I just made a file RealNumber for real number, hope that it can simplify the
bascic problem about real number and provide solutions regarding problems of
infinity, 0.999...,etc. which is explained shortly and indirectly.

--------------------
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.

+-------------+
| Real Number |
+-------------+

n-ary Floating Point Number::= Number represented by a string of digits, the
string may contain a point and/or plus/minus sign.... Two n-ary floating
point number x,y are equal iff their normalized form are identical.

Real Nunmber(ℝ)::= {x| x is represented by n-ary floating point number. The
string of digits of x may be infinitely long }

Note: This definition implies that repeating decimals are irrational number.

Real number is just this simple. The limit theory is a methodology for finding
derivative, nothing to do with what the real number is (otherwise, a definition
like the above must be defined in advance. Otherwise, latter dedution will be
difficult not to contain circular-reasoning.

+--------------------------------------+
| Restoring Interpretation of Calculus |
+--------------------------------------+
Calculus is basically from the area problem of a function: Let F compute the
the area of f. From the meaning of area, we can have:

(F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2) // h is a sufficiently small offset
<=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2

Expected property of F: (1)Error |lhs-rhs| strictly decreases with a tiny offset
h (2)When h=0, lhs=rhs.
Because the h in the lhs cannot be 0, the basic problem of calculus is
finding such a F that satisfies the expected porperty above.

F-expected property(1) can be relaxed, because such a F is still a
acknowledged area function. So, the interpretation above can be specified as:

Lim(h->0) (F(x+h)-F(x))/h = f(x)

Note: (1)The meaning of Lim is the same as the lim in text books but the h
must be explicitly 0 in the final processed result of (F(x+h)-F(x))/h.
Otherwise, the logical conclusion is always 'approximation'.
(2)The above expression is treated as an identity that can be derived,
not a definition (3)Nothing about infinity or infinitesmal is mentioned.

Continuous (function)::= Function f is continuous at a iff f(a) can be
approached from nearby-value f(a+h) as described by ε-δ method..

Note1: Borrowing from the limit theory, the idea can be expressed as
lim(h->0) f(x+h)= f(x)
Note2: For complex problems, most iterative algorithms depend on continuous
property to find the solution.
Note3: P≠NP should have been proved
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof.txt/download
-------------------------------------------------------------------------------

On 2024-03-08 07:56:02 +0000, wij said:

> I just made a file RealNumber for real number, hope that it can simplify the
> bascic problem about real number and provide solutions regarding problems of
> infinity, 0.999...,etc. which is explained shortly and indirectly.
>
> --------------------
> The purpose this text is for establishing the bases for computational
> algorithm.
> This file
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> may be updated anytime.
>
> +-------------+
> | Real Number |
> +-------------+
>
> n-ary Floating Point Number::= Number represented by a string of digits, the
> string may contain a point and/or plus/minus sign.... Two n-ary floating
> point number x,y are equal iff their normalized form are identical.

That is not a definition because "normalized form" is not defined.

--
Mikko

On 8/03/24 08:56, wij wrote:
> I just made a file RealNumber for real number, hope that it can simplify the
> bascic problem about real number and provide solutions regarding problems of
> infinity, 0.999...,etc. which is explained shortly and indirectly.

What was wrong with the Dedekind cut definition? A real number is
defined by two infinite sets of rational numbers so that every rational
number is in one of the sets, and all the numbers in set Less are less
than all the numbers in set Greater.

(If the real number is equal to a rational number, that rational number
still has to be in one of the sets, and Dedekind arbitrarily decides it
should go in set Greater.)

On Fri, 2024-03-08 at 17:15 +0100, immibis wrote:
> On 8/03/24 08:56, wij wrote:
> > I just made a file RealNumber for real number, hope that it can simplify the
> > bascic problem about real number and provide solutions regarding problems of
> > infinity, 0.999...,etc. which is explained shortly and indirectly.
>
> What was wrong with the Dedekind cut definition? A real number is
> defined by two infinite sets of rational numbers so that every rational
> number is in one of the sets, and all the numbers in set Less are less
> than all the numbers in set Greater.
>
> (If the real number is equal to a rational number, that rational number
> still has to be in one of the sets, and Dedekind arbitrarily decides it
> should go in set Greater.)
>

I don't want to go too much into the current real number theory, because
it is waste of time. Whatsoever, two theories, which one is better?

My 'theory' (fact actually) is very simple and intuitive: Whenever a number is
proved not finitely representable in n-ary floating number, it is irrational.

The 'current real number theory' provides more trouble than what it tries to solve.

On Fri, 2024-03-08 at 17:34 +0200, Mikko wrote:
> On 2024-03-08 07:56:02 +0000, wij said:
>
> > I just made a file RealNumber for real number, hope that it can simplify the
> > bascic problem about real number and provide solutions regarding problems of
> > infinity, 0.999...,etc. which is explained shortly and indirectly.
> >
> > --------------------
> > The purpose this text is for establishing the bases for computational
> > algorithm.
> > This file
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download 
> >
> > may be updated anytime.
> >
> > +-------------+
> > > Real Number |
> > +-------------+
> >
> > n-ary Floating Point Number::= Number represented by a string of digits, the
> >    string may contain a point and/or plus/minus sign.... Two n-ary floating
> >    point number x,y are equal iff their normalized form are identical.
>
> That is not a definition because "normalized form" is not defined.
>

Details are omitted. Everybody has idea what it should be.

