0.9 repeater = 1

[Rasmus]

How much time are you going to give me? It would take forever and a day.

No, it takes two iterations to show that there *really* are going to be an endless series of threes in the result. There is no need to keep on wirting down stuff once you have

a) worked out that 3 fits into 10 three times and leaves a rest of 1 and
b) reached a point where you have to divide a number that you have already dome the division for: From that point onwards, you have a repeating sequence of digits.

I'm still pretty sure about the second part of my post you didn't include though:

But that depends on whether you agree that 0,333 ..... is *exactly* 1/3, If you do agree, then you can show that 0,9999.... is exactly the same as 1.

1/3 + 1/3 + 1/3 = 3/3 = 1
0,333.... + 0,333.... + 0,333.... = 0,999....

IFF 1/3 = 0,333... then it is also true that 1 = 0,999....
IF you think there is a difference between 1/3 and 0,333.... then the proof will indeed not convince you.

Just to make clear, I don't see a problem with the multiplication proof that has been quoted a few times already in this thread (the, 10x | x = .999 | etc., one ).

Funny. I do. (Because my mind cannot cope with the idea of extending an endless series by one more digit, let alone anything other than 0)

Do you disagree, then, that 1/3 really equals 0,3333.... ?

If so, have you done the division on paper and do you get a different result?

 

[sol invictus]

You forget that I may have been taught wrong, in a backwater mudhole.

Fair point.

How much time are you going to give me? It would take forever and a day.

Really? I could convince myself that 1/3=.333... with an infinite number of 3s in about 10 seconds. You probably can too. Just start doing the long division and look for a pattern .

I'm still pretty sure about the second part of my post you didn't include though:

I think most people are comfortable with the fact that 1/3=.333.... . If so, one can start from that and use it to convince them that .999... = 1. If not, one needs to start somewhere else.

This isn't about rigor, because there's no problem here for anyone that knows mathematics. It's about convincing other people, and the best way to do that depends on where they start from.

 

[This is The End]

How much time are you going to give me? It would take forever and a day.

<snip>

Really? I could convince myself that 1/3=.333... with an infinite number of 3s in about 10 seconds. You probably can too. Just start doing the long division and look for a pattern .

I should have put joke tags around that one...

 

[This is The End]

Do you disagree, then, that 1/3 really equals 0,3333.... ?

That wasn't the point. It was that I don't see a difference between 1/3 = .333... and 1 = .999... and, like someone said in a previous post, 1/4 = .24999.. = .25. So you shouldn't be using one in the proof of another.

 

[Rasmus]

That wasn't the point. It was that I don't see a difference between 1/3 = .333... and 1 = .999... and, like someone said in a previous post, 1/4 = .24999.. = .25. So you shouldn't be using one in the proof of another.

So I shouldn't use stuff you know and agree with to logicically deduce what it I want to convince you of?

 

[This is The End]

So I shouldn't use stuff you know and agree with to logicically deduce what it I want to convince you of?

Believe me when I say I understand what you are saying, and that, actually, I agree with you. But! whether or not you or I agree about something doesn't really have anything to do with how well it works in a proof.

Even if it was exactly the same the other way around (which I agree with you that it isn't), I wouldn't want to see .999... = 1 used in a proof of .333 = 1/3 either.

Either way, I would concede to you (and Sol) since, I am assuming, I have many more years between myself and any dedicated math study than you... not to mention I was taught by wolves .

 

[gnome]

Believe me when I say I understand what you are saying, and that, actually, I agree with you. But! whether or not you or I agree about something doesn't really have anything to do with how well it works in a proof.

But, it can get rid of that "just bugs me" feeling when the result of a proof seems counter-intuitive.

 

[rwguinn]

anybody that wants to argue about anything less than 1 part in 1e9 is either a biologist, astronomer, or idiot.
Or works for NASA as an engineer.

