of infinite digits, the remainder becomes smaller than
>>>>>>>> any number and becomes zero, so isn't 'non-zero'.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> Is this correct? If so, from the equation above we can deduce
>>>>>>>>> 1/3 - 0.333...= non_zero_remainder
>>>>>>>>
>>>>>>>> Except that at in infinite number of digits, the remainder is goes to
>>>>>>>> zero, as The Reals don't represent infinitesimal values.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>>>>>>>>> 1/3 ≠ 0.333...
>>>>>>>>>
>>>>>>>>> Is there any flaw?
>>>>>>>>
>>>>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
>>>>>>>> result, but as you add more and more digits the value shrinks, and
>>>>>>>> shrinks fast enough that when we get to the infinite number of digits
>>>>>>>> (in the limit) it becomes 0.
>>>>>>>>
>>>>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
>>>>>>>> when you invoke it, you invoke the limit property, and the limit of that
>>>>>>>> remainder is zero, and of 0.999... is 1.
>>>>>>>>
>>>>>>>> ---
>>>>>>>> 1/3= 0.333... + non_zero_remainder ?
>>>>>>> You've narrowed your context a bit too quickly I think.
>>>>>>> I recommend you cast your net wider.
>>>>>>> Consider closure and operator theory.
>>>>>>> You will destroy the rational value in the process.
>>>>>> Q: 1/3= 0.333... + non_zero_remainder ?
>>>>>>
>>>>>> True or false and for what reason (proof)? Idiot.
>>>>> 1/3=0.333....
>>>>>
>>>>> Thee proofs are endless
>>>> 1/3 can be represented as .333... in base 10. Therefore:
>>>>
>>>> .333... * 3 = 1, as 1/3*3=1
>>>
>>> 1/3= 0.333... + non_zero_remainder
>>> 0.333...= 1/3 - non_zero_remainder
>>>
>>> --- Pythagoreans' Code ---
>>> Infinitely approaching means equal. Number too small equals zero.
>>> Any real number can be approached by ratio number.
>> the representation of 0.333... *is not approaching 1/3*, it is merely an abbreviation of
>> 0.33333333333333333333333333333333333333333333333333333333333333333333
>> 333333333333333333333333333333333333333333333333333333333333333333333
>> 33333333333333333333333333333333333333333333333333333333333...
>
> The point is the representation of 0.333... is indeterminant.

Wrong.

> As is written, I assume you mean
> Σ(n=1,∞) 3/10^n, you can try to add the sum infinitely, the result will never be exact 1/3.

no, I mean 0.333... = 3/9, it is an exact number.
you and your summation are silly.

On 12/20/2021 11:47 AM, mitchr...@gmail.com wrote:
> On Sunday, December 19, 2021 at 9:55:01 PM UTC-8, zelos...@gmail.com wrote:
>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
>>>>> Richard says:
>>>>>> When you use long division to divide 1 by 3, the resultant quotient is
>>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
>>>>>> This fact forms the equality:
>>>>>>
>>>>>> 1/3= 0.333... + non_zero_remainder
>>>>>
>>>>> But at the limit of infinite digits, the remainder becomes smaller than
>>>>> any number and becomes zero, so isn't 'non-zero'.
>>>>>
>>>>>>
>>>>>> Is this correct? If so, from the equation above we can deduce
>>>>>> 1/3 - 0.333...= non_zero_remainder
>>>>>
>>>>> Except that at in infinite number of digits, the remainder is goes to
>>>>> zero, as The Reals don't represent infinitesimal values.
>>>>>
>>>>>>
>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>>>>>> 1/3 ≠ 0.333...
>>>>>>
>>>>>> Is there any flaw?
>>>>>
>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
>>>>> result, but as you add more and more digits the value shrinks, and
>>>>> shrinks fast enough that when we get to the infinite number of digits
>>>>> (in the limit) it becomes 0.
>>>>>
>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
>>>>> when you invoke it, you invoke the limit property, and the limit of that
>>>>> remainder is zero, and of 0.999... is 1.
>>>>>
>>>>> ---
>>>>> 1/3= 0.333... + non_zero_remainder ?
>>>> You've narrowed your context a bit too quickly I think.
>>>> I recommend you cast your net wider.
>>>> Consider closure and operator theory.
>>>> You will destroy the rational value in the process.
>>> Q: 1/3= 0.333... + non_zero_remainder ?
>>>
>>> True or false and for what reason (proof)? Idiot.
>> 1/3=0.333....
>>
>> Thee proofs are endless
>
> You don't have a real proof.
> You are just brainwashed about zero.
>
>
> Mitchell Raemsch

