0.9999... = 1?

[† Mute point]

I understand exactly what he's saying, the problem is, it's wrong. Especially in the initial example.. Once you assign a value to X, it has a definite number of digits, how many there are is irrelevant. 0.9, 0.99, 0.999, 0.9999, and so on... they all have one thing in common. Times by 10 and subtract the same value from it, and you get a number that starts with 8, has some 9's, and ends in 1. .9 becomes 9 becomes 8.1, .99 becomes 9.9 becomes 8.91, and so on and so forth. You cannot assign an infinite value to the number of decimals, because in an equation it would have a definite number, no matter how you try to complicate it, that is the truth.

Now, if you were allowed to have the value of the product of 10X change to keep the same number of decimal digits, then your example would be correct, but you can't do that, because same letter variables do not change in the same equation. Take some basic algebra if you don't understand that. Higher math doesn't change the rules just because you can make a complicated equation that looks like it should be correct.

[Dae314]

I can understand that there may be problems with the initial example. However, the convergence theorem is provable, and mathematically sound unlike any of the other theories pushed in this thread. The thing's pretty much air tight. You're not changing any rules with an infinite summation. You're doing an infinite summation. That's not possible unless you do a limit. The actual summation you evaluate is limit of the summation on the expression from n = 1 to i where i is going to infinity. This is a higher math concept that is not captured in algebra, and it is a proven and tried concept. There are no arguments available to you against the convergence theorem.

It doesn't just look correct, it is correct, and you can evaluate it without bending any rules or redefining anything. Click the link in my above post to wolfram's evaluation of the summation. According to Wolfram (you can prove it by hand if you want) 3 different convergence tests confirm that the summation function converges, and when you calculate the convergence value it's 1.

[zx0]

Once again you spout symbols to attempt to prove a point that cannot be true. In plain words, explain how a .9 repeating answer can come about through the natural mathematical laws, not through applying infinity to a number that doesn't come out to be an infinite by any mathematical equation not including infinity. If you can't do this, your point is invalid, as the counter points have been made in terms anyone can understand, and clearly refute everything you've tried to show.

Lemmingllama's proof is valid, it's the same proof I posted earlier. If you don't understand mathematics I suggest you start learning about it rather than spouting nonsense. Just because you don't understand it doesn't mean it's wrong, it means you have a poor grasp of mathematics.

Would you refute that the square root of negative 1 is i (complex number)?

You may have been told in school that you can't do the square root of a negative number, but all your teacher is saying is you don't have the KNOWLEDGE to do the square root of a negative number because you haven't been taught about complex numbers yet.

Some things can't be just dumbed down for any idiot to understand, why do you think you can do whole degrees in mathematics if it could be made so a child could understand easily.

I understand exactly what he's saying, the problem is, it's wrong. Especially in the initial example.. Once you assign a value to X, it has a definite number of digits, how many there are is irrelevant...

You cannot assign an infinite value to the number of decimals, because in an equation it would have a definite number, no matter how you try to complicate it, that is the truth.

No it doesn't. Your understanding of infinite/recurring decimals is wrong. Once you assign a value to x it does NOT have a definite number of digits.

x = 1/7

So how many digits do we have here? It's RECURRING. meaning never ending, infinite.

You have no understanding of infinity, seriously read something and LEARN instead of spouting nonsense. Try picking up a mathematics textbook.

A hotel has an infinite number of rooms. The hotel is full with an infinite number of guests.

Q1. If a person arrives at the hotel looking for a room, how can can we fit them in?

Q2. If an infinite number of guests arrive (at the already full hotel) how can we give each guest a room?

Just to give you the answers to the puzzle for fun

A1. Ask the person in room 1 to move to room 2. The person in room to to move to room 3. The person in room 3 to move to room 4 etc. So every person moves to the next room. There is now space in room 1 for the person.

A2. Ask the person in room 1 to move to room 2. The person in room to to move to room 4. The person in room 3 to move to room 6 etc, so everyone doubles their room number. this leaves all the odd numbered rooms free for the infinite number of extra people to occupy.

The thing you have to remember about infinity is that is NEVER ends.

[Dae314]

I wanna take this post to apologize for my previous posts sounding demeaning or mean. I'm not trying to say that you're stupid or ignorant for not understanding what I'm trying to say or what other people are trying to say. Rather, I'm seeking to try and make clear what is unclear and resolve potential and evident misunderstandings that I see within your posts. I don't expect that any random person I meet knows everything about advanced mathematics, and I respect that math may just not be that person's area of interest. I'm sure that any person I may have made feel bad has a hundred other talents, interests, and knowledge that I don't know/couldn't understand from where I'm at right now. So forgive me if I've made you feel inadequate, I'm just trying to get the correct message across in the most efficient way possible.

