Re: .999... = 1

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
.

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