agree, he has zero brainwashing

wij schrieb am Samstag, 18. Dezember 2021 um 14:40:17 UTC+1:

> Q: 1/3= 0.333... + non_zero_remainder ?
>
> True or false and for what reason (proof)?

Nobody would trust in 3.25 = 1 + 1 + 1 + 1 - 1 + 1 - 1 +-... Why should we trust in pi = 3.1415...? To write pi as a decimal fraction is as impossible as to write 0 as a positive fraction. Same is true for 0.333... There is no remainder, but provably no digit finishes the task, not even if there were ℵo digits. 1/3 is the limit never reached. And never means never, not "in the infinite"!

Regards, WM

On Tuesday, 21 December 2021 at 02:57:30 UTC+8, WM wrote:
> wij schrieb am Samstag, 18. Dezember 2021 um 14:40:17 UTC+1:
>
> > Q: 1/3= 0.333... + non_zero_remainder ?
> >
> > True or false and for what reason (proof)?
> Nobody would trust in 3.25 = 1 + 1 + 1 + 1 - 1 + 1 - 1 +-... Why should we trust in pi = 3.1415...? To write pi as a decimal fraction is as impossible as to write 0 as a positive fraction. Same is true for 0.333... There is no remainder, but provably no digit finishes the task, not even if there were ℵo digits. 1/3 is the limit never reached. And never means never, not "in the infinite"!
>
> Regards, WM

If it is infinite, you are stuck in an infinite loop, you have to break the
loop and find reason/symbol for it.

1/3= 0.333... + non_zero_remainder.

If no non_zero_remainder, you won't have infinite repeating 3.

wij expressed precisely :
> On Tuesday, 21 December 2021 at 00:04:49 UTC+8, Serg io wrote:
>> On 12/20/2021 9:56 AM, wij wrote:
>>> On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
>>>> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
>>>>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
>>>>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com
>>>>>> wrote:
>>>>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
>>>>>>>> Richard says:
>>>>>>>>> When you use long division to divide 1 by 3, the resultant quotient
>>>>>>>>> is accumulated by a repeating '3' (the quotient is 0.333...) and a
>>>>>>>>> remainder '1', This fact forms the equality:
>>>>>>>>>
>>>>>>>>> 1/3= 0.333... + non_zero_remainder
>>>>>>>>
>>>>>>>> But at the limit of infinite digits, the remainder becomes smaller
>>>>>>>> than any number and becomes zero, so isn't 'non-zero'.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> Is this correct? If so, from the equation above we can deduce
>>>>>>>>> 1/3 - 0.333...= non_zero_remainder
>>>>>>>>
>>>>>>>> Except that at in infinite number of digits, the remainder is goes to
>>>>>>>> zero, as The Reals don't represent infinitesimal values.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
>>>>>>>>> 1/3 ≠ 0.333...
>>>>>>>>>
>>>>>>>>> Is there any flaw?
>>>>>>>>
>>>>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
>>>>>>>> result, but as you add more and more digits the value shrinks, and
>>>>>>>> shrinks fast enough that when we get to the infinite number of digits
>>>>>>>> (in the limit) it becomes 0.
>>>>>>>>
>>>>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit,
>>>>>>>> and when you invoke it, you invoke the limit property, and the limit
>>>>>>>> of that remainder is zero, and of 0.999... is 1.
>>>>>>>>
>>>>>>>> ---
>>>>>>>> 1/3= 0.333... + non_zero_remainder ?
>>>>>>> You've narrowed your context a bit too quickly I think.
>>>>>>> I recommend you cast your net wider.
>>>>>>> Consider closure and operator theory.
>>>>>>> You will destroy the rational value in the process.
>>>>>> Q: 1/3= 0.333... + non_zero_remainder ?
>>>>>>
>>>>>> True or false and for what reason (proof)? Idiot.
>>>>> 1/3=0.333....
>>>>>
>>>>> Thee proofs are endless
>>>> 1/3 can be represented as .333... in base 10. Therefore:
>>>>
>>>> .333... * 3 = 1, as 1/3*3=1
>>>
>>> 1/3= 0.333... + non_zero_remainder
>>> 0.333...= 1/3 - non_zero_remainder
>>>
>>> --- Pythagoreans' Code ---
>>> Infinitely approaching means equal. Number too small equals zero.
>>> Any real number can be approached by ratio number.
>> the representation of 0.333... *is not approaching 1/3*, it is merely an
>> abbreviation of
>> 0.33333333333333333333333333333333333333333333333333333333333333333333
>> 333333333333333333333333333333333333333333333333333333333333333333333
>> 33333333333333333333333333333333333333333333333333333333333...
>
> The point is the representation of 0.333... is indeterminant. As is written,
> I assume you mean Σ(n=1,∞) 3/10^n, you can try to add the sum infinitely, the
> result will never be exact 1/3.

