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Topic: Proof 0.999... is not equal to one.
Replies: 194   Last Post: Feb 16, 2017 5:56 PM

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 quasi Posts: 12,052 Registered: 7/15/05
Re: Proof 0.999... is not equal to one.
Posted: May 31, 2007 4:34 AM

On 31 May 2007 01:02:12 -0700, chajadan@mail.com wrote:

>You have defined 0.999... to be a real number without jusitification.
>I can make no such assumption. Each position within 0.999... can be
>expressed as a real number, but the totality, the very infinite nature
>of it, seems to render it a never ending relation more than a specific
>explicit location on the real number line.

So what about sqrt(2)? or Pi?

Are those real numbers?

If so, note that they also have infinite decimals representations,
thus are also "never ending". So why do you accept those infinite
decimals as real but not .999... ?

For that matter, what about the fraction 1/3?

I'm sure you accept 1/3 as a real number.

Do you accept 1/3 = .333... ?

If so, note that .333... really means

3/10 + 3/10^2 + 3/10^3 + ...

which is the sum of an infinite series. Recall that the sum of an
infinite series means, by definition, the limit of the partial sums,
providing the limit exists.

2/3 = .666... = 6/10 + 6/10^2 + 6/10^3 + ...

I'm hoping you agree up to this point.

If so, we have

1/3 = 3/10 + 3/10^2 + 3/10^3 + ...

2/3 = 6/10 + 6/10^2 + 6/10^3 + ...

Adding the left hand sides gives 1.

Adding the right hand sides, term by term gives

9/10 + 9/10^2 + 9/10^3 + ...

Thus,

1 = 9/10 + 9/10^2 + 9/10^3 + ...

Now rewrite the RHS as a decimal ...

1 = .999...

Voila.

quasi

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