0.9999....(recurring) = 1?

George,Y · 2007-12-26 10:12:43

BTW, the part to search the missing 9 employs matching, however, I match all the 9 matches that have a following 9 match, which is much more conclusive than just match equivalent 9's from the first decimal, to the second, to the third...

So how my more conclusive matching technique wrong and your less conclusive just counting matching technique correct? Is it because your way of thinking the only one people shall think in?

George,Y · 2007-12-26 10:22:41

But one more? How can you conclude which set has more?

You are not reading, Ricky.

The conclusion comes from the fact that its subset just has the same. And I did not define it, for I don't like to pre-define or pre-assume things the way mathematicians do.

Again, you use naive matching from left one by one again.

" smallest* such set"
Good, quite interesting, then you agree there are several such sets? Then what distinguishes the smallest one to any larger one? The largest integer? The largest integer+1 ? Come on, I wanna know what is the smallest set missing so to locate the end. Don't wanna talk about it? Ban us from talking about it? That's all what mathematicians can do?

Still, thank you for your advice.

Last edited by George,Y (2007-12-26 10:26:31)

Ricky · 2007-12-26 11:01:09

The conclusion comes from the fact that its subset just has the same. And I did not define it, for I don't like to pre-define or pre-assume things the way mathematicians do.

Then your approach has no rigor. You don't define something yet you wish to draw conclusions from it. Don't you see how wrong that is?

There are numerous problems from trying to define an ordering on sets based upon subsets. I will not go into these in detail, but it mostly stems from this not being a total ordering.

Good, quite interesting, then you agree there are several such sets? Then what distinguishes the smallest one to any larger one? The largest integer? The largest integer+1 ? Come on, I wanna know what is the smallest set missing so to locate the end.

There are certainly many such sets, and some are not equivalent. They are distinct based upon how they are generated. There is an open question in mathematics on what kinds of sets there are. This is the continuum hypothesis and I will leave it to you to look this up.

Two different such sets are the integers and the real numbers. There is no bijection in between them. So to answer your question as to what the integers are missing, it's numbers such as 0.5, pi, and the square root of 2. As a matter of fact, the real numbers are the same size as the power set of the integers.

George,Y · 2007-12-26 11:14:18

"
{1, 2 , 3, 4, 5, ...}
matching by +1=
{2, 3, 4, 5, 6, ...}
above plus the set {1}=
{1, 2, 3, 4, 5, 6,...}#
"

OK, this is my defination, {2,3,4, 5,6...} is just defined by a set whose elements are just 1 larger than corresponding ones in the original set {1, 2, 3, 4,...}. Or just suppose you add 1 to each of the elements in {1,2,3,4,5,...}

Is it clear to you?

Last edited by George,Y (2007-12-26 11:37:31)

Ricky · 2007-12-26 12:19:11

No George. We are talking about the definition of larger.

George,Y · 2007-12-26 13:39:43

Hehe, you simply don't accept {1, 2, 3, 4, 5, 6,...}# has one more element than {1,2,3,4,5...} because you suppose since they both have infinite amount, they are the same. Don't you?

George,Y · 2007-12-26 14:02:35

It is larger because {2,3,4,5,...} then absorbs a new element that is not in it-it has grown larger.

Suppose {1,2,3,4,...} has an amount of elements, no matter finite or infinite, but stable,
And then {2, 3, 4, 5,..} has the same amount because it's only a "+1" operation of the former one,
And it absorbs a new member, hasn't it grown larger? Larger means more. It is definately having a more member than {2,3,4,...} which has the same amount of members with the old version {1,2,3,4,...}, and this is the reason why it has more.
c>b, b=a, hence c>a, transivity.
Do you wanna define the transivity doesn't exist?

Or you simply deny it because you read somewhere that infinity can be large, infinity can be small, infinity+1=infinity? Well if it is the case, then the amount of elements of {1,2,3,4,...} can be varying as well as infinite. Good! If you accept it can vary but at the same time it stays the same thing infinite, you are conceptulizing an infinite amount that is docile, and that can grow larger or smaller. And how do I tell from the larger "it" and the smaller "it". My method would be if one "it" not only contains everthing another "it" contains, but also contains something the other doesn't, it is thus more and larger.(Can you give a more fundamental more correct defination for "larger" than this?)

