Re: Euclidean Geometry in Schools




On 26/06/2005 12:42, William Elliot wrote:

> From: Jean-Claude Arbaut <jean-claude.arbaut@xxxxxxxxxxx>
>> Herman Rubin wrote:
>
>>> The teachers seem unable to unlearn the idea that numbers are
>>> strings of decimal digits, to be manipulated. They cannot
>>> understand that the representation by decimal digits is a
>>> derived representation and a convenience, not at all basic.
>
>> But real numbers *are* (infinite) strings of digits!
>
> They are not. You have to include a decimal point and you have to
> acknowledge that different strings of digits can represent the same real
> number. Thus strings of digits aren't the reals for if they were, there'd
> be a one-one correspondence between them. No, at best a string of digits
> can be construed as representation of a power series that converges in
> [0,1]. So a string per se, without decimal point convention, leaves one
> without strings for numbers greater than 1 and less than 0. The dictum
> real numbers are infinite strings of digits is misleading naivete.
>
> Whoops, forgot to consider "are (infinite) strings". Then alas,
> 1/2 = 5 = 50 = 500 = 5000 = 500... = 499...
> is infinitely many different series of digits, and so should be infinitely
> many different numbers. No problem, the real 499... infinitesimally
> smaller than the real 5 and the real 50 infinitesimally larger than the
> real 5 and the real 500 infinitesimally larger than the real 50 and ....
> It's easy to explain away why 499... is infinitesimally smaller than 5,
> but students are not going to get it why 50 infinitesimal larger than 5
> and 500 is infinitesimal larger than 50 and .... This is of course is
> made simple when you compare 0 with 00, 000,... and 00... and with, hm
> what series of digits is infinitesimally smaller than 0? But as you say,
> "But real numbers *are* (infinite) strings of digits!" you'll have no
> difficultly showing me strings of digits for -1 and 999,999.999, now will
> you?
>
> Now confoundingly confusing is 1/2 = 5, for a series of digits is just
> that, a series of digits, while a decimal representation of a number has a
> decimal point whence the common sensical
> 1/2 = 0.5 = 0.50 = 0.500 = 0.500... = 0.499....
> So again, the naive notion of a series of digits flounders for not even be
> a decimal representation (which aren't of themselves the reals, but merely
> inattentive representations.)

Sigh. "Real numbers may be represented as finite or infinite strings of
digits, with sometimes a decimal point". I answered in another post that the
correct characterization is given by properties. Basically, all Q
properties, plus "every majorized subset of R has a sup".
And they should be constructed from Q with Cauchy series or Dedekind cuts. I
like the latter, but Cauchy series can be generalized, so the idea seems
better.

.

Relevant Pages

  • Re: Cantor and the binary tree
    ... >>>List all the infinite binary sequences with a bijection to the integers. ... to be referring to the "proof" of uncountability of the reals. ... By traversing the list diagonally ... the sense that there are as many digits in each number as numbers in the list. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... and diagonal traversal does not cover all strings. ... Do the math, and stop playing bad logic games, and declaring nonexistent differences between the finite and infinite. ... Any such list is exponentially longer than it is wide in digits. ... If they are a larger set than the naturals, then that is a valid conclusion, perhaps, but to say they can't be enumerated like the naturals, is just wrong. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... length n as initial strings followed by the expansion of pi after the nth ... long decimal expansion followed by an infinite expansion. ... for producing successive digits of pi without end. ... There are no other reals at the end of or besides this list. ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... > with an infinite number of procedures that produce .2. ... That is the foundation which math grows out of. ... continue generating all the digits. ... for as long as the real generating processes produces reals to work with. ...
    (sci.math)
  • Re: Math errors in python
    ... > They don't even share three digits beyond the decimal point. ... Uh, "constructive reals", such as those you can find at ... constructively interesting" subset that is _countably_ infinite. ... coefficients fit comfortably in memory (with space left over for some ...
    (comp.lang.python)