Thread: Does 0.9 repeated = 1?

Originally Posted by blade5x

0.999... goes to 1 at infinity. Infinity cannot be reached, therefore 0.999... does not equal 1.

Fractions represent decimals to infinity, that's why it works out where some people where able to make it seem that 0.999... equals 1.

but how come 9/9 equals both 1 and 0.9 rec. then?

Originally Posted by Universal Mind

No. So, is there such thing as 0.000...1?

i dont see what that has to do with it, but it doesnt matter anyway because i was just clearing up the fact that it is possible to have no numbers inbetween two numbers.

Originally Posted by slash112

i dont see what that has to do with it, but it doesnt matter anyway because i was just clearing up the fact that it is possible to have no numbers inbetween two numbers.

Somebody else was talking about 0.000...1, and I asked you if such a number is possible because you said there is no such thing as a step past infinity. I was trying to bring you into that subdiscussion, and anybody else who might want to take a stab at it.

Originally Posted by Universal Mind

There is an infinitely small boundary between them. If two rectangles share a side, there is nothing separating them, but they are still separate rectangles. The fact that nothing separates two numbers does not prove that they are the same number.

I believe I see a problem with this paradox. There are 2 rectangles, let's suppose 2 units high each. You argue that the sides are touching, there is continuity, yet there still exists a space between them. The height is 4 here. Basically you're saying that the height of the first box is from 0 to 2, while the second box would have to be 2.0...1 to 4.0...1, to distinguish the border. Your circumstances claim the height is 4.0...1.

For our purposes, it is the same.

Originally Posted by Universal Mind

Somebody else was talking about 0.000...1, and I asked you if such a number is possible because you said there is no such thing as a step past infinity. I was trying to bring you into that subdiscussion, and anybody else who might want to take a stab at it.

you know something, i never thought about it, and i never realsed someone brought it up, this brings a whole other side to the arguement. let me think about it for a while, i would instictivly say yes it is possible, but i need to think logically...

Originally Posted by TimeStopper

I believe I see a problem with this paradox. There are 2 rectangles, let's suppose 2 units high each. You argue that the sides are touching, there is continuity, yet there still exists a space between them. The height is 4 here. Basically you're saying that the height of the first box is from 0 to 2, while the second box would have to be 2.0...1 to 4.0...1, to distinguish the border. Your circumstances claim the height is 4.0...1.

For our purposes, it is the same.

The first box is 0-2, and the second box is 2-4. The boundary has no width, which apparently means it has a width of 0.000...1, which is 0.

This is one fucked up universe.

Can infinity be reached?

The 9's on the piece of paper never get to a point where the number has a value of 1, yet they do.

But that's exactly what the ~ symbol means anyway; that infinity has been reached.

Again, a number is not a process. It is not a matter of anything ever being 'reached' because the number has one value which does not increase. The tilde shows us that infinity has been reached. You only need to put a couple of 9s on paper.

You've got to bear in mind that mathematics is an abstract system. You can do things which don't have any intuitive meaning in reality whatsoever, such as raising numbers to the power of zero, or taking the root of a negative. When we answer the question 'what is the root of minus 1' and answer 'i'; i is just a symbol. It does not have some kind of objective reality at the heart of the universe or anything like that: it is just a symbol which we arbitrarily define as being the answer to that question, not because that is 'true' in reality, but because the particular system that arises is incredibly useful.

At least in every case you know of.

What could you possibly mean?

Such a proof about the nature of numbers applies to every pair of different numbers you can possibly think of. There's only one case.

Originally Posted by TimeStopper

I believe I see a problem with this paradox. There are 2 rectangles, let's suppose 2 units high each. You argue that the sides are touching, there is continuity, yet there still exists a space between them. The height is 4 here. Basically you're saying that the height of the first box is from 0 to 2, while the second box would have to be 2.0...1 to 4.0...1, to distinguish the border. Your circumstances claim the height is 4.0...1.