I am OK with this docile thing. Since then we can discuss which part is docile, which is not. I can say that any element that can be displayed in an number is not docile. Suppose you substract 123456789123456789 from any larger "it", it's no longer the same and thus the smaller "it" should have 123456789123456789 as well. And then you have to admit the docile part, the difference, can only be some member or members that cannot be displayed as a number, and what is it (are they)? Demon?

Ricky · 2007-12-26 14:03:10

George, you just don't understand set theory, and I don't have the patience to try to help you with it when you fight every step of the way and attempt to speak for me putting words in my mouth in the process, all while claiming that I'm censoring you.

Learn mathematics. Then you can feel free to criticize it. Till then, you are criticizing something which you do not understand, not something which is wrong.

George,Y · 2007-12-26 14:14:06

The same thing for equal and larger in my earlier argument
Amount of elements of {1, 2, 3, 4, ...} = Amount of elements of {2, 3, 4, 5,...}
Because every member of the latter gets 1 added from the former.
(You can say I cannot do this, but please tell from where, since I know any number in the former set can be added by 1)

And {1,2,3,4,5,...}# is larger than {2,3,4,5,...} because it is generate by duplicating the latter and adding in a new element, an element does not belong to the latter. Thus it is larger according to my defination in common sense. What's your problem with this?

And the last part, transivity. Do you want to say that it is not transible for infnity?
Infinity+1=Infinity?
Then what's your infinity exactly? It's not stable? It's docile? It can be added in something and stays the same? Riduculous as it is, I have a refutation for this already in the previous post.

Ricky · 2007-12-26 14:22:06

Thus it is larger according to my defination in common sense. What's your problem with this?

Common sense and infinity can never be used together. Mathematics isn't about common sense. It's about formal definitions and rigor. People tried to use common sense and ran into problems.

Which set is larger, the set of irrationals or the set of integers? What about primes or irrationals? What about the Gaussian integers and prime numbers congruent to 3 modulo 4? What does your common sense definition say about these set sizes?

George,Y · 2007-12-26 14:22:47

Ricky wrote:

George, you just don't understand set theory, and I don't have the patience to try to help you with it when you fight every step of the way and attempt to speak for me putting words in my mouth in the process, all while claiming that I'm censoring you.
Learn mathematics. Then you can feel free to criticize it. Till then, you are criticizing something which you do not understand, not something which is wrong.

You are talking about something like if you don't know everything I have already have known, then you are not qualified to talk to me and discuss anything.

Then you are eliminating discussion or raising a protection, since few one has learn everything another knows. If this is the case, no senator can critisize the president since they know what?! No journalist can critisize someone since they know what?

Or I just stay here, claim I am right, list a tedious philosophy book list for you, and say you are not entitled to say I am wrong before finishing those books?

How do you feel?

BTW, I knew Cantor set earlier than you did, if you could remember.

George,Y · 2007-12-26 14:34:09

Ricky wrote:

Thus it is larger according to my defination in common sense. What's your problem with this?
Common sense and infinity can never be used together. Mathematics isn't about common sense. It's about formal definitions and rigor. People tried to use common sense and ran into problems.
Which set is larger, the set of irrationals or the set of integers? What about primes or irrationals? What about the Gaussian integers and prime numbers congruent to 3 modulo 4? What does your common sense definition say about these set sizes?

Perhaps you are just asking for trouble. Before constructing these sets, My logical argument has already stopped having an infnite set of all integers. And irrationals with infinite decimals may not exist at all.

And after all, we live in a common sense world, not in a maths fairy tale world. Many mathematical problems are just a waste of time since they are created so different than the real world and thus never get a chance to apply to it. (Infinity and surreals for example)

Ricky · 2007-12-26 14:40:12

Check this out. We have the set of natural numbers:

{1, 2, 3, ...}

Now you are saying that this is "bigger" than the set of all natural numbers that are greater than 1:

{2, 3, 4, ...}

However, this is not true. Remember, numbers are just symbols. We could use letters for numbers, so the first set would look like:

{a, b, c, d, e, f, g, h, i , az, aa, ab, ac, ad}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Where 'a' stands for '1', b stands for '2' and so on. Now keeping in mind that numbers are just symbols, I will write:

one := 2
two := 3
three := 4
four := 5

So for example, when I say "one" I write the symbol 2. Now we have the set:

{1, 2, 3, 4, ...}

Being rewritten as the set:

{2, 3, 4, 5, ...} //Exactly the same set as above, written with different symbols. Remember, it starts at 1, but I'm just writing the number 1 in a funny way

And thus, {1, 2, 3, 4, ...} is larger than itself. Contradiction. Your "larger by 1" does not work George.