[† Mute point]

1/7 does not have go on into infinite, because it is 1/7, as in one seventh. Solving it does repeat however, because it cannot truly be solved, there is always something left over.

Zx0, you are lucky I'm a tolerant person, or I would report that abusive post and laugh as you got banned from the forums.

Once again however, I must point out that, no matter how you swing it, .9 repeating cannot exist naturally, so when you assign it to the equation, it has an end. When it has an end, you can solve for it, when you can solve for it, you see that 10x-x = 8.91, or 8.991, or so on down the line. It does not equal 9, nor does it end up equaling 1.

Unless you can explain your point without complicated theorems and symbols that mean nothing to the average person, you have nothing.

A good example of a number like what you are discussing here is pi, 22/7, It doesn't repeat, but has been calculated into the thousands of decimal places, and still hasn't been solved. There is an end to it somewhere, just because you can't find it doesn't mean it doesn't exist.

[Dae314]

Are you trying to argue that just because there's no fraction to represent 0.999... that you can't use a real number like that? I believe 0.999... falls under the category of irrational numbers which just means that there's no fraction of integers that can represent it (you've said this I know). However that doesn't mean you can't have irrational numbers exist and use them. You might not be able to use them like how lemming tried to do what with multiplying by 10 and all (it's debatable what would actually happen) but that's why it's not an air tight proof (as you're saying). However I don't think it works the way you're describing it. Just because you stuck 0.999... in an equation doesn't mean it suddenly has an end. Infinity has no end. It goes on forever. There are however different sizes of infinity. If you want to argue that taking 1 decimal off of 0.999... changes that size of infinity sufficiently to make it a smaller infinity than the original then you might be able to argue that 9.99... - 0.999... wouldn't equal 9 because the second infinity is larger than the first. That's a hard point to argue though since you're basically taking infinity - 1 which is still infinity .

I'm not sure why you're stuck on explaining this without theorems and symbols. You want an air tight proof, but you're confining that proof to be within only the realm of math that you know. That's not how it works especially when you aren't a mathematician yourself. The air tight proof you're asking for is the convergence theorem. Instead of asking for a simplified proof, learn how to read the proof we're giving you (it's not that hard to understand summations) especially because it's not some proof we're making up this thing's been accepted by the math community .

Complex notation and symbols and stuff actually exist to abstract away from all the plumbing that goes into the complex idea that the notation or symbol represents. Think of it kinda like how you would think of multiplication. You don't want to have to write out all those digits and + signs so you condense things into a * b. Now that you can see things in a simple concise way but still understand the theory of what it does you can use your new tool and make it do something odd like a * (b + c). Sure you can write that all out as addition, but that's really long and complicated and the idea behind it is hidden in the plumbing. If you're more programming oriented think of the basic math you know as assembly language. Then think of the notation and junk that comes later as high order languages. High order language commands represent several assembly language commands that are useful to have mixed in a certain way. Since you can capture an idea more concisely with a high order language you can make more complicated ideas on top of that concise idea just as you would with math. Yes, you could've done the same thing with assembly language, but then it'd be convoluted, and the overarching idea would be hidden by all the junk that goes into describing it. The same goes for summations and limits. They're ideas from a higher level of math that encapsulate an idea so that you can expand its uses and create even more complex ideas.

[zx0]

1/7 does not have go on into infinite, because it is 1/7, as in one seventh. Solving it does repeat however, because it cannot truly be solved, there is always something left over.

Zx0, you are lucky I'm a tolerant person, or I would report that abusive post and laugh as you got banned from the forums.

Once again however, I must point out that, no matter how you swing it, .9 repeating cannot exist naturally, so when you assign it to the equation, it has an end. When it has an end, you can solve for it, when you can solve for it, you see that 10x-x = 8.91, or 8.991, or so on down the line. It does not equal 9, nor does it end up equaling 1.

Unless you can explain your point without complicated theorems and symbols that mean nothing to the average person, you have nothing.

A good example of a number like what you are discussing here is pi, 22/7, It doesn't repeat, but has been calculated into the thousands of decimal places, and still hasn't been solved. There is an end to it somewhere, just because you can't find it doesn't mean it doesn't exist.

My post was not meant to be abusive, i'm sorry if it came off like that. I'm just getting a bit frustrated as you have no excuse for ignorance as there are 3 of us here trying to educate you and you are just not accepting it.

Also, pi is not 22/7. 22/7 is a recurring decimal, pi is not; it's an irrational number.

22/7 = 3.142857 recurring, pi = 3.14159265...

People only use 22/7 as an approximation to pi, as you can see it only works to 2dp.

There are many proofs that pi is irrational (https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational) Most of which only require basic maths so you should be able to understand that pi is irrational and therefore will repeat or terminate.