For our purposes, it is the same.

that is very true, which brings us back to the arguement of wether 0.000...01 is possible.

it the amount of zeros could be infinity minus 1, that would make it possible, infact it just seems too obvious, so i dunno.

Originally Posted by Xei

But that's exactly what the ~ symbol means anyway; that infinity has been reached.

That is where things get insane, though real.

Originally Posted by Xei

Again, a number is not a process. It is not a matter of anything ever being 'reached' because the number has one value which does not increase. The tilde shows us that infinity has been reached. You only need to put a couple of 9s on paper.

From left to right, it increases. That does not mean the number itself increases. It is what it is. But each 9 digit after the decimal adds value to the value of the number that precedes it. The paradox is that the additions result in the reaching of infinity even though how they do it seems impossible and is unexplainable, as far as I can tell.

Originally Posted by Xei

You've got to bear in mind that mathematics is an abstract system. You can do things which don't have any intuitive meaning in reality whatsoever, such as raising numbers to the power of zero, or taking the root of a negative.

Raising a number to the power of 0 makes sense. It means the number is represented 0 times in that instance but the term does not change the value of anything multiplied by it. That makes the term have to have a value of 1. However, the square root of a negative number is imaginary. Imaginary numbers are hypothetical figures that are not considered part of reality. It is a matter of applying real rules to unreal numbers. 1 is a real number, so it should make sense no matter how it is approached or dissected.

Originally Posted by Xei

When we answer the question 'what is the root of minus 1' and answer 'i'; i is just a symbol. It does not have some kind of objective reality at the heart of the universe or anything like that: it is just a symbol which we arbitrarily define as being the answer to that question, not because that is 'true' in reality, but because the particular system that arises is incredibly useful.

Reality does not have to apply completely to imaginary numbers.

Originally Posted by Xei

What could you possibly mean?

I am talking about the limits of inductive reasoning while admittedly playing devil's advocate.

Originally Posted by Xei

Such a proof about the nature of numbers applies to every pair of different numbers you can possibly think of. There's only one case.

So far...

I agree that 1 = 0.999... and that there are no numbers between them. But questioning reality is always fun.

Originally Posted by Universal Mind

The 9's on the piece of paper never get to a point where the number has a value of 1, yet they do.

At least part of the paradox is probably saying never when talking about infinity. By definition, the end of infinity is "never".

Er.. my head hurts now.

By the way, I don't thing that rectangles analogy is really applicable, since a real number has no width. More like two lines sharing a side.

Originally Posted by RedfishBluefish

At least part of the paradox is probably saying never when talking about infinity. By definition, the end of infinity is "never".

But when it is also an "ever", it is paradoxical.

Originally Posted by RedfishBluefish

By the way, I don't thing that rectangles analogy is really applicable, since a real number has no width. More like two lines sharing a side.

Good point. But, uh, how can I rationalize against that? Then why is it that a 10 foot wide wall is so much wider than a 1 foot wide wall?

Okay, I am going to stop playing devil's advocate right about now. I don't like arguing stuff I don't agree with. That is why I don't practice law.

i dont really want to believe that the two numbers are the same, but you cant argue with the proof. e.g. the thing i said earlier, about 1/9 = 0.1 rec. * 9 = 0.9 rec. and 1/9 * 9 = 9/9 even thouh 9/9 = 1 usually, so it equals two things. so they must be the same.

From left to right, it increases. That does not mean the number itself increases. It is what it is. But each 9 digit after the decimal adds value to the value of the number that precedes it. The paradox is that the additions result in the reaching of infinity even though how they do it seems impossible and is unexplainable, as far as I can tell.

Well if you still can't grasp this intuitively then you'll have to just accept it.

This is almost always the case in more advanced mathematics; many results you cannot comprehend at all, but as long as you have used the correct rules for shunting symbols, then you cannot argue with it. It is so because you have proven it, and that is that.

