It's actually true. I just learned about this and it makes all the sense in the world. My mind was blown by this a while ago. Good ole' Kollada.

Chill man, I wasn't reffering to you, but to somebody else, MDMAlliance to be exact. I do agree that I sounded arrogant and condescending but it was a reaction; you guys should understand that it's fine to doubt something until you're shown proof, but to claim the opposite of something that is well known and understood in mathematics after being told so that the thing is well known makes you look infinitely (yes I use this word on purpose) more untolerable than, say, flat Earth people.

If you're this desperate to see rigorous proof, check out the book Principles of Mathematical Analysis by Walter Rudin. It's a rigorous introduction to analysis, and there's a whole section on deriving the set of real numbers from the set of rationals.

You may also want to check Michael Spivak's Calculus. The first chapter treats numbers basically as abstractly as you can consider for an introduction.

Yes the number is for all practical purposes 1.  However by it infinately aproaches bet never reaches 1.  If you were to write the relationship you are still left with

1 > 0.99999999...

Much like limits at infinity that asymptotically aproach but never reach a value.

Thats the thing... you can not depict infinity graphically or by writing a large number. So you are correct in the sense that you can't plot a coordinate system and show infinity. You can only depict finite graphs and numbers. And for every finite number 0.999999......9 there is a solution of a number 0.0000001

But 0.9999.....9 is not equal to 0.9 period, no matter how many 9s you add. And there is no last digit. Because even the number 1 can be depicted as 1.00000000 with infinite 0's.

Look at this problem from this point of view.

We think that 0.9 period + A = 1

We assume that A is something like 0.0000000000000000...00001. How can we express this number? (10^-oo). And what is the value of 10^-oo? 0

What is the value of 10^-(100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)?

Although the number is extremely small, but it is not 0. And I can add finetely more 0's and it would still be > 0.

10^-oo = 0 and 0.9 period = 1.

the step 'subtract x' is a trick. you are treating x as a constant at the start of the proof and a variable on the 'subract x' step. so when you say subtract x it means subtract 0.99999 therefore your proof doesn't really hold up mathematically. its a trick

x = 0.999... and therefore 'subtract x' is the same thing as 'subtract 0.999...'. You can operate on both sides of an equation; the form you operate on each side is up to you to decide as long as both forms are equal. Besides, your point about treating x as a constant instead of a variable doesn't make much sense. x is x and it has a known value; what's the problem?

Oh I get it. You're just trying to mess with us.

i'm not messing with you. the problem with his proof is he is treating x as a variable. but it is a constant as he has stated in the first step. you can't just suddenly decide x is a variable in the middle of a proof when he has stated it's a constant at the start of a proof. In a proof you must define x to a degree for example x is any positive integer. he defined x as 0.99999. nothing else. his proof is completely wrong.

http://en.wikipedia.org/wiki/User:ConMan/Proof_that_0.999..._does_not_equal_1 

If you think the proof is wrong, please do point out the step that's wrong and why. All you have is an ambiguous statement about what's wrong but if there's something wrong, you should point it out exactly because it's not obvious. What does a variable even mean to you? And what about the other proofs? Also, have you studied mathematics? Besides high school level, that is.

1 = 100%
0.9 = 99%
0.99 = 99%
0.999 = 99%
0.998 = 99%
0.997 = 99%
0.99999999999999997 = 99%
0.98999 = 98%
1.99 = 199%

instead of confusing yourself, you use percentage relative to the subject that you are counting, how many of it as a whole instead of how many of it there is.

First of all, I would like to say that 0.99… equaling 1 is pretty much a fact. Search any serious mathematics sources and ask any mathematician and the overwhelming majority will agree that this is true. Of course this does not prove that it is true. But I’m just bringing this up because some people appear to be under the illusion that 0.99… obviously isn’t 1 and anyone who thinks it is obviously is uneducated. I would just like to say that that is not the case and most educated people, people good at math do agree with this thread. So those who disagree with this thread should at least know that they are in the minority.

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It seems that many people are not understanding what 0.999… really is. Some people are saying it’s getting close to 1 but never reaches it. This is false based on the fact that it’s not getting close to anything. It’s not getting anywhere; no more than 9 is “getting” to 10. 0.99… does have a value and that value doesn’t hover from close to 1 to even closer to 1. That doesn’t make sense. It has one static value which does not magically change on its own.