On 03/08/2024 05:18 PM, wij wrote:
> On Fri, 2024-03-08 at 17:15 +0100, immibis wrote:
>> On 8/03/24 08:56, wij wrote:
>>> I just made a file RealNumber for real number, hope that it can simplify the
>>> bascic problem about real number and provide solutions regarding problems of
>>> infinity, 0.999...,etc. which is explained shortly and indirectly.
>>
>> What was wrong with the Dedekind cut definition? A real number is
>> defined by two infinite sets of rational numbers so that every rational
>> number is in one of the sets, and all the numbers in set Less are less
>> than all the numbers in set Greater.
>>
>> (If the real number is equal to a rational number, that rational number
>> still has to be in one of the sets, and Dedekind arbitrarily decides it
>> should go in set Greater.)
>>
>
> I don't want to go too much into the current real number theory, because
> it is waste of time. Whatsoever, two theories, which one is better?
>
> My 'theory' (fact actually) is very simple and intuitive: Whenever a number is
> proved not finitely representable in n-ary floating number, it is irrational.
>
> The 'current real number theory' provides more trouble than what it tries to solve.
>
>

Heh.

Do you know what the "current real number theory" is?

Do you know that "Dedekind cut" is already only a
sort of poor-man's simplification of "real number"?

Do you know what "interval arithmetic" is?

Are you aware of extended-precision arithmetic and Karatsuba?

Do you know that the numbers you're talking about,
"floating point numbers", already are defined and
only are what they are?

Now, ir-rational has at least two definitions. One is
mathematical, and the only thing it means is, "real,
but not a ratio". It's well known that in any given
positional notation with a fixed radix, that many
rational numbers have infinite expansions.

So, when you say "n-ary", and it means "I checked all
the finite or co-prime radices", that would help, but,
there are infinitely many of those.

The other definition of ir-rational, follows from since
rational in ratio also meant rational in reason,
and ir-rational meant otherwise.

What you got there is "floating-points" and "not-a-numbers".

Of course that's fine to use that in your code as it
only makes sense to you. Yet, the best way to make
sense usually is to use the correct definitions and
with their usual meanings.

You have a long way to do to catch up with Archimedes
Plutonium, whose entire "real numbers" fit in a 10x10 square.

Most people learn probably in school or from reading,
if they even bother, the formalism, of real numbers,
as, "equivalence classes of series with the property
of Cauchy that they converge", with those being simply
as all the infinite expansion, finite before the radix
and infinite after.

This has the property of completeness that, as a set,
of them being points, that the set of them, contains
the least upper bound of the set, which your finite
floating-point numbers don't.

Then, of course there are other notions like, "clock
arithmetic", "odometer reals", that range [0,1] or
empty -> full, are, "continuous", that "it's a model
and any old model of the linear continuum which is
continuous".

Then another sort of notion is that it's "whatever
is the smooth to the grainy".

Then, each of these has finite discretizations,
whatever is grainy to the smooth, whatever is
dots in a row, for example the rationals the
ordered field vis-a-vis the complete ordered field,
the usual.

So anyways if they're extended precision floating
point numbers, or for example fixed-point numbers
with unbounded exponents for "scientific numbers",
though usually those live in a world with mathematical
constants like e and pi, that's what they are.

I mean, if you make a floating point file,
isn't it the same thing?

Then, finite element analysis of course,
for difference methods, sure, they can help
you compute all sorts of things that could
be close enough, but, people using "properties
of real real numbers", can compute arithmetic codings,
still faster than that.

Look into getting a slide-rule, as close as you can
read it, it offers pretty much the scientific numbers
there extended for as long as it is.

Maybe an abacus.

Anyways there are interesting things in
"current real number theory", like there
being at least three continuous domains,
like line-reals, field-reals, signal-reals,
each with their own establishments of extent,
density, completeness, and measure, then as
about how they all live their elements in or
on or about the linear continuum.

Then the idea is, besides that for example,
"symbolic algebra" can do all the things with
real numbers that mathematicians can do,
that there result being able to find even
more interesting facts about the "large" numbers,
and what result the "small" numbers, and about
all the sequences of 0's and 1's there are, at all.

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

So, with rationals having a finite representation
in at least one radix, base, and irrationals not,
it still results that the least-upper-bound, of
those rationals, isn't necessarily rational,
that it's not saying much to say that rationals
have a finite length in at least one among co-prime
radices.

Anyways if you're just saying that a given rational
number has a finite representation in a given integer
base, radix, just say that and that's what it is and
people might say "how clever of you to notice a mathematical
fact", then about what's better is basically what you
hope to show. What you got there just seems to be
a fact about rational numbers, that while it's so,
that given any real number, isn't necessarily demonstrable.

Of course, in mathematics, that kind of has to be
in relations of numbers, "quantifiable".

