Poll: 0.999... = 1

[Jaime_Wolf]

Actually, it didn't :-/ That proof works.

Third last step, at 2a(a-b) = a(a-b). a = b, so dividing by (a-b) is impossible. Yet they still do. So no.

The third last step is still valid, because a-b = 0, so both sides equal 0.

You could just as easily say

532531521215a(a-b) = a(a-b)
532531521215 = 1

which is equally as wrong.

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

You're doing it wrong, too. At step 4, 9x = 9, that is not true.

9x = 8.99999999



Edit: I'm not even a math major, but the amount of ignorant false intellectualism in this thread is about to make me cry.

Dont mock these people, this proof is correct. Let me write it differently for you.

EDIT: I see what you did there... your saying x = 1 because 0.999 rec = 1. So 9x SHOULD be 8.999

1/3 = 0.333 recurring. times 0.33 recurring by three. You know have 0.9999 recurring logically. However you have 3/3 if written as a fraction. This shows that 3/3 = 0.999 recuring AND 3/3 also = 1. Therefor 0.999 reccuring must also = 1.

No infinites there, just fractions of an irational nature. These are logical. These work. You cannot tell me that 3/3 does not = 1 but that 1/3 = 0.33333 recuring. These are both true. Leading to the logical conclusion that 0.99999 recurring = 1.

As has been repeatedly stated, that proof is extremely problematic in that most only accept that 1/3 = .333... because they've been conditioned to do so. The equations don't actually explain how or why .000...=1 in any serious way.

You need analytic proofs or proofs from the construction of the real numbers to really shed light on the question. I think every such well-known proof has already been posted and people should really be looking at those rather than trying to cling to these algebraic proofs.

 

[Subzerowings]

If x equals 0.999...
Then doesn't 10x equal 9.999....9990?

 

[Lyx]

Infinity is a concept, not a number, not a process.

Last time i checked, concepts can be processed (though, i guess that depends on how one defines "concept")

 

[Lukeje]

No. Your intuitive understanding of infinity is not the general mathematical understanding of infinity. It is you that are redefining infinity to fit your intuitive understanding.

In that case, i rather trust my "intuitive definition", which is based on actual observation what happens and logical considerations about what our minds CAN mean by saying "infinity", rather than some invented claims without proof (unless one considers an axiom a proof - i don't, and never will).

You've observed infinity?

I have observed what happens when the concept of infinity is applied And that is plain repetition. The "program" doesn't even contain the "instruction" to make the leap which you'd like it to make. Which is why infinity cannot exist - it doesn't ever make any final statement... it just repeats without ever "returning".

If the program stops in finite time, then it has not reached infinity. Thus you cannot have observed infinity.

 

[Lyx]

No. Your intuitive understanding of infinity is not the general mathematical understanding of infinity. It is you that are redefining infinity to fit your intuitive understanding.

In that case, i rather trust my "intuitive definition", which is based on actual observation what happens and logical considerations about what our minds CAN mean by saying "infinity", rather than some invented claims without proof (unless one considers an axiom a proof - i don't, and never will).

You've observed infinity?

I have observed what happens when the concept of infinity is applied And that is plain repetition. The "program" doesn't even contain the "instruction" to make the leap which you'd like it to make. Which is why infinity cannot exist - it doesn't ever make any final statement... it just repeats without ever "returning".

If the program stops in finite time, then it has not reached infinity. Thus you cannot have observed infinity.

Exactly my point

 

[Knusper]

I read this in the newspaper a few years back, and the evidence they had for it then was:

1/3 = 0.3333333...
therefore: 3/3 (which equals 1) = 0.99999...

and so therefore you have proof that 0.999... = 3/3 = 1

Also, can you think of a number which is bigger than 0.999... but smaller than 1? No? Well then they must be the same.

 

[Lukeje]

No. Your intuitive understanding of infinity is not the general mathematical understanding of infinity. It is you that are redefining infinity to fit your intuitive understanding.

