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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

For example, a circle with infinite radius is a straight line.

Hmm, that example sounds so familiar. Who on earth did you get it from?

So there is no end to 0.999...

Right? (He asks tentatively)

Yep.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,591

Ricky wrote:

For example, a circle with infinite radius is a straight line.

Hmm, that example sounds so familiar. Who on earth did you get it from?

Yes, it was you, here: http://www.mathsisfun.com/forum/viewtop … 939#p30939

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

If you believe that .99999999999... is 1.00000000000..., then you don't believe in different values for infinity.

You think infinity is a constant, which ofcourse it is not!!

**igloo** **myrtilles** **fourmis**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

If you believe that .99999999999... is 1.00000000000..., then you don't believe in different values for infinity.

I'm not entirely sure what you mean, could you reword this?

But for different values of infinity, it's not even a number. So saying it has a "value" is like me asking you what country is north of the north pole! It just doesn't make sense.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

Ricky wrote:

George, I agree that approaching is different than being. But that's not what we are talking about. 0.111... isn't approaching anything. It's a notation, and nothing more. It means "0.1 with an infinite number of ones after it." Now, I think you are trying to prove that 1/9... is different than "0.1 with an infinite number of ones after it." Now if that is so, then please tell me which decimal place does not have a '1'.

What's the difference between [0,1) and [0,1]?

One's closed, one's neither closed nor open.

"0.1 with an infinite number of ones after it."- you are using reached infinite. well, if infinite can be reached, infinitesmalls can reach 0 accordingly. what is Δs/Δt when Δt EXACTLY reaches 0??

Why do i say real numbers are MOBILES?

Cantor used a special defination to class rational numbers and irrational numbers into the same category:

He defined √2 as a set containing all rational numbers p satisfying p[sup]2[/sup]<2 , since "=" is invalid.

To make homogenousity(similarity), real 1 is certainly defined as another set containing all rational numbers p satisfying p<2 , where "=" cannot be used either.

Further Explaination

Between 2 rational numbers exist another

We can easily verify this by drawing an axis, setting unit 1, then using basic geometric skills (parallel lines) to point out any rational number on the axis, and finally point out the midpoint-another rational number Rational Numbers are STABLE.

Real Numbers as mobiles

but for a real number 1, you cannot point it out on an axis only according to its defination. The only way you can solve it is to say it's idendical to the rational 1,

and point out the latter instead.

But how do you arrive at real 1 is idendical to rational 1?

The trick necessary is given any rational number between the two, it can be concluded in real 1's set. The set "enlarges" as if a variable approaches to its limit.

so when proving a real 1 = a rational 1, a mistake to equal "approaching" to "being" seems unavoidble.

Let alone "between 2 real numbers exist another", this time they both can move, what else could i say?

*Last edited by George,Y (2006-03-20 00:26:44)*

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**Ricky****Moderator**- Registered: 2005-12-04
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But how do you arrive at real 1 is idendical to rational 1?

By definition, the rationals are a subset of the reals, and thus, any rational number is also a real number of the same name.

Don't see what this has to do with anything though.

"0.1 with an infinite number of ones after it."- you are using reached infinite. well, if infinite can be reached, infinitesmalls can reach 0 accordingly.

No I'm not. It's goes on infinitely long, and thus is never ending. You can never reach then end. Why do you say that I am?

I will ask you again, if you think that 1/9 is not 0.1 with an infinite amount of 1's behind it (i.e. 0.111...), then tell me which number digit in 1/9 is not a 1 or where it ends.

what is Δs/Δt when Δt EXACTLY reaches 0??

0/0, an indeterminate form. What does this have to do with anything?

*Last edited by Ricky (2006-03-20 03:29:31)*

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

No I'm not. It's goes on infinitely long, and thus is never ending. You can never reach then end. ----------

Thus i am not sure if the result 0.1 0.11 0.111 0.11...1 (k 1's), =>0.11...11 (k+1 1's) gotten from mathematical induction could catch up your number, for digits of your number never ends, my never ends, either.

Similar to your argument, infinity-infinity is an indeterminate form i don't know if my infinite 1's could equal yours.

i don't know how to compare with my step by step growing number with your existing never ending "number", mathematical induction doesn't garantee this, what it does say is divided to a finite digit, it can be divided to 1 more digit.