As Dae314 said, you can't explain complicated mathematics in simple terms to the average person. This is why I have spent years at university learning about mathematics. I didn't pick up this knowledge in a 30 minute class in school when I was 13.

[† Mute point]

My post was not meant to be abusive, i'm sorry if it came off like that. I'm just getting a bit frustrated as you have no excuse for ignorance as there are 3 of us here trying to educate you and you are just not accepting it.

Also, pi is not 22/7. 22/7 is a recurring decimal, pi is not; it's an irrational number.

22/7 = 3.142857 recurring, pi = 3.14159265...

People only use 22/7 as an approximation to pi, as you can see it only works to 2dp.

There are many proofs that pi is irrational (https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational) Most of which only require basic maths so you should be able to understand that pi is irrational and therefore will repeat or terminate.

As Dae314 said, you can't explain complicated mathematics in simple terms to the average person. This is why I have spent years at university learning about mathematics. I didn't pick up this knowledge in a 30 minute class in school when I was 13.

Anything can be explained in simple terms, I'm not ignorant at all, as I fully understand exactly what you are saying. What I'm telling you, is that because your number does not repeat according to natural mathematical law, you cannot apply infinite to it as a variable in an equation. And since this forum is mostly made up of average people, you are being quite ignorant of them if you fail to explain your point in terms they can understand. And if you are incapable of explaining the point in terms that an average person could understand, you are not qualified to teach the subject, and by extension, not qualified to debate on it.

The math I know is not relevant, but goes far beyond simple equations. What I am telling you all you need to do, is defend your point, whether I agree with it or not. I'm acting as any good professor would in telling you that you need to explain it, not just in ways the scientific community, or mathematical geniuses can comprehend, but in ways the layman can at least relate to. In Dae's latest post, he's come a lot closer to doing this than any of the rest of you have managed. So please, continue, without name calling and assumptions based on your perceptions of other's education.

[Dae314]

There's a few things holding me back from explaining things here more deeply. The main thing being, which proof are you attempting to ask clarification for? We keep talking about complicated symbols so I think you're talking about the convergence theorem, but I see scattered references to the initial proof which is algebraic and can't really be simplified any further. I just want to clarify before I launch into a long explanation of all the math behind the wrong proof.

The next thing is that while I can explain to you the convergence theorem, it will require me to go into an explanation of limits and summations (since that proof uses an infinite series) along with some convergence tests. My inner hacker mentality tells me not to drag the topic so far off from the main point and to just expect the people reading here to look up what they don't know (something which I'm attempting to get anyone asking a question to do).

The last thing that's holding me back from explaining the proof is what I referred to in my previous post. In breaking down the plumbing behind the pretty symbols, I'm releasing a torrent of compressed ideas that could (and often does) wash people away. Basically they get lost in the plumbing behind the system. It's the reason why, when I attempt an explanation of a computer to someone I don't go to the logic gate level, I stop at the main component level and just say stuff like "and the CPU handles calculations!!!"

[† Mute point]

The Algebraic formula is the one I'm disputing as true, because according to the most basic algebraic principles it cannot be true. The convergence is solid, but choosing a different way to prove something is not the same as proving the initial example. I want Lemming to stand up and prove his example, because without applying an artificial construct from a higher level of mathematical possibility, there is nothing that can create the situation he is calculating from.

[zx0]

The Algebraic formula is the one I'm disputing as true, because according to the most basic algebraic principles it cannot be true. The convergence is solid, but choosing a different way to prove something is not the same as proving the initial example. I want Lemming to stand up and prove his example, because without applying an artificial construct from a higher level of mathematical possibility, there is nothing that can create the situation he is calculating from.

There is nothing to prove that is not already there.

As I understand it your argument is that when you multiply an infinite decimal by 10 that you have 1 less digit after the decimal point. That is not the case, there are an infinite number of decimals after the decimal point. As we have said before, infinity - 1 = infinity.

For the record infinity minus a billion is still infinity also.

Maybe I can demonstrate it in a way you won't refute by showing that the logic in the first post is correct.

Would you agree that 1/3 = 0.3 recurring?

Let x = 0.3 recurring

10x = 3.3 recurring

10x - x = 3.3... - 0.3...

9x = 3

x = 1/3

So you can see that multiplying 0.3 recurring by 10 did not result in one less digit after the decimal point.

Another one we can do is 3.142857142857 recurring

Let x = 3.142857142857 recurring

Here we have to multiply by 1,000,000

1,000,000x = 3142857.142857142857 recurring

1,000,000x - x = 3142857.142857142857... - 3.142857142857...

999,999x = 3142854

x = 3142854/999,999

x = 22/7

Hopefully now you can see that the method used is correct, and so 0.9 recurring = 1

In the same way 0.49999... = 0.5

x = 0.49...

10x = 4.9...

100x = 49.9...

100x - 10x = 49.9... - 4.9...