Because you don't understand what a "Series" is. It is summed all at
once, not in steps. Steps and order can be used for better and better
rational approximations.

On 12/20/2021 12:57 PM, WM wrote:
> wij schrieb am Samstag, 18. Dezember 2021 um 14:40:17 UTC+1:
>
>> Q: 1/3= 0.333... + non_zero_remainder ?
>>
>> True or false and for what reason (proof)?
>
> Nobody would trust in 3.25 = 1 + 1 + 1 + 1 - 1 + 1 - 1 +-... Why should we trust in pi = 3.1415...? To write pi as a decimal fraction is as impossible as to write 0 as a positive fraction. Same is true for 0.333... There is no remainder, but provably no digit finishes the task, not even if there were ℵo digits. 1/3 is the limit never reached. And never means never, not "in the infinite"!
>
> Regards, WM
>
>

WM and his fake math, his own self deceptive math.

WM never took an algebra course!

just use mod(1/3)

1/3 = 1 mod(1/3)

On 12/20/2021 11:16 AM, wij wrote:

> The point is the representation of 0.333... is
> indeterminant.
> As is written, I assume you mean Σ(n=1,∞) 3/10^n,
> you can try to add the sum infinitely,
> the result will never be exact 1/3.

Σ(n=1,∞) 3/10^n and 0.333... are not indeterminate,
but they are not determined the same way 0.3, 0.33, 0.333,
and so on are determined.

0.33 is determined by a finite sequence of operations.
3 = 1+1+1
10 = 1+1+1+1+1+1+1+1+1+1
0.33 = ((3/10)+3)/10

Because 1 is unique and each operation produces
a unique value for unique arguments,
0.33 determines a unique point.

0.333... refers to the unique point that is in each
of the intervals
[0.3, 0.4]
[0.33, 0.34]
[0.333, 0.334]
....

We know that there is a unique point there.

----
We know that there can't be _two_ points x,y in each of
those intervals because these are points on a line
without infinite and without infinitesimal points.

The way we express that is, for any two distances b,c
on the line, there is some finite natural k
such that k*c > b

| Assume that there are two points x < y in each interval.
| | Then there is some finite k, 1 < k*(y-x)
| Because k is finite,
| there is some last power of 10, 10^m =< k
| such that 10^m*(y-x) =< 1 and 1 < 10^(m+1)*(y-x)
| | However, x and y are both inside intervals of length
| 1/10, 1/10^2, 1/10^3, ...
| For each finite n, 10^n*(y-x) < 1
| That is contradicted by 1 < 10^(m+1)*(y-x)

Therefore,
there aren't _two_ points in each interval.