In that case, i rather trust my "intuitive definition", which is based on actual observation what happens and logical considerations about what our minds CAN mean by saying "infinity", rather than some invented claims without proof (unless one considers an axiom a proof - i don't, and never will).

You've observed infinity?

I have observed what happens when the concept of infinity is applied And that is plain repetition. The "program" doesn't even contain the "instruction" to make the leap which you'd like it to make. Which is why infinity cannot exist - it doesn't ever make any final statement... it just repeats without ever "returning".

If the program stops in finite time, then it has not reached infinity. Thus you cannot have observed infinity.

Exactly my point

So what are you arguing? Just that the differentiation between 0.(9) and 1 makes no difference in the real world, and is as such just an intellectual exercise? I don't think anyone would deny that.

 

[Woodsey]

Yes.

We had this thread come up so many times at once a few months ago that anyone left not understanding why it is should be shot spoken to.

 

[Lyx]

Exactly my point

So what are you arguing? Just that the differentiation between 0.(9) and 1 makes no difference in the real world, and is as such just an intellectual exercise? I don't think anyone would deny that.

I'm argueing that the only thing someone has ever seen of infinity, is the process towards "infinity" - for a reason

(the unquoted infinity and quoted "infinity" have different meanings)


P.S.: That for practical applications (i.e. computations in programs to do something in the real world), there is no difference between 1 and 0.999... stopped at sufficient accuracy, i'd never deny. The discussion is irrelevant for practical applications in tech.... it does however have strong conceptual rammifications.... things like mathematical points for example are used a lot in physics - and as a result, physics claims that there are objects in reality that contain infinities.

 

[quhouv vhqwiufv qwiu]

Actually, it didn't :-/ That proof works.

Third last step, at 2a(a-b) = a(a-b). a = b, so dividing by (a-b) is impossible. Yet they still do. So no.

The third last step is still valid, because a-b = 0, so both sides equal 0.

You could just as easily say

532531521215a(a-b) = a(a-b)
532531521215 = 1

which is equally as wrong.

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

You're doing it wrong, too. At step 4, 9x = 9, that is not true.

9x = 8.99999999



Edit: I'm not even a math major, but the amount of ignorant false intellectualism in this thread is about to make me cry.

Dont mock these people, this proof is correct. Let me write it differently for you.

EDIT: I see what you did there... your saying x = 1 because 0.999 rec = 1. So 9x SHOULD be 8.999

1/3 = 0.333 recurring. times 0.33 recurring by three. You know have 0.9999 recurring logically. However you have 3/3 if written as a fraction. This shows that 3/3 = 0.999 recuring AND 3/3 also = 1. Therefor 0.999 reccuring must also = 1.

No infinites there, just fractions of an irational nature. These are logical. These work. You cannot tell me that 3/3 does not = 1 but that 1/3 = 0.33333 recuring. These are both true. Leading to the logical conclusion that 0.99999 recurring = 1.

As has been repeatedly stated, that proof is extremely problematic in that most only accept that 1/3 = .333... because they've been conditioned to do so. The equations don't actually explain how or why .000...=1 in any serious way.

You need analytic proofs or proofs from the construction of the real numbers to really shed light on the question. I think every such well-known proof has already been posted and people should really be looking at those rather than trying to cling to these algebraic proofs.

You can use long division to prove that 1/3 = 0.333...

 

[skeliton112]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

However 0.999... < 1 as 1x10 = 10
0.999...x10 = 9.999...
Therefore 0.999...(not equal)1

SEE HERE I AM STILL RESPONDING.

Your argument is a logical fallicy. You're saying .99... doesn't equal 1 because .99... doesn't equal 1.

9.99... = 9 + .99...
9.99... = 9 + 1 (since .99... = 1, as he showed in his proof)
9.99... = 10

Dammit saw through it. Well ummm

Ok fine ill stop trying to dispove it. I already know it is true.