Raising a number to the power of 0 makes sense. It means the number is represented 0 times in that instance but the term does not change the value of anything multiplied by it. That makes the term have to have a value of 1. However, the square root of a negative number is imaginary. Imaginary numbers are hypothetical figures that are not considered part of reality. It is a matter of applying real rules to unreal numbers. 1 is a real number, so it should make sense no matter how it is approached or dissected. Reality does not have to apply completely to imaginary numbers.

2^3 means 2 multiplied by itself 3 times.
2^2 means 2 multiplied by itself 2 times.
2^1 means 2 multiplied by itself 1 time (even this is already on shaky terms with regards to experience)
2^0 means 2 multiplied by itself 0 times. It is very hard to give any physical meaning to this.
And even if you can; what is 2^-1, physically? 2 multiplied by itself -1 times? How can you multiply 2 by 2 -1 times?

The stuff you say about imaginary numbers shows that you have very little understanding of what maths is, and I often hear it from people who think they know what an imaginary number is but have never actually been taught about them or done any calculations involving them. Maths is a formal system with which we can sometimes see isomorphisms with reality, but maths itself is not real. A 'REAL number' does not mean a number which is real, it means a number in the logical system with no IMAGINARY part.

An IMAGINARY number does not mean a number which is not real, it is simply a technical term for a number with no REAL part.

A complex number has an IMAGINARY part and a REAL part.

Let me try to explain via matricies. The follow matrix is isomorphic to a 1:

1 0
0 1

The following matrix is isomorphic to i:

0 -1
1 0

What you're essentially saying is that
1 0
0 1
has physical existence but
0 -1
1 0
doesn't.

Which is clearly mad. Both of these things form a coherent system which can sometimes be used to model reality, but neither REAL numbers nor IMAGINARY numbers are 'real' in the sense you mean; having physical existence.

This is what Calculus is for. Take a limit of some function, at infinity. That's what it is at infinity, in this case 1. And the limit is just what the function continuously approaches, but never reaches.

There's a reason when you take a Calc exam these types of questions ask "what is the limit of f(x) as n->∞?" and not "what is f(x) at n=∞?"

I believe I see a problem with this paradox. There are 2 rectangles, let's suppose 2 units high each. You argue that the sides are touching, there is continuity, yet there still exists a space between them. The height is 4 here. Basically you're saying that the height of the first box is from 0 to 2, while the second box would have to be 2.0...1 to 4.0...1, to distinguish the border. Your circumstances claim the height is 4.0...1.p\

I like this one, but I believe you can just say the borders have a width "dx" or "dy" - infinitesimally thin.

There's no paradox here in my opinion. The only paradox I see here is everyone is trying to use simple things like fractions/decimals and whole numbers to explain infinities.

Originally Posted by Xei

Well if you still can't grasp this intuitively then you'll have to just accept it.

This is almost always the case in more advanced mathematics; many results you cannot comprehend at all, but as long as you have used the correct rules for shunting symbols, then you cannot argue with it. It is so because you have proven it, and that is that.

I'm glad you finally admit it.

Originally Posted by Xei

2^3 means 2 multiplied by itself 3 times.
2^2 means 2 multiplied by itself 2 times.
2^1 means 2 multiplied by itself 1 time (even this is already on shaky terms with regards to experience)
2^0 means 2 multiplied by itself 0 times. It is very hard to give any physical meaning to this.
And even if you can; what is 2^-1, physically? 2 multiplied by itself -1 times? How can you multiply 2 by 2 -1 times?

You just have to accept it, right? But I gave meaning to the exponent of 0. Is that what set off your asshole response? I knew it would be coming in right about the time you felt stumped. So predictable. Get help, seriously.

First of all, 2^-1 is not paradoxical. It presents no apparent contradictions we can't explain away. What does it mean? It means 2 represented -1 times, which is 2 represented 1 time in reciprocal form. 2^-1 = 1/2. The negativity of exponents creates reciprocality due to the geometric aspect and not simple left movement on a number line, which is arithmetic.