The finite-ratio requirement, excluding infinite and
infinitesimal points, is called the Archimedean property.
For that reason (since I am not a historian) I think
it was well know to and accepted by classical Greeks.

----
Also,
there aren't _zero_ points in each interval.

In the case of 0.333... this is easy to show.
1/3 is in
[0.3, 0.4]
[0.33, 0.34]
[0.333, 0.334]
....

There is no less than one point which 0.333... determines
and there is no more than one point which 0.333...
determines -- just as for each of 0.3, 0.33, 0.333, ...
but the reasoning is different, more involved for 0.333...

We want to be able to claim that, in general, for any
infinite decimal, there aren't zero points in each of
the intervals for that infinite decimal.

| Assume the opposite.
| Assume there is some infinite decimal 0.abcdefg...
| for which there _are_ zero points in all of
| [0.a, 0.a+1/10]
| [0.ab, 0.ab+1/100]
| [0.abc, 0.abc+1/1000]
| [0.abcd, 0.abcd+1/10000]
| ...
| | Consider a continuous function f: [0,1] -> ℝ
| By "continuous", I mean that, for each x in [0,1],
| there is some neighborhood B_x for which x' ∈ B_x
| keeps f(x') within some specified distance from f(x).
| | What we expect from a continuous function f is that,
| if f(0) = 0 and f(1) = 1, f should cross every point
| between 0 and 1. "No teleporting" is the rule.
| | If there are _zero_ points in all of
| [0.a, 0.a+1/10]
| [0.ab, 0.ab+1/100]
| [0.abc, 0.abc+1/1000]
| [0.abcd, 0.abcd+1/10000]
| ...
| | then we can define f(x) = 0 for x < at least one
| interval, and f(x) = 1 for x > at least one interval.
| That function f satisfies our definition of "continuous"
| at every point in [0,1]
| (Every point is < or > some interval.)
| But f teleports over all the points between 0 and 1.
| This is not what we mean by "continuous".
| | We repair this flaw in our description of a continuous
| function by requiring that there are enough points
| at which f can be continuous for this scenario to be
| avoided. This requirement is called Dedekind completeness.

For any infinite decimal, there aren't zero points in
each of the intervals for that infinite decimal.
This follows from Dedekind completeness.
We require Dedekind completeness because we do not
want our "continuous" functions teleporting.

----
For any infinite decimal,
there aren't _zero_ points in each of the intervals for
that infinite decimal, and
there aren't _two_ points in each of the intervals for
that infinite decimal.

Each infinite decimal determines a unique point,
but NOT in the same way that a finite decimal determines
a unique point.

FromTheRafters schrieb am Montag, 20. Dezember 2021 um 20:25:37 UTC+1:

> Because you don't understand what a "Series" is. It is summed all at
> once, not in steps.

The usual fraud.

> Steps and order can be used for better and better
> rational approximations.

A series is a sequence of partial sums. Which one cannot be summed by steps? None. The limit exists, but is never reached. Not even by all once, because all will fail.

Regards, WM

on 12/20/2021, WM supposed :
> FromTheRafters schrieb am Montag, 20. Dezember 2021 um 20:25:37 UTC+1:
>
>> Because you don't understand what a "Series" is. It is summed all at
>> once, not in steps.
>
> The usual fraud.
>
>> Steps and order can be used for better and better
>> rational approximations.
>
> A series is a sequence of partial sums.

The summation of an infinite sequence's terms. Addition is commutative,
so the order doesn't matter. The sequence of partial sums is used to
determine what limit, if any, it has.

On 12/20/2021 3:57 PM, WM wrote:
> FromTheRafters schrieb am Montag, 20. Dezember 2021 um 20:25:37 UTC+1:
>
>> Because you don't understand what a "Series" is. It is summed all at
>> once, not in steps.
>
> The usual fraud.
>
>> Steps and order can be used for better and better
>> rational approximations.
>
> A series is a sequence of partial sums.

Fail. by oversimplification.

> Which one cannot be summed by steps? None.

sequences are not series. many cannot be summed by steps, but do converge on a value.

