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Real Number --- Merely numbers whose digits can be infinitely long

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  Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-04-29 04:42 +0800
    Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-04-29 11:57 +0100
      Re: Real Number --- Merely numbers whose digits can be infinitely long Andy Walker <anw@cuboid.co.uk> - 2024-04-29 13:35 +0100
      Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-04-30 11:03 +0800
        Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-04-29 22:02 -0700
        Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-01 22:58 +0100
          Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 09:33 +0800
            Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 18:38 -0700
              Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 10:03 +0800
                Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 20:38 -0700
                  Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 20:46 -0700
                    Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-01 21:52 -0700
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-01 22:40 -0700
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Andy Walker <anw@cuboid.co.uk> - 2024-05-02 09:36 +0100
                        Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-02 13:22 -0700
                    Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 12:46 +0800
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 13:51 -0700
                        Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 08:43 +0800
                          Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 18:02 -0700
                            Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 09:44 +0800
                              Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 19:58 -0700
                                Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-02 20:23 -0700
                                  Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-04 10:36 -0700
            Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-02 23:03 +0100
              Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 08:41 +0800
                Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-04 00:02 +0100
                  Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-04 08:28 +0800
                    Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-05 10:14 +0100

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#104244

wij <wyniijj5@gmail.com> writes:
> On Thu, 2024-05-02 at 18:02 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> [snip]
>> > Nothing is different from the math. you understand (except several corner cases
>> > which you will never need to worry about).
>> 
>> So there's nothing novel in your notation, and I needn't waste any more
>> time asking questions about it that you're unwilling and/or unable to
>> answer.
>> 
>> Is that a fair summary?
>> 
>> (I'll ignore the "corner cases" you allude to.)
>
> No, nothing novel there changed the usual usage of 'fixed point number'.

I have no idea whether that was an attempt to answer my question.  If it
was such an attempt, it failed.

The term "fixed point number" is quite distinct from both "rational
number" and "real number".  I don't know which you're trying to define.
Fixed point numbers, in my experience, are a computer representation,
not a mathematical abstraction.  But your subject header talks about
real numbers.

Feel free to attempt to clarify if you're so inclined.  Or don't.

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */

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#104245

On 05/02/2024 07:58 PM, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
>> On Thu, 2024-05-02 at 18:02 -0700, Keith Thompson wrote:
>>> wij <wyniijj5@gmail.com> writes:
>>> [snip]
>>>> Nothing is different from the math. you understand (except several corner cases
>>>> which you will never need to worry about).
>>>
>>> So there's nothing novel in your notation, and I needn't waste any more
>>> time asking questions about it that you're unwilling and/or unable to
>>> answer.
>>>
>>> Is that a fair summary?
>>>
>>> (I'll ignore the "corner cases" you allude to.)
>>
>> No, nothing novel there changed the usual usage of 'fixed point number'.
>
> I have no idea whether that was an attempt to answer my question.  If it
> was such an attempt, it failed.
>
> The term "fixed point number" is quite distinct from both "rational
> number" and "real number".  I don't know which you're trying to define.
> Fixed point numbers, in my experience, are a computer representation,
> not a mathematical abstraction.  But your subject header talks about
> real numbers.
>
> Feel free to attempt to clarify if you're so inclined.  Or don't.
>

https://en.wikipedia.org/wiki/Numerical_tower

The line-reals are naturals n/d, 0 <= n <= d, d goes to infinity,
it has extent, density, completeness, measure [0,1], measure 1.0.

The field-reals are the equivalence classes of sequences that
are Cauchy, and that's the standard definition of the complete
ordered field, including when Dedekind cuts (of rationals) won't do.

The signal-reals are as that the rationals are huge,
when doubling them results a continuous domain.


It gets involved doubling and halving measures and spaces
and real non-standard analytical character and new results
in numerical series and methods in the Cantor space or
"2 ^ omega" of each of these different models of real numbers.
Then standard ordinary set theory is left consistent by just
making another result in set theory in function theory.

That _always_ exists.

