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

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On Thu, 2024-05-02 at 13:51 -0700, Keith Thompson wrote:
> 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?
> 

Nothing is different from the math. you understand (except several corner cases
which you will never need to worry about).

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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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