90x = 45

x = 0.5

We have proven the case, and if you don't get it you should read all of the links I have posted. The crux of your argument seems to be based on a misunderstanding of infinity.

And no teacher tries to jump in and explain things without laying the foundations as you expect us to do.

[lemmingllama]

Mute, I understand that you are saying that there is no possible way to get 0.999... using only whole numbers, since it would have to be an infinate over an infinate. I just used that since it is much more relatable for people, plus some may not believe that the infinate series is the same as 1. Please just sub in something that can be made with whole numbers, like 1/3 being 0.333...

As a side note, we can also see that 3/3 is 0.999... Since it is equal to one, and any number over itself is 1(excluding 0 and infinity). But basically this is multiplying 1/3 by 3, same as multiplying 0.333... by 3, making it 0.999... So it is possible to get this.

[† Mute point]

Zx0, you are once again making assumptions based on your perception of my education, which far exceeds this point. What you all are presenting here would be the equivalent of a thesis statement, which a good professor would attack relentlessly until you proved it. Because if you can't defend your paper to him/her, you can't defend it to the rest of the community either.

In your current example, 1/3, the reason there are infinite decimals is because it is a number that naturally has infinite decimals, it is an unsolvable equation if you are not allowed to use the "repeating" or "infinite" principal.

Neither of your examples would have an end result of being 1 the way .9 repeating would if it could be used in a simple equation. and here is why.

X = .333333333..........

10X= 3.3333333333.......

10X - X = 3

9X (When found by the previous answer, not through actual multiplication) = 3

x=1/3

So basically... You posted a series of equations to prove what everyone already knows. Good job.

Incorrect again Lemming, for when you add that final third, there is nothing left over when dividing to create the decimal. 1/3 = .3 with a remainer of 1. 2/3 = .6 with a remainder of 2. 3/3 simply = 1

[lemmingllama]

Incorrect again Lemming, for when you add that final third, there is nothing left over when dividing to create the decimal. 1/3 = .3 with a remainer of 1. 2/3 = .6 with a remainder of 2. 3/3 simply = 1

Well think of it. 0.3 repeating is 1/3. So multiply 0.3 repeating gives you 0.9 repeating, right? Cause 0.2 * 3 = 0.6, so likewise a 0.3 * 3 = 0.9 and a 0.333... * 3 = 0.999... What we are trying to say is that 0.9 repeating is a whole, just expressed in a funny way. So 3/3 equals 1 and 0.999... since they are the same thing. Same thing with 1.000... repeating then a 1 at the end. See, if we look at 1/3 just as 0.33, then 2/3 would be 0.66 and 3/3 would be 1. However, to make this whole thing true then that means that the last third must equal 0.34. Its the same thing with 0.999... repeating, just that we cant reach that last little bit since the threes will go on forever and we cant reach the end.

[† Mute point]

Multiplying 1/3 by 3 gives you 1. Not .999.....

Multiplying .3333..... by 3 on the other hand would, except that wasn't what you were doing.

Anyway, I'm tired of this, so I'll just go ahead and say it. All of the equations shown, except for the Convergence theorem (With an e, not a u Lemming, spelling is important, even in math) were entirely wrong for "proving" this point. The convergence theorem was exactly right for this. What I was trying to get you to do, was think outside of your math book, and come up with a way to explain this concept to an average person, without being completely wrong like you were in the initial example. Since you failed to do this, I can only conclude that you read about this in your math book, and posted it without a true understanding of the concept.

Dae I believe understands the concept very well, he came the closest to getting the point of what I was saying you should do. And unlike the rest of you, he did it without calling anyone an idiot.

[lemmingllama]

And unlike the rest of you, he did it without calling anyone an idiot.

I do understand that I was not explaining this as well as I could have, but it is difficult since I would explain this with numbers and it is difficult to do over a keyboard. Also, since we dont not know what your level of mathematics are, we had to assume at some points that you would already know certain things.

However, I do take offense that you are lumping me into a category without actually reading my posts. I never used foul language or called people names except their own username in a post.

[Dae314]

I never did like the first proof. You're kinda assuming things about infinity that may or may not be correct. I thought Mute was trying to refute the convergence theorem, which is why I was arguing, but I'm with mute that there are some things that just aren't clear about the algebraic proof.

It's a pretty well known idea that subtracting infinity from infinity is undefined. What you're trying to do with 9.999... - 0.999... is say that there is a 1-to-1 and onto relationship between the number of 9's after the decimal point in the first part (9.99...) and the number of 9's after the decimal point in the second part (0.999...). However, you can't assume that because there are an infinite number of 9's after the decimal point in both parts. In order for that expression to even out and become 9, you'd have to have some way to prove that the two infinities are the exact same size (lolwut but infinity doesn't have a "size"?). As it is, that subtraction is indeterminate.