Re: Problems I have with 1.999...=2


Kirby Cook wrote:
> The flaw in the oft-cited second "proof", or demostration, which
tries
> to show that there is no number between 1.999... and 2, may be found
in
> the law of the reciprocal, by analogy and inference. Say, for
instance,
> that there is no number between the infinite sum (difference)
1-.999...
> and zero.Taking it step by step, 1-.9 has a reciprocal of 1/(1-.9) or

> 10; 1-.99 has a recirpocal of 100, and so on. So, if you assert that

> there is no number between 1-.999... and zero, then you are also
> asserting that there is no number between 1000... and a point often
> termed the point at infinity. And how many of you will buy that?

1000... is not a number. If you were to define it in the obvious way
--- i.e., the limit of the sequence 10, 100, 1000, ..., then it would
be infinity.

>
> It should be obvious, I hope, that I am directing my remarks not to
> those who carefully assert that 1.999... =2 is merely sloppy
shorthand
> for the idea that 1.999... *approaches* 2. No, my remarks are for
those
> hopelessly superstitious minds that claim that "equals means equals"
in
> this case.

There is no superstition involved. The number 1.999... is, BY
DEFINITION, the LIMIT of the sequence

1.9, 1.99, 1.999, 1.9999 and so on.

By your own admission, this limit is 2. So 1.999... = 2.

Perhaps your discomfort comes from never having seen a sound
mathematical treatment of these matters.

Hope this helps,

Lasse
---
(@remove.for.spam.maths.warwick.ac.uk)

.

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