 

[Coldie]

@Jaime

I disagree. .999... is a process, not a number. I have never seen anything else, and i cannot even MEAN anything else... this is not just a matter of "imagination".... the value which you claim infinity is, is not constructable in our minds. It also isn't constructable in a machine. In fact, it NOWHERE is constructable.

It does not exist. All that exists is a looped process.

It exists in math. N-dimensional lattices exist in math, but you can't construct one in your mind. Just because something may be incomprehensible, does not mean it doesn't exist. 0.(9) is a number. Some of its representations are infinite. There are no processes there, they just are.

Math is axiomatic and absolute. If a system or a theory says something works in a certain way, then it just does, within that system. There is no intuition, there is nothing to comprehend, there is only Math and its laws, as defined by the System's Postulates. If you deny an axiom and substitute your own, you create a new system with a new ruleset. If you do it as a part of a proof, the proof is invalid in the original system and therefore irrelevant.

Sufficiently advanced math is indistinguishable from magic.

 

[Jaime_Wolf]

@Jaime

I disagree. .999... is a process, not a number. I have never seen anything else, and i cannot even MEAN anything else... this is not just a matter of "imagination".... the value which you claim infinity is, is not constructable in our minds. It also isn't constructable in a machine. In fact, it NOWHERE is constructable.

It does not exist. All that exists is a looped process.

Then you disagree with the question, not the result (since the answer of an incoherent question cannot be true or false). Assuming you take the view that this is an erroneous question, you should probably realize that you are, in the end, arguing that all real numbers are only processes (Values having multiple representations stems from the nature of constructing representations of real numbers. So you can, for any real number, create a number system that forces you to conclude that the values are processes). As soon as you take such a view, you either have to abandon a notion of equality altogether or you have to say that they're equal. So nothing is really ever gained.

More simply, .999... is very definitely a number. The fact that you can't conceive of it as a number is a problem with your own understanding of mathematics. To say that one can't mean anything else is downright false, since I certainly mean a number when I say .999...

You're also verging on deeper epistemological questions, which would derail this thread and which I don't really want to try to deal with given a naive audience on the internet. Suffice it to say that claiming that infinity doesn't "exist" as anything other than a process is an extremely problematic view to hold.

 

[Lyx]

Math is axiomatic and absolute. If a system or a theory says something works in a certain way, then it just does, within that system. There is no intuition, there is nothing to comprehend, there is only Math and its laws, as defined by the System's Postulates. If you deny an axiom and substitute your own, you create a new system with a new ruleset. If you do it as a part of a proof, the proof is invalid in the original system and therefore irrelevant.

Sufficiently advanced math is indistinguishable from magic.

That is part of my dislike for maths and how it is abused to for real-world modelling tasks, in which it doesn't belong. See my P.S. in my previous post. If weird rules just affect stuff inside a theoretical system - fine. If those rules result in massive logical breaks when making explanations about reality, not fine.

 

[brunothepig]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

No, no no no. The problem with that proof is the whole infinity thing. Infinity is just a concept. Basically, 10x0.999.. should equal 9.999..8
Because infinity is supposed to be, well, forever, the 8 is kinda ignored in that proof, but it should be on the end of that never ending number. You see why it's a problem?
OT: I think my stance on this is rather obvious. 0.9 recurring doesn't equal 1. It is in fact, 1x10^(-infinity) less.

 

[Lukeje]

Math is axiomatic and absolute. If a system or a theory says something works in a certain way, then it just does, within that system. There is no intuition, there is nothing to comprehend, there is only Math and its laws, as defined by the System's Postulates. If you deny an axiom and substitute your own, you create a new system with a new ruleset. If you do it as a part of a proof, the proof is invalid in the original system and therefore irrelevant.

Sufficiently advanced math is indistinguishable from magic.

That is part of my dislike for maths and how it is abused to for real-world modelling tasks, in which it doesn't belong. See my P.S. in my previous post. If weird rules just affect stuff inside a theoretical system - fine. If those rules result in massive logical breaks when making explanations about reality, not fine.

