Re: .999... = 1
- From: "Ken Quirici" <kquirici@xxxxxxxxx>
- Date: 31 Aug 2005 07:29:48 -0700
David C. Ullrich wrote:
> On 29 Aug 2005 07:54:02 -0700, "Ken Quirici" <kquirici@xxxxxxxxx>
> wrote:
>
> >[...]
> >
> >I'm confused about this
>
> That's correct.
>
> >(and vaguely disturbed).
> >
> >Given that any irrational can be represented by a sequence of
> >rationals that approach arbitrarily close, and similarly, that any
> >rational can be approached by a sequence of rationals that approach
> >arbitrarily close (both by let's say Cauchy sequences - which
> >I take to be a particular way of 'approaching arbitrarily close'),
> >it would seem to me that
> >
> >.99999....
> >
> >represents, not 1, it's 'limit', but rather a particular Cauchy
> >sequence of values approaching 1.
>
> Notation means what the definition _says_ it means, not what
> it seems to you it represents. In fact that _meaning_ of
> the notation 0.999... is the limit of the sequence
> 0.9, 0.99, 0.999, etc. Saying it seems to you it represents
> something else is simply incoherent - you're saying that
> it seems to you that a _definition_ is wrong.
>
> >The notion of setting the sequence equal to its limit
>
> I didn't set any sequences equal to their limits here.
> 0.999... is not a sequence, it is a single number.
>
> >makes sense
> >because for all practical calculations to which the sequence would
> >be put in mathematics, the results would be identical if you used
> >the limit rather than created infinite sequences of calculations
> >tending to the result obtained just by going directly to 1.
> >
> >I.e., mathematics would be hopelessly bogged down if you didn't
> >replace any Cauchy sequence approaching a limit by that limit and
> >consider yourself done.
> >
> >However, to me, .99999... represents a sequence that tends to 1,
> >rather than 1.
> >
> >If you want to calculate using the limit of the sequence, use 1.
> >If you want to investigate the sequence, use the sequence, not
> >1. It may be that in some investigations the sequence is as
> >important as the number it tends to.
> >
> >Hope this makes some sense.
>
> None whatever. Why in the world would you think that
> what it _seems_ to you a given notation represents
> has anything to do with reality? Definitions are
> correct, by definition.
>
> You're making exactly as much sense as if I'd
> said that the notation a + b denotes the sum of
> a and b, and you said it seems to you that in fact
> a + b represents the product of a and b.
>
> >I can't wrap my mind around the
> >notation .99999.... = 1. It's apples and oranges, at least
> >to my benighted mathematical logic.
> >
> >Thanks.
> >
> >Ken
>
>
> ************************
>
> David C. Ullrich
Yes. It must be so. 0.9999.... is the limit of a sequence of sums,
and that limit = 1. so 0.9999... = 1.
0.3333... = 1/3 also, but for some reason that is less troubling.
Thanks.
Ken
.
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