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1=0.9999...

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Rigil Kent
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« on: January 31, 2012, 08:44:19 AM »

What the hell?! Huh?

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Rigil Kent
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« Reply #1 on: January 31, 2012, 09:18:28 AM »

Well, well ... apparently it does!
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Mefiante
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« Reply #2 on: January 31, 2012, 09:31:44 AM »

Yes, it’s true.  The proof shown in the OP is the standard one.  Another proof can be framed in terms of the sum of a convergent geometric series with the ratio of consecutive terms equal to 1/10 = 0.1 (exact).  (As an aside, this counterintuitive fact illustrates that infinity in mathematics is not as simple a concept as one might think.)

A mathematically less rigorous way to think about it is as follows:

1/3 = 0.333333…

Then 1/3 + 1/3 + 1/3 = 0.333333… + 0.333333… + 0.333333… = 0.999999…

But 1/3 + 1/3 + 1/3 = 3/3 = 1.

Therefore 0.999999… = 1.  QED.

'Luthon64
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Rigil Kent
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« Reply #3 on: January 31, 2012, 09:49:12 AM »

A mathematically less rigorous way to think about it is as follows:

1/3 = 0.333333…

Then 1/3 + 1/3 + 1/3 = 0.333333… + 0.333333… + 0.333333… = 0.999999…

But 1/3 + 1/3 + 1/3 = 3/3 = 1.

Therefore 0.999999… = 1.  QED.

'Luthon64

Very convincing for sure. It's somehow more intuitive than the proof in the OP.

Thanks, Mefiante. Cool
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cyghost
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« Reply #4 on: January 31, 2012, 10:14:40 AM »

I saw this first with dawkin's scale discussions  Wink

So I follow the first proof untill here:

a = 0.999...
10a = 9.999...
10a = 9 + 0.999...
10a = 9 + a

why is 9a now 9 please?
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brianvds
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« Reply #5 on: January 31, 2012, 10:24:41 AM »

I saw this first with dawkin's scale discussions  Wink

So I follow the first proof untill here:

a = 0.999...
10a = 9.999...
10a = 9 + 0.999...
10a = 9 + a

why is 9a now 9 please?

From your last step above:

10a = 9 + a
=> 10a - a = 9 + a - a (subtract a from both sides)
=> 9a = 9

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cyghost
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« Reply #6 on: January 31, 2012, 10:28:34 AM »

Thank you, briandvds, I follow that Smiley
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