George,Y · 2007-12-26 14:47:11

What I hate about Reals is that in order to justify that the residuals have been eliminated or that the irrationals do exist, it abandons the defination from common sense of basic numbers. 1 is no longer 1, 2 is no longer 2, they are defined as a set of all the junky rationals and irrationals smaller than themselves in Cantor's defination.

Who cares about all the numbers smaller than 2? I am just eating 2 apples, 1+1 apples in essence, not eating a combination of -Pi, -e, -1, 0, 1.5, 1.999... apples what soever! Because you have trouble justifying you have eliminated the residual then I have to use such an unnatural idea?

I hate this unnatural thing so much that I don't wanna waste a minute or a braincell on it. Still, thank you for reminding me to read a book on set. I know in order to offend the orthodoxy maths I may need to know it much better than any defenders. That's quite twisted but perhaps I should put up with it?!

Ricky · 2007-12-26 14:49:24

As soon as I give you a specific example on why your definition of larger does not work, you don't want to waste another "braincell" on it. Go figure.

George,Y · 2007-12-26 15:04:22

"Remember, numbers are just symbols"

Numbers are not just so. Don't belittle them as Cantor and the mathematicians after him does, plz.

They stands for meanings.

2 stands for solid 1 and 1.
If you have bought two sandwiches and the counter only gives you one, will you be outraged or not? Or will s/he say 1 and 2 are the same, just symbols?

one := 2
two := 3
three := 4
four := 5

When writting such a thing, you are defining
2 as the single unit,
and 3 as the single unit + the single unit.
Hence essentially your new 2 means the samething the old 1 does. It can take on a different symbol but its ESSENCE stays the SAME.

Back to your game
well the new
{2,3,4,5...} stands for the same thing the old
{1,2,3,4...} stands for,
it should be compared to the old {1,2,3,4..} by what old symbols it means or in both new symbols they mean and both are fair. Don't try to play a translation or a word puzzle by simply and wrongly mixing them up.

If I say a MIT is smaller than Massachussets Institute of Technology, do you agree?
Well because the letters of the former is less than the latter.

But essentially they stands for the same thing!

You can use 1 2 3 4 or 2 3 4 5 or whatever. But you should know the meanings behind the letters, that 2=1+1=single+single or 3=2+2=single+single. Don't try to blend them together or you are just fooling around saying something like one dime is smaller than five cents because one is smaller than five.

Last edited by George,Y (2007-12-26 15:05:30)

Ricky · 2007-12-26 15:20:13

George, are you telling me that:

{2, 3, 4, 5, ...}

Has a different size than:

{2, 3, 4, 5, ...}

When I interpret the second as the numbers "one", "two", "three", and so on under my new definition for what the symbols mean?

George,Y · 2007-12-26 15:24:24

A 2 in {2,3,4,5...} created by you
is not the same 2
in
{1,2,3,4...}
And they are not equal unless you switch
{1,2,3,4,...} to {2,3,4,5...} in new defination.

Since they are not equal, you cannot say {2,3,4...} in {1,2,3,4,...} is the same as {2,3,4,...} created by new defination and thus you cannot apply my defination of "larger" and disprove it.

Or I can put it in something you feel easy to understand.
{1,10,1000,...} in mode 10
isn't the same as
{1,10,1000,...} in mode 2
and what you did is just like to equal the two.

George,Y · 2007-12-26 15:26:54

My defination stays on a real term basis instead of a nominal term basis.
I clarify this now, don't cheat.

Ricky · 2007-12-26 15:27:32

So you are trying to tell me that a set's size depends on what I interpret it's elements values are. That's just plain ridiculous!