> The limit exists, but is never reached.

in your world of Bad Math.

> Not even by all once, because all will fail.

.... and so you reject Calculus.

>
> Regards, WM

måndag 20 december 2021 kl. 17:16:45 UTC+1 skrev wij:
> On Tuesday, 21 December 2021 at 00:04:49 UTC+8, Serg io wrote:
> > On 12/20/2021 9:56 AM, wij wrote:
> > > On Monday, 20 December 2021 at 16:18:52 UTC+8, Chris M. Thomasson wrote:
> > >> On 12/19/2021 9:54 PM, zelos...@gmail.com wrote:
> > >>> lördag 18 december 2021 kl. 14:40:17 UTC+1 skrev wij:
> > >>>> On Saturday, 18 December 2021 at 20:43:02 UTC+8, timba...@gmail.com wrote:
> > >>>>> On Saturday, December 18, 2021 at 3:47:40 AM UTC-5, wij wrote:
> > >>>>>> Richard says:
> > >>>>>>> When you use long division to divide 1 by 3, the resultant quotient is
> > >>>>>>> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > >>>>>>> This fact forms the equality:
> > >>>>>>>
> > >>>>>>> 1/3= 0.333... + non_zero_remainder
> > >>>>>>
> > >>>>>> But at the limit of infinite digits, the remainder becomes smaller than
> > >>>>>> any number and becomes zero, so isn't 'non-zero'.
> > >>>>>>
> > >>>>>>>
> > >>>>>>> Is this correct? If so, from the equation above we can deduce
> > >>>>>>> 1/3 - 0.333...= non_zero_remainder
> > >>>>>>
> > >>>>>> Except that at in infinite number of digits, the remainder is goes to
> > >>>>>> zero, as The Reals don't represent infinitesimal values.
> > >>>>>>
> > >>>>>>>
> > >>>>>>> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > >>>>>>> 1/3 ≠ 0.333...
> > >>>>>>>
> > >>>>>>> Is there any flaw?
> > >>>>>>
> > >>>>>> The fact that while for a FINITE number of digtis, you have a non-zero
> > >>>>>> result, but as you add more and more digits the value shrinks, and
> > >>>>>> shrinks fast enough that when we get to the infinite number of digits
> > >>>>>> (in the limit) it becomes 0.
> > >>>>>>
> > >>>>>> Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
> > >>>>>> when you invoke it, you invoke the limit property, and the limit of that
> > >>>>>> remainder is zero, and of 0.999... is 1.
> > >>>>>>
> > >>>>>> ---
> > >>>>>> 1/3= 0.333... + non_zero_remainder ?
> > >>>>> You've narrowed your context a bit too quickly I think.
> > >>>>> I recommend you cast your net wider.
> > >>>>> Consider closure and operator theory.
> > >>>>> You will destroy the rational value in the process.
> > >>>> Q: 1/3= 0.333... + non_zero_remainder ?
> > >>>>
> > >>>> True or false and for what reason (proof)? Idiot.
> > >>> 1/3=0.333....
> > >>>
> > >>> Thee proofs are endless
> > >> 1/3 can be represented as .333... in base 10. Therefore:
> > >>
> > >> .333... * 3 = 1, as 1/3*3=1
> > >
> > > 1/3= 0.333... + non_zero_remainder
> > > 0.333...= 1/3 - non_zero_remainder
> > >
> > > --- Pythagoreans' Code ---
> > > Infinitely approaching means equal. Number too small equals zero.
> > > Any real number can be approached by ratio number.
> > the representation of 0.333... *is not approaching 1/3*, it is merely an abbreviation of
> > 0.33333333333333333333333333333333333333333333333333333333333333333333
> > 333333333333333333333333333333333333333333333333333333333333333333333
> > 33333333333333333333333333333333333333333333333333333333333...
> The point is the representation of 0.333... is indeterminant. As is written, I assume you mean
> Σ(n=1,∞) 3/10^n, you can try to add the sum infinitely, the result will never be exact 1/3.

It is not indeterminant and the infinite sum, the series, is equal to 1/3 exactly and provably so.