Also, 2^-1 does not represent "multiply 2 by 2 -1 times". An exponent represents the number of times a number is represented, not how many times it is multiplied by itself. A number with an exponent of 1, for example, is represented 1 time, not multiplied by itself 1 time. A number with an exponent of 2 is multiplied by itself 1 time since it is represented 2 times.

Originally Posted by Xei

The stuff you say about imaginary numbers shows that you have very little understanding of what maths is.

I have taught a lot of math courses, and I wrote an algebra textbook. You need to give up on your desperate rationalization. It is dishonest and pathetic, along with being just really shitty. See if you can talk about math and not me. Do you think you can do that? I will stick to the subject if you will. Let's see what you can do.

Originally Posted by Xei

Maths is a formal system with which we can sometimes see isomorphisms with reality, but maths itself is not REAL. A 'real number' does not mean a number which is REAL, it means a number in the logical system with no IMAGINARY part.

The real numbers are in fact real. The imaginary numbers are in fact imaginary. If real numbers are not real, why didn't they just make pi = 3? Humans invented the symbols but discovered the numbers. Math is completely logical, and we do not decide on its rules or its components, only its symbols.

Originally Posted by Xei

An IMAGINARY number does not mean a number which is not real, it is simply a technical term for a number with no REAL part.

... which makes it not real. The square root of -1 does not exist. However, you are looking at 1 post right now.

Originally Posted by Xei

Let me try to explain via matricies. The follow matrix is isomorphic to a 1:

...

Which is clearly mad. Both of these things form a coherent system which can sometimes be used to model reality, but neither REAL numbers nor IMAGINARY numbers are 'real' in the sense you mean; having physical existence.

A matrix is a human system for listing and working with numbers. They are not numbers themselves. They involve real operations, but they are not naturally existing operations in and of themselves.

I did not say real numbers have physical existence. I said they are real. Matter is not the only form of reality.

Now tell me... Why didn't they just make pi = 3?

Originally Posted by blade5x

That's what it is at infinity, in this case 1. And the limit is just what the function continuously approaches, but never reaches.

"At infinity" is paradoxical because infinity is not a number. "At infinity" is like "at the end of space".

The "but never reaches" situation is what is at the root of the paradox. It cannot reach, so it seems, yet it does reach.

Originally Posted by Universal Mind

Perhaps that is not always true.

It is.

Devil's advocate there. I am convinced that 1 = 0.999..., and my part in this thread isn't about whether or not the two figures are equal. I am just saying that it is a paradox that has not been completely explained.

But it isn't a paradox. It's just an alternate way of writing '1'. They are equal, and you can use them interchangeably.

What is 0.00000...1? The figure suggests that there is something on the other side of infinity. Does it not? It definitely suggests that there is something, which is not nothing. That is part of the paradox. Yes, your math is accurate, but notice the paradox.

It's a reversed pattern of 0.10... It means infinite amount of zeros and then a '1'. Which makes it zero BECAUSE there is nothing on the other side of infinity.

You are just reasserting your conclusion and not dissecting the paradox. I am calling into question certain aspects of the fact that the two figures are equal, not asking you to state again that they are equal. Telling me that one is a way of writing the other adds nothing to your conversation with me except a reassertion of the fact that I am trying to dissect.

Then you might as well dissect why the numbers .5 and 1/2 are the same number.

You can't give me any explanation other than they are interchangable and their values are equal, and then I could go on to say that there must then be some mystical 'other kind' of number that determines true value, right?

Well, no. There isn't. They're the same because they are equivalent. They are two representations of the same value, that is it.

The border has no width, yet it is a border. The border between the rectangles is just as wide as the border between 0.999... and 1?

Devil's advocate again.

No, there is no border. They aren't two rectangles sharing a side, they're a rectangle that's been labelled twice.

You are now officially on my troll watch list.