So what about it?

On Sat, 2024-03-09 at 09:20 +0800, wij wrote:
> On Fri, 2024-03-08 at 17:34 +0200, Mikko wrote:
> > On 2024-03-08 07:56:02 +0000, wij said:
> >
> > > I just made a file RealNumber for real number, hope that it can simplify the
> > > bascic problem about real number and provide solutions regarding problems of
> > > infinity, 0.999...,etc. which is explained shortly and indirectly.
> > >
> > > --------------------
> > > The purpose this text is for establishing the bases for computational
> > > algorithm.
> > > This file
> > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download 
> > >
> > > may be updated anytime.
> > >
> > > +-------------+
> > > > Real Number |
> > > +-------------+
> > >
> > > n-ary Floating Point Number::= Number represented by a string of digits, the
> > >    string may contain a point and/or plus/minus sign.... Two n-ary floating
> > >    point number x,y are equal iff their normalized form are identical.
> >
> > That is not a definition because "normalized form" is not defined.
> >
>
> Details are omitted. Everybody has idea what it should be.
>

That passage of wording (and other minor detail) was modified:

n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a point and/or plus/minus sign...(details omitted).
Two n-ary fixed-point number x,y are equal iff their normalized form are
identical.

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

On 2024-03-09 01:20:24 +0000, wij said:

> On Fri, 2024-03-08 at 17:34 +0200, Mikko wrote:
>> On 2024-03-08 07:56:02 +0000, wij said:
>>
>>> I just made a file RealNumber for real number, hope that it can simplify the
>>> bascic problem about real number and provide solutions regarding problems of
>>> infinity, 0.999...,etc. which is explained shortly and indirectly.
>>>
>>> --------------------
>>> The purpose this text is for establishing the bases for computational>
>>> > algorithm.
>>> This file> >
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>>
>>> may be updated anytime.
>>>
>>> +-------------+
>>>> Real Number |
>>> +-------------+
>>>
>>> n-ary Floating Point Number::= Number represented by a string of digits, the
>>>    string may contain a point and/or plus/minus sign.... Two n-ary floating
>>>    point number x,y are equal iff their normalized form are identical.
>>
>> That is not a definition because "normalized form" is not defined.
>>
>
> Details are omitted. Everybody has idea what it should be.

It should be a sequence of digits followed by a point followed by antoher
sequence of digits. The first sequence of digits must be finite and it
may be enpty. The sequence of digits must not start with the zero digit.
The second sequence of digits may be empty, finite, or infinite. If finite,
it must not end with the zero digit. If infinite, there must be inifinitely
many digits different from the digit nine.

Is this what you mean?

--
Mikko

wij <wyniijj5@gmail.com> writes:

> On Fri, 2024-03-08 at 17:15 +0100, immibis wrote:
>> On 8/03/24 08:56, wij wrote:
>> > I just made a file RealNumber for real number, hope that it can simplify the
>> > bascic problem about real number and provide solutions regarding problems of
>> > infinity, 0.999...,etc. which is explained shortly and indirectly.
>>
>> What was wrong with the Dedekind cut definition? A real number is
>> defined by two infinite sets of rational numbers so that every rational
>> number is in one of the sets, and all the numbers in set Less are less
>> than all the numbers in set Greater.
>>
>> (If the real number is equal to a rational number, that rational number
>> still has to be in one of the sets, and Dedekind arbitrarily decides it
>> should go in set Greater.)
>
> I don't want to go too much into the current real number theory, because
> it is waste of time. Whatsoever, two theories, which one is better?

If they are isomorphic (i.e. they describe structures with exactly the
same properties) then your first task is to prove that. If they are
not, then you have no business using the term "real number" for yours.
Unless of course you are just a Usenet crank who thinks they know better
than everyone else.

--
Ben.

On 9/03/24 02:18, wij wrote:
> On Fri, 2024-03-08 at 17:15 +0100, immibis wrote:
>> On 8/03/24 08:56, wij wrote:
>>> I just made a file RealNumber for real number, hope that it can simplify the
>>> bascic problem about real number and provide solutions regarding problems of
>>> infinity, 0.999...,etc. which is explained shortly and indirectly.
>>
>> What was wrong with the Dedekind cut definition? A real number is
>> defined by two infinite sets of rational numbers so that every rational
>> number is in one of the sets, and all the numbers in set Less are less
>> than all the numbers in set Greater.
>>
>> (If the real number is equal to a rational number, that rational number
>> still has to be in one of the sets, and Dedekind arbitrarily decides it
>> should go in set Greater.)

> I don't want to go too much into the current real number theory, because
> it is waste of time. Whatsoever, two theories, which one is better?

The Dedekind cut definition says more about the numbers themselves, and
the finite representation definition says more about the way you can
write them.

> Whenever a number is
> proved not finitely representable in n-ary floating number, it is irrational.
True. It is equivalent to saying that every rational number can be
written as a/b where a and b are integers.

> Note: This definition implies that repeating decimals are irrational
number.

All repeating decimals can be finitely represented as n-ary floating
point numbers, for example "one third" is finitely represented as "0.1"
in trinary.