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#104335

On 05/02/2024 08:23 PM, Ross Finlayson wrote:
> On 05/02/2024 07:58 PM, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>>> On Thu, 2024-05-02 at 18:02 -0700, Keith Thompson wrote:
>>>> wij <wyniijj5@gmail.com> writes:
>>>> [snip]
>>>>> Nothing is different from the math. you understand (except several
>>>>> corner cases
>>>>> which you will never need to worry about).
>>>>
>>>> So there's nothing novel in your notation, and I needn't waste any more
>>>> time asking questions about it that you're unwilling and/or unable to
>>>> answer.
>>>>
>>>> Is that a fair summary?
>>>>
>>>> (I'll ignore the "corner cases" you allude to.)
>>>
>>> No, nothing novel there changed the usual usage of 'fixed point number'.
>>
>> I have no idea whether that was an attempt to answer my question.  If it
>> was such an attempt, it failed.
>>
>> The term "fixed point number" is quite distinct from both "rational
>> number" and "real number".  I don't know which you're trying to define.
>> Fixed point numbers, in my experience, are a computer representation,
>> not a mathematical abstraction.  But your subject header talks about
>> real numbers.
>>
>> Feel free to attempt to clarify if you're so inclined.  Or don't.
>>
>
> https://en.wikipedia.org/wiki/Numerical_tower
>
> The line-reals are naturals n/d, 0 <= n <= d, d goes to infinity,
> it has extent, density, completeness, measure [0,1], measure 1.0.
>
> The field-reals are the equivalence classes of sequences that
> are Cauchy, and that's the standard definition of the complete
> ordered field, including when Dedekind cuts (of rationals) won't do.
>
> The signal-reals are as that the rationals are huge,
> when doubling them results a continuous domain.
>
>
> It gets involved doubling and halving measures and spaces
> and real non-standard analytical character and new results
> in numerical series and methods in the Cantor space or
> "2 ^ omega" of each of these different models of real numbers.
> Then standard ordinary set theory is left consistent by just
> making another result in set theory in function theory.
>
> That _always_ exists.
>
>

The infinite expressions and completions and closures
and the inductive and deductive and non-inductive and
anti-inductive in the infinite expressions, for the
infinite limits and continuum limits, of course has
that mathematics has great examples of the results of
deductive inference as the abductive over inductive inference,
as what arrives at the continuum has at least these three
different and distinct definitions, that arrive together
as inter-compatible if not inter-changeable, that is
quite better than even the standard way today, as it
really shows that mathematics has these various laws
of large numbers for their various regularities for
their various rulialities for their various common
consequences of each their fixed completions,
why it is so that line-reals, field-reals, signal-reals
are more replete the complete linear continuum, and
as about the integer continuum, the linear continuum,
and the long-line continuum, in all the objects of
the real analysis.

So, a premier mathematician of this age, knows these things.

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#104231

wij <wyniijj5@gmail.com> writes:

> On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> wij <wyniijj5@gmail.com> writes:
>> 
>> > Got your idea.
>> 
>> It's not my idea.  It's a standard notation.
>> 
>
> There is no standard notation for formal grammar. I adopt the idea using either
> ',' or '|'. <fixed_point_number> is simple, average readers know what it should be.
>
>
>> > I'll try use '|' exclusively. Thanks for the suggestions:
>> > 
>> >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>> >      <wnum>::= 0
>> >      <wnum>::= <nzd> { 0 | <nzd> }
>> >      <frac>::= { 0 | <nzd> } <nzd>
>> >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>> > 
>> >     Ex: 78, -12.345, 3.1414159
>> 
>> So what's the point of defining these strings that represent a subset of
>> the rationals?
>> 
>
> <fixed_point_number> is a super set of rationals.

Give an example <fixed_point_number> that is not rational.

If you can it should be one of the examples you give since you are
obviously using the EBNF notation is a new, non-standard way.  Giving
three examples that at conventional rational numbers is a huge missed
opportunity to show what you really mean.

-- 
Ben.

