0.9999999... = 1... or DOES it?!

[cornyjohn]

There are tons of proof out there that 0.99999999 = 1.

1/3 = .33333333....
1/3 * 3 = .33333333.... * 3
3/3 = .99999999....
1 = .99999999....

What's your opinion on the "proof" people show for 0.99999999 = 1? Personally, I believe it shows that we simply cannot express 1/3 to 100% accuracy.

Just thought I'd throw this at you. Debate's been rather dead lately.

[Wired Cous]

I always thought that this could never be exactly accurate and was just expressed that way. So I agree with you, Cornyjohn...I'll see if I can find some articles on this.

And yes, debate has been dead...

Articles:
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://en.wikipedia.org/wiki/0.999...
http://www.faqs.org/faqs/sci-math-fa...bers/0.999eq1/

One can show that this limit is 9/10 + 9/100 + 9/1000 ... using Analysis, and a proof really isn't all that hard (we all believe it intuitively anyway); a reference can be found in any of the Analysis texts referenced at the end of this message. Then all we have left to do is show that this sum really does equal 1:
Proof: 0.9999... = Sum 9/10^n
(n=1 -> Infinity)

= lim sum 9/10^n
(m -> Infinity) (n=1 -> m)

= lim .9(1-10^-(m+1))/(1-1/10)
(m -> Infinity)

= lim .9(1-10^-(m+1))/(9/10)
(m -> Infinity)

= .9/(9/10)

= 1

umm.....corny, I don't get these sums very well....
and are they even right?

[sparty]

I wouldn't say .9999 repeating equals one. Because .33333 repeating can never actually get a real number to equal 1/3 of 1. So you can't multiplay .3 repeating to think that it equals 1/3. Hopefully you understand what I'm trying to say.


.3 repeating can't be multiplied by 3 accurately.

[t-nodz]

let's look at this realistically

if we were to see, a baseball perhaps, that was 9999999999% complete, but there was a thread of one of the laces missing...

As humans, we would say it was all of a baseball, because it is virtually impossible to tell otherwise

when a number is that close to one, it is, merely because it is virtually impossible to tell eitherway, if something is 100%, or .9999999999%

[cornyjohn]

Originally Posted by sparty

I wouldn't say .9999 repeating equals one. Because .33333 repeating can never actually get a real number to equal 1/3 of 1. So you can't multiplay .3 repeating to think that it equals 1/3. Hopefully you understand what I'm trying to say.


.3 repeating can't be multiplied by 3 accurately.

I'm just throwing that out there. Of course it doesn't equal one, but the fraction is off by a tiny part, so you get the anomalous answer of 0.9999999 = 1.

And actually, if you really want to think about the logic behind that, we shouldn't multiply, but divide.

1/3 = 0.9999999
0.99999999 = 0.99999999

But of course, it wouldn't be incorrect to follow the multiply by three method, as order of operations state that you would be allowed to do so and should give the correct answer.

Like I said, it's all about the fact that 1/3 of 1 can't be expressed by our number system.

[Kovalchuk]

Originally Posted by cornyjohn

There are tons of proof out there that 0.99999999 = 1.


3/3 = .99999999....


What's your opinion on the "proof" people show for 0.99999999 = 1? Personally, I believe it shows that we simply cannot express 1/3 to 100% accuracy.

Just thought I'd throw this at you. Debate's been rather dead lately.

I dont understand how 3/3 = .999999999999? Isent any number divided by itself 1?

[cornyjohn]

Originally Posted by Wii_4_PE

I dont understand how 3/3 = .999999999999? Isent any number divided by itself 1?

By following order of operations, we should see that 1/3 * 3 = .3333333 * 3, correct?

3/3 = .99999999999 -> simplify the fraction
1 = .999999999999

Quite simple, when you think of it, but it just means our decimal system cannot correctly and 100% accurately depict 1/3.

[Fletch627]

1=1
.999999999=.999999999

No. .9999999 does not equal 1.

[cornyjohn]

Well, yes, but some people would argue with a signpost about it.

It's more or less an anomalous equation. You won't really see results like that elsewhere.

[sparty]

Actually I'm sort of lost in this but .3 repeating multiplied by 3 wouldn't equal .9 repeating but it would equal 1. So 1 equals 1. I'm not sure if you can say 3 * .3 repeating = .9 repeating.