 

[Rasmus]

Either way, I would concede to you (and Sol) since, I am assuming, I have many more years between myself and any dedicated math study than you... not to mention I was taught by wolves .

Oh, I don't know a whole lot of math ....

anyway, the way I see it is this:

My proof works if you agree with the premises.
i.e. if you understand that any why there are an infinite number of threes after the comma for 1/3 it follows that 0,999.... is exactly the same as one.

If you dissagree with the premises I have to options:

Chose a different proof or convince you of the premises I am using.

But when you agree that 1/3 = 0,333 .... then any proof will do, even if it is utterly trivial.

 

[marplots]

I wonder if the same mechanism that gives the 'feeling' that 1/3 does not equal 0.3333... also gives problems with other types of constructs.

Does 1/4 equal 4/16? Or is there some nuanced difference? Are they different ways of writing the same thing or is there some qualia difference?

Maybe the distinction that instinctive math is pointing to is similar to the way I might be perceived differently if you called me "Bill" or "Dad." It might be hard for someone to glom onto the fact that calling X, "one" or calling it 0.999... doesn't change what X is.

Isn't it part of traditional magical thinking that the naming of a thing has some influence on that thing? So, when I call you "Polonius" or "Two-bits" or "Fat Cheeks Mcgee" I am somehow altering what you are? I'm searching for a psychological reason why some people don't accept the proofs offered. Are they more vulnerable to buying products with prices that end in .99?

 

[Third Eye Open]

Ahh, I remember this I also remember when I didn't understand the Monty Hall problem. That was such a satisfying 'click' in my brain when I finally did

 

[marplots]

Yes! That click. The last time I felt it was here on the JREF forum reading the Downwind Faster Than the Wind thread.

 

[Raze]

How about this:

if 0.99999999... is a subset of 1, and 1 is a subset of 0.999999999..., then 0.999999999... = 1.


Well, it is pretty obvious that 0.999999... ≤ 1, so 0.9999999... ⊆ 1; that is every element of 0.9999999... is also an element 1.


Now how do you show that 1 ⊆ 0.9999999...?

One way I have seen this done is that every element of 1 is some rational number, a/b. Is there any rational number in the set from 0 to 0.9999999... that is not also in 1, and is there any rational number in the set from 0 to 1 that is not in 0.9999999...?

If the answer is no, then [0, 0.999999...] = [0, 1], and since the intervals are continuous, then it seems logical to me that [0+c, 0.999999....] = [0+c, 1], c < 0.9999999...., all the way up until {0.9999999...} = {1}.

But I lost my direction here.


It's already been asked in the thread "Is there any rational number between 0.9999999... and 1?"

So how about this:


Is there any rational number in the interval [0, 1] that is not in [0, 0.9999999....].

If the answer to that question no, then that means the following, I think:

There is no element in {0, ... 1} that is not also in {0, ... 0.999999....}, which further means that {0, ...1} ⊆ {0, ...0.999999...}, and since it is also true that every element in {0,...0.999999...} is in {0, ... 1}, then {0,...0.999999....} = {0, ...1}, and by the argument above, also {0.999999....} = {1}.

And since these two final sets only have one non-empty element, and both sets are equal, then 0.99999... = 1.



Or something like that.


But the point I'm getting at is, why not this way:

if 0.9999999...⊆ 1 and 1 ⊆ 0.99999999.... then 0.9999999.... = 1.

 

[jsfisher]

Is there any rational number in the interval [0, 1] that is not in [0, 0.9999999....].

All the interesting stuff would have to occur between 0.999... and 1, so why not simplify the question: Is there a rational number, X, that is 0.999... < X < 1.

If you prefer it expressed as an interval, is (0.999..., 1) non-empty?

 

[jsfisher]

How about this:

if 0.99999999... is a subset of 1, and 1 is a subset of 0.999999999..., then 0.999999999... = 1.