Can you explain where this leads to a logical break in our explanation of reality please?

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

No, no no no. The problem with that proof is the whole infinity thing. Infinity is just a concept. Basically, 10x0.999.. should equal 9.999..8
Because infinity is supposed to be, well, forever, the 8 is kinda ignored in that proof, but it should be on the end of that never ending number. You see why it's a problem?
OT: I think my stance on this is rather obvious. 0.9 recurring doesn't equal 1. It is in fact, 1x10^(-infinity) less.

What you're missing is that 1x10^(-infinity) is defined to be equal to zero.

 

[Lyx]

Math is axiomatic and absolute. If a system or a theory says something works in a certain way, then it just does, within that system. There is no intuition, there is nothing to comprehend, there is only Math and its laws, as defined by the System's Postulates. If you deny an axiom and substitute your own, you create a new system with a new ruleset. If you do it as a part of a proof, the proof is invalid in the original system and therefore irrelevant.

Sufficiently advanced math is indistinguishable from magic.

That is part of my dislike for maths and how it is abused to for real-world modelling tasks, in which it doesn't belong. See my P.S. in my previous post. If weird rules just affect stuff inside a theoretical system - fine. If those rules result in massive logical breaks when making explanations about reality, not fine.

Can you explain where this leads to a logical break in our explanation of reality please?

How many infinities (including mathematical points) can you spot in physics? Why is it that the maths which use those do work, and yet, the infinities themselves have never been observed directly? Here's a hint: perhaps the most reasonable use of a mathematical point, for calculating reality, is like a "position-marker" (while never actually using it infinitely precise... its just there in the models, and then in practice gets enough precision as necessary).

P.S.: I could go into more detail, about how numbers and the ranges in-between, are directly derived from how we "address" things in our perception - and thus, actually come from something very intuitive and imaginative... and how this mechanic resulted in a lot of misunderstandings.... including the wave/particle dualism... but this would derail the thread too much.

 

[smithy_2045]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

No, no no no. The problem with that proof is the whole infinity thing. Infinity is just a concept. Basically, 10x0.999.. should equal 9.999..8
Because infinity is supposed to be, well, forever, the 8 is kinda ignored in that proof, but it should be on the end of that never ending number. You see why it's a problem?
OT: I think my stance on this is rather obvious. 0.9 recurring doesn't equal 1. It is in fact, 1x10^(-infinity) less.

Which is equivalent to zero. Which means there is no difference. Which means .999... = 1

 

[Piflik]

I have a similar proof to the one with 0.9999... = 1

See if you can find the mistake and you have the same mistake for the other one...

1-1 = 0
(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1) = 0
(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+..... = 0
1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+..... = 0
1 = 0

See...perfectly correct and logical...

Spoiler: The mistake

The mistake is how people deal with infinity. One cannot write an infinite number of ciphers, so we stop at a certain point and just hint at infinity (by using '...' for example). But if you want to do real math with infinite numbers, you cannot just hint at infinity, you must take it into account completely.

So to proof that 0.99999 =/= 1 we write 0.999999...99 instead (the last two ciphers symbolize the theoretical 'end' of the infinite number)

x = 0.999999...99
10x = 9.999999...90
10x-x = 9.99999...90 - 0.999999...99
9x = 8.9999999...91
x = 0.999999...99

Here you go.

In my example there is a -1 still waiting at the end of infinity that I omitted.

 

[ultimateownage]

No it's not, but the difference between 0.9*[footnote]the * represents recurring.[/footnote] and 1 is 0.0*1 and since there are infinite 0's then the difference is infinitely small.
Anyone with a basic understanding of maths can tell the difference between equal to and infinitely close to.
Also, this topic is used too much.

On a side note, the last thread I saw of this had the most stupid post I saw in a long time. It went something like a=1 b=0.9* and abc=2 therefore ab=2 and b=1.