Re: .9999... = 1 proof

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 12/28/04 

Date: 28 Dec 2004 00:37:39 GMT

Lasse wrote:
> Absolute convergence does not enter into this - the sum of two
> convergent series converges (to the correct limit).

Only if you're somewhat careful about how you define the sum, since
you can't just pick arbitrary pairs of terms to construct the new
series. Even when you are using a definition for which it holds, it's
a property that has to be proved.

> The problem with the .999... =?= 1 people is that they don't
> understand what .999... is actually supposed to be (i.e., a limit,
> and thus a real number), and yet talk as if they do.

That is a major problem, yes. Another problem is that the
representation is unique for *almost* all real numbers, so it's not
surprising that people might be a bit put out when they come across
two different representations in the system and are told that they
represent the same number. Especially when one of those
representations is almost never used in practice.

Then there's the counterintuitive property that the limit of a
sequence can have different properties from every element in that
sequence. That one was a stumbling block for a lot of very smart
people over the centuries, so it's no great surprise that even today
some might find it hard to get past.

> Of course it is quite possible to define the rational numbers
> represented by an eventually repeating decimal expansion without
> having to refer to limits. Then .9999... = 1 follows immediately
> from the definition.

Indeed it does.

- Tim

Relevant Pages

  • Re: p^2 = a^2+b^2+c^2
    ... > as the sum of three positive squares; ... [Note that the representations above are to be taken without regard to ...
    (sci.math)
  • Re: p^2 = a^2+b^2+c^2
    ... Representations of integers as sums of squares. ... problem of representations by the sum of squares. ... but the text itself, with its many problems and research topics, would ...
    (sci.math)
  • Re: Representation by quadratic form
    ... As for the number of representations, ... So counting the number of representations of 2t+1 by your form is ... equivalent to counting the number of representations of 4t+2 as a sum ... of 3 squares, with the third variable even. ...
    (sci.math.research)
  • Egyptian Harmonic Numbers
    ... I am wondering about this variation on Egyptian fractions. ... etc) to represent the rational r as a sum of H's? ... and the m's are positive integers, ... then no rational r has no representations as such a sum. ...
    (sci.math)
Quantcast