George,Y · 2007-12-26 15:36:15

They can be without a value. But the meaning should stay consistent.

suppose
{a,b,c,d}
Now like you did, a foreign language just transform
a->b
b->c
c->d

And now you have
{b,c,d}
and does it lack one a?
No, it lacks one old d.

And does it equal to the subset {b,c,d} from the old set?
No it equals to or it is equavalent to the subset {a,b,c} in the old set.

Do you agree the above or not?
Or are you lost in translation?

Ricky · 2007-12-26 15:37:44

The difference between your mapping is that it is not 1-1 because you implicitly mapped d to d. So both c and d map to d. On the other hand, my map is both 1-1 and onto.

You can not come up with a 1-1 and onto map which goes between two sets of different sizes.

George,Y · 2007-12-26 15:53:11

"1 because you implicitly mapped d to d. So both c and d map to d."

You forge this up, dude.

" my map is both 1-1 and onto. "

Alright, let's stay with your mapping.
{2,3,4,5,...}#
How many have you mapped?
The same amount of
{1,2,3,4,...}?

And {2,3,4,5,...}# has lacked one symbol of 1 in it?
And yet it has the same amount.
And so it has one more symbol than the original subset {2,3,4,5,...} otherwise the mapping cannot be completed in the same amount.
And what is it? Isn't it the transformation of the last symbol in the old {1,2,3,4,...}?
And I can apply the same methodology to find the last symbol, just by comparing
{2,3,4,...} and a {2,3,4,...}# (which just have one more symbol than the former).

Unless you deny you have mapped all of the members in old {1,2,3,4,...} into the new {2,3,4,5,...}, which simply makes your "disproof" unvalid.

George,Y · 2007-12-26 15:57:00

What you did is just see
{2,3,4,5.,..} #
and
{1,2,3,4,5,...}

you have seen 2,3,4,5,... in both, and you then conclude that they are the same? Don't hide the demon in the elipse, face it, find whether two elipses omit the same thing, plz.

Ricky · 2007-12-26 16:43:06

Neither of those posts made any sense, in both the English and mathematical sense. What is your objection to my relabeling method?

Math Is Fun Forum

#876 2007-12-26 10:12:43

Re: 0.9999....(recurring) = 1?

#877 2007-12-26 10:22:41

Re: 0.9999....(recurring) = 1?

#878 2007-12-26 11:01:09

Re: 0.9999....(recurring) = 1?

#879 2007-12-26 11:14:18

Re: 0.9999....(recurring) = 1?

#880 2007-12-26 12:19:11

Re: 0.9999....(recurring) = 1?

#881 2007-12-26 13:39:43

Re: 0.9999....(recurring) = 1?

#882 2007-12-26 14:02:35

Re: 0.9999....(recurring) = 1?

#883 2007-12-26 14:03:10

Re: 0.9999....(recurring) = 1?

#884 2007-12-26 14:14:06

Re: 0.9999....(recurring) = 1?

#885 2007-12-26 14:22:06

Re: 0.9999....(recurring) = 1?

#886 2007-12-26 14:22:47

Re: 0.9999....(recurring) = 1?

#887 2007-12-26 14:34:09

Re: 0.9999....(recurring) = 1?

#888 2007-12-26 14:40:12

Re: 0.9999....(recurring) = 1?

#889 2007-12-26 14:47:11

Re: 0.9999....(recurring) = 1?

#890 2007-12-26 14:49:24

Re: 0.9999....(recurring) = 1?

#891 2007-12-26 15:04:22

Re: 0.9999....(recurring) = 1?

#892 2007-12-26 15:20:13

Re: 0.9999....(recurring) = 1?

#893 2007-12-26 15:24:24

Re: 0.9999....(recurring) = 1?

#894 2007-12-26 15:26:54

Re: 0.9999....(recurring) = 1?

#895 2007-12-26 15:27:32

Re: 0.9999....(recurring) = 1?

#896 2007-12-26 15:36:15

Re: 0.9999....(recurring) = 1?

#897 2007-12-26 15:37:44

Re: 0.9999....(recurring) = 1?

#898 2007-12-26 15:53:11

Re: 0.9999....(recurring) = 1?

#899 2007-12-26 15:57:00

Re: 0.9999....(recurring) = 1?

#900 2007-12-26 16:43:06

Re: 0.9999....(recurring) = 1?

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