0.999999[r] = 1

[mobobuff]

I just love seeing all the different ways believers try to demonstrate that it's true, and all the ways non-believers try to show that it's false. I haven't been totally convinced yet, but I'm leaning towards true, once you throw in a little semantic dispute.

So.

0.99999999[repeating]* = 1

Discuss.

*repeating = infinity

 

[Machupo]

0.9999999[r] * 100 = 99.99999999[r]

99.99999999[r] - 0.99999999[r] = 99

99/99 = 1

 

[Cawchy87]

i don't understand your logic behind those 2 seemingly unrelated equations.

 

[blahblah99]

This topic has been beaten to death already.

Here's the stupid man's proof:

If you take 1/3 + 2/3, you get 3/3 which equals 1.

Yet, 1/3 = 0.33333... and 2/3 = 0.66666.... , and 0.3333.. + 0.6666... = 0.9999.., which must equal 1 according to the first statement.


And here's a more mathematical proof:

Let x=0.99999.....repeating infinitely.

Then 10x = 9.9999.... inifinitely.

10x - x = 9.9999.... - 0.999999 = 9x = 9

x must equal 1. But we let x=0.99999.... in the first statement, so this proof contradicts itself, therefore, 0.999999.... = 1.

 

[NewBlackDak]

0.99999[r] != 1 as far as I'm concerned.
However, if you think of it in something measurable, you'll soon realize that the difference between 0.99999[r] and 1 becomes immeasurable.

 

[Sahakiel]

Proving .99999... = 1 is akin to proving general relativity.

Explaining it in layman's terms is similar to explaining the true meaning behind the above sentence.

That's all.

 

[CTho9305]

Originally posted by: Sahakiel
Proving .99999... = 1 is akin to proving general relativity.

Explaining it in layman's terms is similar to explaining the true meaning behind the above sentence.

That's all.

I think anyone who knows enough math to understand, or has the ability to follow simple reasoning already gets it, and the rest of the people here (well, Off Topic) have demonstrated themselves to be incapable/unwilling to accept provable facts. The discussion of a fact is a waste of time when those who don't agree can't understand proofs.

 

[duragezic]

Yes I don't see the problem with understanding the proof(s)??

Convergent series people... It's not that hard!

 

[Cawchy87]

Originally posted by: blahblah99
This topic has been beaten to death already.

Here's the stupid man's proof:

If you take 1/3 + 2/3, you get 3/3 which equals 1.

Yet, 1/3 = 0.33333... and 2/3 = 0.66666.... , and 0.3333.. + 0.6666... = 0.9999.., which must equal 1 according to the first statement.


And here's a more mathematical proof:

Let x=0.99999.....repeating infinitely.

Then 10x = 9.9999.... inifinitely.

10x - x = 9.9999.... - 0.999999 = 9x = 9

x must equal 1. But we let x=0.99999.... in the first statement, so this proof contradicts itself, therefore, 0.999999.... = 1.

That makes a lot more sence, thanks! (i have only taken 2 years of pre-cal so i haven't learnt proofs yet)

 

[CTho9305]

Originally posted by: Cawchy87
That makes a lot more sence, thanks! (i have only taken 2 years of pre-cal so i haven't learnt proofs yet)

...and here I was, thinking you were a math geek and your name was related to Cauchy-Riemann equations. Guess that makes me the math geek

 

[Cawchy87]

wo, for sure not. It sounds like my last name... thats what some people call me and i thought it was catchy.

 

[mobobuff]

Yes, I'm surely a believer. It all just makes me ponder on the depths and mysteries of the term "infinity". To have no end just boggles my mind to, well, no end.

Space, for example, must be infinity itself. You wouldn't just run into a wall. Even if you did, what's on the other side of the wall? Are we all just living in a universe in a ball on a cat's collar?

Let's talk about infinity.

 

[Cogman]

infinity is such a cool word. If you have an infinitly small thing, then you have nothing. Such is the case here. the distance, by the deffinition of .999999999[r], betwean .9999999[r] and 1 is infinitly small hence, no diffrence. infinity is not that hard to understand.

 

[PowerEngineer]

Originally posted by: blahblah99

Here's the stupid man's proof:

If you take 1/3 + 2/3, you get 3/3 which equals 1.

