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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory
Subject: Re: Real Number --- Merely numbers whose digits can be infinitely long
Date: Thu, 02 May 2024 13:51:13 -0700
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wij <wyniijj5@gmail.com> writes:
> On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
>> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
>> > wij <wyniijj5@gmail.com> writes:
>> > > On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>> > > > wij <wyniijj5@gmail.com> writes:
>> > > > > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> > > > > > wij <wyniijj5@gmail.com> writes:
>> > > > [...]
>> > > > > > >      <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.
>> > > > [...]
>> > > > 
>> > > > An extraordinary claim.
>> > > > 
>> > > > Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>> > > > your <fixed_point_number> notation?
>> > > > 
>> > > 
>> > > I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>> > > Substitute the n in n-ary with the q in p/q, every p/q is representable 
>> > > by <fixed_point_number>.
>> > > And, the rule of <frac> can generate infinitely long fractions, read it carefully!
>> > 
>> > That kind of notation almost universally refers to *finite* sequences of
>> > symbols.
>> > 
>> > If you intend it to be able to specify infinite sequences, that's fine,
>> > but it's not inherent in the notation you've presented.  I also wonder
>> > how an infinitely long <frac> can have <nzx> as its last element.
>> > 
>> > So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>> > 
>> > I presume that the "n-ary" base can be any integer greater than or equal
>> > to 2, and that the digits can range from 0 to n-1.  That means you'll
>> > need arbitrarily many distinct symbols for the digits in large bases.
>> > That's all fine, but it would be good to state all this explicitly.
>> > 
>> > There are already perfectly good mathematical methods for constructing
>> > the integers, the rationals, and the reals.  Your method of using base-n
>> > notation to *define* the reals and/or rationals seems superfluous.  It
>> > can probably be done consistently, but I fail to see how it's useful.
>> 
>> And something I thought of immediately after I posted the above:
>> 
>> You need to use different bases to represent all rational numbers, but
>> the base isn't part of your notation.  Your grammar matches "0.1", but
>> how do I know whether than's 1/10, 1/3, or 1/1729?
>> 
> Do you use different bases to represent all rational numbers?

We're not talking about how *I* represent rational numbers.

You've presented a grammar for "<fixed_point_number>", and you claim (I
presume) that it's useful as a way to represent at least all rational
numbers.  A given string may or may not satisfy the grammar; if it does,
it represents some real number.  Is that correct so far?

The string "0.1" satisfies your grammar.  What number does it represent?
How is someone supposed to know what base "n" is being used?

The string "0.2" with n=10 and the string "0.1" with n=5 (assuming we
have some way of specifying the value of n) both represent the same
number.  Every rational number (perhaps other than 0) has infinitely
many representations in your notation, with different values of n.  Were
you aware of that?  Is it a deliberate feature of your system?

And I see you ignored another question: Given your assertion that <frac>
can be infinitely long, can <wnum> also be infinitely long?  Isn't this
the kind of thing that you need to state explicitly?

>> 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
>> and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
>> indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
>> equal without using the real numbers that you're trying to *define*?
>> 
> How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or base-16
> if you also did not say it explicitly?
>
>> Or are you assuming that real numbers already exist, and you're defining
>> this notation on top of that?  If so, what's the point?
>
> Your request is valid but not practically reasonable.

I have no idea what you mean by that.  I did not make a request.
I asked several questions about the notation you're presenting.

How is your notation useful?  Do you believe it to be an improvement on
the other existing methods of defining the rational and real numbers?

-- 
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 */