0.999… can be written as the sum of an infinite series, which is one static number. It is the sum of the infinite sequence (9/10) + (9/10)(1/10)^2 + (9/10)(1/10)^3….As I have already stated, this is equal to 9/10 divided by 1-9/10, which is 1 (read about the sum of convergent geometric series). I think the thing most people have trouble understanding is the fact this number is the result after you add an infinite amount of the numbers in the sequence. People are saying things like “you can never add enough numbers to get to 1, because you can always add another 9.” But that isn’t what 0.999…is. There are already an infinite number of 9s so adding more would not make sense. I am not talking about 0.9999 or 0.9999999 or even 0.99999999999999, I’m talking about infinite 9s and what its value would be.

Some people may be upset because they may be thinking “you can’t add an infinite number of values.” Sure you can; people do it all the time. Think about the sequence 1+1+1+1+1…. What would be the sum of an infinite amount of 1s? I am not talking about the sum of a thousand 1s or a million 1s. I am talking about infinite 1s. What would their sum be? Most people have no problem with this and would kindly say the answer is infinity. Sure, infinite 1s cannot be written down on paper or expressed visually, but with mathematics we can be sure that an infinite amount of 1s does equal infinity. Not a billion and not a trillion; neither of these would be infinite 1s. Infinite 1s would equal infinity.

Going back to 0.99…. No we cannot write an infinite amount of nines. We can’t visualize an infinite amount of nines. We cannot physically construct an infinite amount of nines. But like adding an infinite amount of nines, we can use mathematics to add an infinite amount of nines and we can be sure of its value. And we can represent that number with the appropriate symbols. Yeah, if you physically add nines, we will only get closer to 1 and will never reach 1. But we can’t physically reach infinity. That doesn’t mean we don’t know what will happen if we did theoretically add an infinite amount of nines. It would be equal to that number that you approach (which is 1), when you add a finite amount of nines. But when you add an infinite amount of nines, you REACH that number.

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To all the people saying 0.99… is less than 1, I have one question: Give me a number higher than 0.99… and less than 1. As I have already stated, if two numbers are different, then they will have not just one, but an infinite amount of other numbers between them. I’m only asking you to give me one number. One number higher than 0.99… and one number lower than 1. It  shouldn’t be too difficult if they are in fact different numbers. And no, you can’t say 0.99… is the next number lower than 1, because there is no “next” number after any other number.

And before you say “infinite isn’t a number” so we can’t tell. That shouldn’t matter. Just because a number expressed (in decimal notation) has an infinite amount of digits doesn’t mean we can’t find numbers higher than it or lower than it. And it certainly doesn’t mean we can’t see if it’s the same as another number. Take pi for instance. Let’s say someone said pi is equal to 3.141. You could easily see that this is false by simply finding more precise value of pi. We could figure out that pi goes to 3.1415, so it’s between 3.141 and 3.142. We can get more precise and see that pi goes to 3.141592, meaning it’s between 3.141592 and 3.141593. We could play this game forever and there would never be a number that’s equal to pi, because pi would always be higher or lower. Same with 0.333 or 0.258714285714…. or any other infinite decimal unless the number you compare it to is equal.

You can even do that with 0.999… Say someone said was equal to 0.99999 was the number that was higher than 0.999…. and lower than 1. We could easily disprove this by getting more precise with 0.999… and seeing that it equals 0.99999999, which is higher than 0.9999. We could play this game forever and we could NEVER find a value higher than 0.9999 AND lower than 1. This would never happen. The fact that there is no number higher than 0.999…. and lower than 1 is proof that they are of the same value. Again, if there is no number higher than 0.99... and lower than 1, then they are equal. If they weren't equal, there would be an infinite number of values between them. And just because 0.9999 has an infinite amount of digits, that shouldn't harm anything.

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The main reason 0.999… equals 1 is because it works. It doesn’t break any mathematic rules at all. You can manipulate it all you want and it would never break anything. In fact, 0.999…equaling 1 would make things wonderful. It would make 0.333…. (or 1/3) perfectly one third of 1 in decimal notation. It makes 0.111111 (or 1/9) perfectly one ninth of 1 in decimal notation. We can multiply it, divide it, add it, subtract, do whatever we want to it, and 0.999…=1 would not have ANY harmful implications.