Yet, 1/3 = 0.33333... and 2/3 = 0.66666.... , and 0.3333.. + 0.6666... = 0.9999.., which must equal 1 according to the first statement.

That's a good one that I hadn't heard before!

 

[Cogman]

I was thinking about it. And really the diffrence betwean .999[r] and 1 is .000[r]1 (at the end of an infinit amount of zeros is a one), and because infinity is never ending you will never hit the 1 at the end, hence there is no diffrence.

 

[Amorphus]

Wait, so is
0.0000.....1 = 0? assuming an infinite number of zeroes.

You can "prove" things mathematically (like the researchers "proved" that bees could not fly in the oft-cited story), but the issue of .999...=1 is a philosophical question. Can something be so insignificantly small such that it is actually nothing?

 

[CTho9305]

Originally posted by: Amorphus
Wait, so is
0.0000.....1 = 0? assuming an infinite number of zeroes.

You can "prove" things mathematically (like the researchers "proved" that bees could not fly in the oft-cited story), but the issue of .999...=1 is a philosophical question. Can something be so insignificantly small such that it is actually nothing?

The way numbers are defined, it's provable.

 

[unlockthesource]

Originally posted by: Amorphus
Wait, so is
0.0000.....1 = 0? assuming an infinite number of zeroes.

You can "prove" things mathematically (like the researchers "proved" that bees could not fly in the oft-cited story), but the issue of .999...=1 is a philosophical question. Can something be so insignificantly small such that it is actually nothing?

I think the problem is that our lives are based on limits and, as a result, the notion of a "non-limit" is truly disturbing and hard to digest. I dont believe that the universe is infinite. We just dont know the limit yet. (Please take a moment to think about just how right and wrong that statement was)

 

[Dinominant]

OK, infinity:
Think of a pipe in a void. This pipe is infinite units long. You are in this void. You grab onto this infinite units long pipe. You will not instantly jolt in one direction. "This pipe is infinite units long", that means that it has a constant length and that it isn't 'growing' as most imagine it, it is already 'there'.

So, 0.999[r] = 1
Well, most people image that further down the string, that there are more and more 9's being tagged on the end. So when they imagine the difference between the 0.999[r] and 1, they imagine 0.000[r] with a 1 tagged on the end. But in this case you must remember that 0.999[r] is a constant and that there is no end to that sting, making it equal to one.

therefore 0.999[r] equals 1

 

[blahblah99]

Please, if you offer a contradiction, then show a mathematical proof instead of an opinionated one.

 

[CTho9305]

Originally posted by: blahblah99
Please, if you offer a contradiction, then show a mathematical proof instead of an opinionated one.

That's like asking John Ashcroft to present evidence against evolution, instead of saying "it's not very likely for humans to evolve".

 

[Dinominant]

OK, here is a valid proof aquired from Dr. Math:

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

Here is another one:

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1

 

[Cogman]

I just thought about it, and an infinity .000 with a one tagged on the end really does not work, because the zeros are infinit there will never actually be and end for the one to be tagged on to. Anyone following what im saying?

 

[imgod2u]

Originally posted by: Amorphus
Wait, so is
0.0000.....1 = 0? assuming an infinite number of zeroes.

You can "prove" things mathematically (like the researchers "proved" that bees could not fly in the oft-cited story), but the issue of .999...=1 is a philosophical question. Can something be so insignificantly small such that it is actually nothing?

Seeing as numbers aren't a natural thing, I don't think so. The number system is a human construct. So it is perfectly reasonable for people to define it whichever way they wish, provided it reflects observations. The issue with 0.999[r] and 1 being the same is simply a case in which two representations happen to imply the same quantity. So we define it as such.

 

[cquark]

Originally posted by: imgod2u
Seeing as numbers aren't a natural thing, I don't think so. The number system is a human construct. So it is perfectly reasonable for people to define it whichever way they wish, provided it reflects observations.

Mathematics is a human construct and I agree up to a point: there's no need for mathematics to reflect the world. There's a need for science to use the parts of mathematics that reflect scientific observations, but there's no reason for mathematics to be limited to the parts useful to the sciences.

In any case, attempting to apply your intuition about finite sets and sequences to infinite sets and sequences will almost always lead you astray, as it does in this example of infinite decimal sequences. If you want another counterintuitive example, how about the fact that the size (cardinality) of the set of all integers is the same as the size of the set of just the odd integers?