Just in case you or somebody else STILL doesn't get the paradox, I will explain it another way. And I am not calling into question whether the two figures are equal, so don't waste your time by telling me they are just two ways of writing the same number. That fact is exactly what I am trying to explain, not counter. Understand?

No, because this is math and ultimate truth in it is represented by using itself to demonstrate that something works.

As long as you can show that x = y, then it is true. There is nothing else here to look at.

Okay, imagine the number 0.999... written on a piece of paper that goes forever. With every next digit, the number represented up to that digit is a little closer to 1. So how far along the number is there a point when the digits up to that point equal 1? A trillion light years? Quadrillion to the octillionth power light years? It never happens, ever, obviously.

Exactly. They aren't they same way of writing the number, obviously. They both represent the same value. 0.9... Will never 'become' 1, but it is of the same value as 1, because that's how maths works.

Infinity has no end. So the number can never reach 1, ever. Right? So the stretched out number that never can possibly get to 1 gets to 1 because that is the number that the entire number is. It can't reach 1, but it reaches 1.

Considering the very specific issue I have raised, why are the two figures representative of the same number, which they are? And please don't tell me what a converging geometric series is. That is the very thing I am calling into question.

You aren't getting it.

.9... DOESN"T reach one. It isn't 1. It's .9... However, they both represent the same value.

This is the same reason that 1/2 never 'becomes' 2/4 or 2^-2 1/4.

They are the same, again, because:

1. there are no boundaries between the numbers on a number line
2. they represent the same value

Originally Posted by A Roxxor

But it isn't a paradox. It's just an alternate way of writing '1'. They are equal, and you can use them interchangeably.

Hey troll, you know I have addressed that about 10 times in this thread, unless you are not reading my posts. I addressed you specifically yesterday when I said I am trying to dissect your assertion and not counter it. Your response causes your argument to be circular. Look into it.

Originally Posted by A Roxxor

It's a reversed pattern of 0.10... It means infinite amount of zeros and then a '1'. Which makes it zero BECAUSE there is nothing on the other side of infinity.

The way the number is written suggests that there is something on the other side of infinity.

Originally Posted by A Roxxor

Then you might as well dissect why the numbers .5 and 1/2 are the same number.

Okay, you are definitely trolling. You are repeating Drew's argument that was already thoroughly covered. The rest of your post is about countering the claim that the two numbers are not equal, and I have thoroughly and repeatedly explained that I am not taking issue with the claim that the numbers are equal except to point out the paradoxical nature of the fact while agreeing that it is a fact. Please do something with your life. Thanks.

*Spews out Raspberry Tea from mouth* A Roxorr isn't banned?!? No offense, but I thought that you were...

Originally Posted by spockman

*Spews out Raspberry Tea from mouth* A Roxorr isn't banned?!? No offense, but I thought that you were...

His last incarnation was, and his new one probably will be any minute now.

I'm glad you finally admit it.

Admit what?

I said if you can't intuitively accept it then you'll just have to accept the proof.

I personally have no problem with it and see no paradox.

You just have to accept it, right? But I gave meaning to the exponent of 0. Is that what set off your asshole response? I knew it would be coming in right about the time you felt stumped. So predictable. Get help, seriously.

I'm trying have a reasoned discussion here. There was nothing remotely inflammatory about my response. Why are you responding like that?

First of all, 2^-1 is not paradoxical. It presents no apparent contradictions we can't explain away. What does it mean? It means 2 represented -1 times, which is 2 represented 1 time in reciprocal form. 2^-1 = 1/2. The negativity of exponents creates reciprocality due to the geometric aspect and not simple left movement on a number line, which is arithmetic.

Also, 2^-1 does not represent "multiply 2 by 2 -1 times". An exponent represents the number of times a number is represented, not how many times it is multiplied by itself. A number with an exponent of 1, for example, is represented 1 time, not multiplied by itself 1 time. A number with an exponent of 2 is multiplied by itself 1 time since it is represented 2 times.