Well, it is pretty obvious that 0.999999... ≤ 1, so 0.9999999... ⊆ 1; that is every element of 0.9999999... is also an element 1.

By the way, I really have no idea what you mean by "is a subset of 1" or what you think are the elements of 1.

 

[W.D.Clinger]

By the way, I really have no idea what you mean by "is a subset of 1" or what you think are the elements of 1.

What he wrote would almost make sense if you think of .9999.... and 1.0 as Dedekind cuts^WP (represented by the rationals less than .9999... and 1.0 respectively), but I doubt whether that was what he meant because he didn't consider rationals less than 0.

 

[Molinaro]

All the interesting stuff would have to occur between 0.999... and 1, so why not simplify the question: Is there a rational number, X, that is 0.999... < X < 1.

If you prefer it expressed as an interval, is (0.999..., 1) non-empty?

That's the point I made earlier in the thread. There is no such X, hence they cannot be different real numbers.

 

[jsfisher]

What he wrote would almost make sense if you think of .9999.... and 1.0 as Dedekind cuts^WP (represented by the rationals less than .9999... and 1.0 respectively), but I doubt whether that was what he meant because he didn't consider rationals less than 0.

You are correct! I hadn't let my mind wander in that direction. It seems like such a long way around your elbow to get to your thumb. People have enough trouble with simple limits; Dedekind cuts will move them further from the goal.

 

[DallasDad]

I'm going to play Devil's Advocate here, just for fun.

The reason the multiplication proof fails is that it employs an undefined operation. While moving the decimal point is a well-known shortcut for multiplying by 10, it's just that: A shortcut. It relies on the underlying algorithm, which works correctly in most cases.

The underlying algorithm, however, fails for repeating decimals. Go back to grade school (or your favorite mathematical philosophy treatise) and find the proper algorithm for multiplying two numbers. One starts by lining up the right-most digits. Since there is no right-most digit on a repeating decimal, it is insusceptible to multiplication (or addition or subtraction, for that matter) using the standard algorithm.

Knowing that the standard algorithm fails, the shortcut for it, which relies on the underlying principles, must also fail. Therefore the proof by multiplication is invalid on its face, QED.

The only reasonable product of 10 * 0.999... is 10 * 0.999... The expression cannot be manipulated further.

One may infer that the repeating decimal can be treated like a non-repeater, and one can further infer the results of operations upon it using normal logic; however, one cannot prove it because the low-level operations required for mathematical proofs simply don't exist.

 

[W.D.Clinger]

The only reasonable product of 10 * 0.999... is 10 * 0.999... The expression cannot be manipulated further.

Untrue. For example, one can prove that 0.999... = 1.0, then simplify the product 10 * 0.999... to 10 * 1, and then simplify it further to 10.

One may infer that the repeating decimal can be treated like a non-repeater, and one can further infer the results of operations upon it using normal logic; however, one cannot prove it because the low-level operations required for mathematical proofs simply don't exist.

Not sure what you're saying here. If you're saying that some of the alleged proofs given in this thread are incorrect, then you're right. If you're saying that all of the alleged proofs given in this thread are incorrect, then you're wrong.

The following posts give correct proofs or arguments that would be correct when the details are filled in:

http://www.internationalskeptics.com/forums/showpost.php?p=186318&postcount=5
http://www.internationalskeptics.com/forums/showpost.php?p=186322&postcount=6
http://www.internationalskeptics.com/forums/showpost.php?p=186339&postcount=9
http://www.internationalskeptics.com/forums/showpost.php?p=186466&postcount=13
http://www.internationalskeptics.com/forums/showpost.php?p=186707&postcount=38
http://www.internationalskeptics.com/forums/showpost.php?p=186822&postcount=43
http://www.internationalskeptics.com/forums/showpost.php?p=187490&postcount=69

In my opinion, posts #13, #38, and #43 seem to be the most detailed of the correct proofs that had been given before this thread was resurrected.