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#104233

On Thu, 2024-05-02 at 23:03 +0100, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
> 
> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > 
> > > > Got your idea.
> > > 
> > > It's not my idea.  It's a standard notation.
> > > 
> > 
> > There is no standard notation for formal grammar. I adopt the idea using either
> > ',' or '|'. <fixed_point_number> is simple, average readers know what it should be.
> > 
> > 
> > > > I'll try use '|' exclusively. Thanks for the suggestions:
> > > > 
> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
> > > >      <wnum>::= 0
> > > >      <wnum>::= <nzd> { 0 | <nzd> }
> > > >      <frac>::= { 0 | <nzd> } <nzd>
> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
> > > > 
> > > >     Ex: 78, -12.345, 3.1414159
> > > 
> > > So what's the point of defining these strings that represent a subset of
> > > the rationals?
> > > 
> > 
> > <fixed_point_number> is a super set of rationals.
> 
> Give an example <fixed_point_number> that is not rational.
> 
> If you can it should be one of the examples you give since you are
> obviously using the EBNF notation is a new, non-standard way.  Giving
> three examples that at conventional rational numbers is a huge missed
> opportunity to show what you really mean.
> 

If you don't like it, you can read it as anything.
Read it as a joke and keep believing 0.333...∉ [0,1/3) and
 1/3 ≠ 0.333... + nonzero_remainder

I mean strongly hold that believe, don't change in your life-time.

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#104277

wij <wyniijj5@gmail.com> writes:

> On Thu, 2024-05-02 at 23:03 +0100, Ben Bacarisse wrote:
>> wij <wyniijj5@gmail.com> writes:
>> 
>> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> > > wij <wyniijj5@gmail.com> writes:
>> > > 
>> > > > Got your idea.
>> > > 
>> > > It's not my idea.  It's a standard notation.
>> > > 
>> > 
>> > There is no standard notation for formal grammar. I adopt the idea using either
>> > ',' or '|'. <fixed_point_number> is simple, average readers know what it should be.
>> > 
>> > 
>> > > > I'll try use '|' exclusively. Thanks for the suggestions:
>> > > > 
>> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>> > > >      <wnum>::= 0
>> > > >      <wnum>::= <nzd> { 0 | <nzd> }
>> > > >      <frac>::= { 0 | <nzd> } <nzd>
>> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>> > > > 
>> > > >     Ex: 78, -12.345, 3.1414159
>> > > 
>> > > So what's the point of defining these strings that represent a subset of
>> > > the rationals?
>> > > 
>> > 
>> > <fixed_point_number> is a super set of rationals.
>> 
>> Give an example <fixed_point_number> that is not rational.

Can you give an example <fixed_point_number> that is not rational?  If
not, why not?

-- 
Ben.

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#104280

On Sat, 2024-05-04 at 00:02 +0100, Ben Bacarisse wrote:
> 3.1414159

3.1414159... (infinitely long)

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#104360

wij <wyniijj5@gmail.com> writes:

> On Sat, 2024-05-04 at 00:02 +0100, Ben Bacarisse wrote:
>> 3.1414159
>
> 3.1414159... (infinitely long)

That's not a string produced by your grammar for <fixed_point_number>.
(Also, your quoting and attributions are incorrect.)

You've cut all the context, presumably because you don't want people to
see the question you can't answer.  Here's the context with the question
you are not able to answer restored:

You: <fixed_point_number> is a super set of rationals.

Me: Give an example <fixed_point_number> that is not rational.

Me again (because you didn't answer): Can you give an example
<fixed_point_number> that is not rational?  If not, why not?

Your eventual reply is not a string produced by your grammar.  Here is
the grammar again since you cut it from your reply.

  <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
  <wnum>::= 0
  <wnum>::= <nzd> { 0 | <nzd> }
  <frac>::= { 0 | <nzd> } <nzd>
  <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary

"3.1414159" is a string produced by this grammar (but, anyway, it's
rational).  None of the other characters ("... (infinitely long)") can
form part of a <fixed_point_number>.

-- 
Ben.

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