Yes sorry I meant 2^3 means 2 multiplied by itself 3-1 times, etecetera.

How can you 'represent' a number a negative amount of times? There is very little intuitive meaning you can give to this.

2^5 = 2 * 2 * 2 * 2 * 2
2^4 = 2 * 2 * 2 * 2
2^3 = 2 * 2 * 2
2^2 = 2 * 2
2^1 = 2

2^-1 = ?

The fact is negative numbers are just as 'unreal' as imaginary numbers. You can't physically represent a negative number of objects.

You need to give up on your desperate rationalization. It is dishonest and pathetic, along with being just really shitty.

What on Earth is wrong with you?

I wholeheartedly mean everything I say, otherwise I would not say it. You think I have some sort of agenda here?? My view also happens to agree with every piece of literature about the nature of mathematics I've ever read. I suggest you read some books about formal systems such as Godel Escher Bach or The Emperor's New Mind.

The real numbers are in fact real. The imaginary numbers are in fact imaginary. If real numbers are not real, why didn't they just make pi = 3? Humans invented the symbols but discovered the numbers. Math is completely logical, and we do not decide on its rules or its components, only its symbols.

Maths does not have monopoly on logic actually.

There are true statements about numbers which cannot be proved with maths.

The first thing you should have been taught when learning about IMAGINARY numbers is that IMAGINARY is a technical term, not the adjective imaginary. The nomenclature comes from an old prejudice against French mathematics. It is now completely redundant.

pi is something which results from Euclid's postulates. Euclid's postulates are a model, and are actually physically wrong to high degrees of precision. But using Euclid's postulates and calculus, we can calculate that the number pi is equal to

4/1 - 4/3 + 4/5 - 4/7...

This is the only value that pi can have. It is not arbitrarily defined.

... which makes it not real. The square root of -1 does not exist. However, you are looking at 1 post right now.

Yes, but I'm not looking at the number '1' right now, am I?

You are using '1' to describe objects in physical reality.

Equally I could describe various physical phenomena using i (and could not without).

I could also use 'quite long' to describe your post. This doesn't mean that 'quite long' exists.

A matrix is a human system for listing and working with numbers. They are not numbers themselves. They involve real operations, but they are not naturally existing operations in and of themselves.

1 and
10
01
are exactly the same thing. How can you claim that one way of representing it is real yet another is not? I can perform exactly the same operations upon it and get exactly the same answers.

I did not say real numbers have physical existence. I said they are real. Matter is not the only form of reality.

Okay, real numbers do not have physical existence. Neither do imaginary numbers. This is obvious. But this 'other reality' which you hint at; why does i not belong there whilst 1 does?

Originally Posted by Xei

2^5 = 2 * 2 * 2 * 2 * 2
2^4 = 2 * 2 * 2 * 2
2^3 = 2 * 2 * 2
2^2 = 2 * 2
2^1 = 2

2^-1 = ?

ok, 2 to the power of -1 is 1/2, any idiot knows that.

and by the way, from a post you made earlier aswell, 2 to the power of 0 is 1. anything to the power of 0 is 1, any idiot knows that aswell.

ok, 2 to the power of -1 is 1/2, any idiot knows that.

and by the way, from a post you made earlier aswell, 2 to the power of 0 is 1. anything to the power of 0 is 1, any idiot knows that aswell.

Yes, any idiot clearly does.

Originally Posted by Xei

Yes, any idiot clearly does.

yes, and you didnt.

where do you come from and what age areyou, because if you are over 14/15 and live in britain then you should really know that, because its in all of britains school ciriculum, dont know about anywhere else though

I know exactly what the answer is, thank you. You have just completely failed to comprehend what my post is about.

I got an A in A Level maths a year early and I managed to pass interview and get an offer to read maths at Cambridge university, which has the best and hardest maths course in the world.

So yes, I know what 2^-1 is.

._.