*Last edited by George,Y (2006-03-23 02:38:45)*

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**Ricky****Moderator**- Registered: 2005-12-04
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Thus i am not sure if the result 0.1 0.11 0.111 0.11...1 (k 1's), =>0.11...11 (k+1 1's) gotten from mathematical induction could catch up your number, for digits of your number never ends, my never ends, either.

Two real numbers are equal when there exists no real number between them. By inspection, I would have to say there exists no real number between your version of 0.111... and mine. There is nothing you can add to one to make it the other. So they are equal.

Similar to your argument, infinity-infinity is an indeterminate form i don't know if my infinite 1's could equal yours.

Sounds like a bijection may be needed. Your 1's are definitely countable. Are mine?

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**George,Y****Member**- Registered: 2006-03-12
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What do you mean by Two?

You assume your number is real.

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**Ricky****Moderator**- Registered: 2005-12-04
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George, now you are going into the absurd.

0.111.... has no complex part, and thus, it must be part of the reals.

And I'm not sure if I can explain two in any other meaningful way than 1+1.

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**George,Y****Member**- Registered: 2006-03-12
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0.111...123 is a number too, uh??

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**George,Y****Member**- Registered: 2006-03-12
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baaba wrote:

The proof fails because of circular logic. For the proof to be valid, it must be known that 0.999... is a rational number, however the only way to know that is to prove that 0.999... equals 1 in another way, thus making the "easy" proof obsolete.

Now i couldn't agree with it more

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**Ricky****Moderator**- Registered: 2005-12-04
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George,Y wrote:

0.111...123 is a number too, uh??

If ... means for infinity, then 0.111...123 = 0.111...

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**George,Y****Member**- Registered: 2006-03-12
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Any way, you need to add a belief first.

it's like believing convergence before solving a limit out.

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**George,Y****Member**- Registered: 2006-03-12
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Ricky wrote:

And I'm not sure if I can explain two in any other meaningful way than 1+1.

well, i may 1+1 in some way.

but 0.111... can be explained as infinite numbers adding together (different from varying variable), if you don't insist on interpreting it as drawings of 1s

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**God****Member**- Registered: 2005-08-25
- Posts: 59

0.9999... = 1-k

Doubling, 1.99999...(8) = 2-2k

If k is not zero, there is a number 2-k between 2-2k and 2

Halfing, there is a number 1 - k/2 between 1-k and 1

Therefore, if 0.9999... isn't 1, then it isn't the largest real number less than one either, which leads me to ask, if 0.999... isn't the largest real number less than 1, then what is?

*Last edited by God (2006-04-02 14:50:08)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

I get a kick out of this topic. It is really funny!!

My current opinion is if infinity exists in our minds, then it's value is multi-faceted.

MIF says it's fully grown. Maybe both are true.

I guess I'm not very logical on this subject, but I tend to side with the opinion that

infinity is more special than a single dinky thing, that's why I like to let it take on different

values. Not a value like a constant, but a huge value that is hard to describe and is not

always the same, perhaps changing by whim.

I guess I could be persuaded that it is simply not a number because it is larger than numbers.

Then I would say it doesn't exist, even in our minds.

So what we are talking about is infact a recursive definition of getting larger and larger and doing

this until it is fully grown, says MIF. I wonder why he said that. Anyway. Maybe we can only grasp the definition

of infinity, but cannot grasp infinity itself?

Disregard this entire discussion as it is just silliness...

**igloo** **myrtilles** **fourmis**

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

e=1

Proof

e= (1+1/n)[sup]n[/sup] where n is fully grown

n is fully grown, thus 1/n=0 the base is now 1

from induction and imagination we know 1[sup]∞[/sup]=1

∴e=1

*Last edited by George,Y (2006-04-03 17:08:46)*

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**George,Y****Member**- Registered: 2006-03-12
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**The poor of induction **

Russel once said this story

a hen saw her master giving her food one day.

she saw the same thing the 2nd day, the 3rd day...

after many many days, she concluded an ultimate truth based on induction

- he will always give me food.

the next day, she was *killed* by her master.

This is a philosophical attack on all human knowledge, and a underpinning concept of many films, such as *Matrix*.

Here i say, imagining infinity is an art, you can have whatever imaginations as you like.it's like Plato's idea. there isn't too much sense...

*Last edited by George,Y (2006-04-03 17:21:05)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Okay, here's something to ponder.

1/9 = 0.111111...

2/9 = 0.222222...

3/9 = 0.333333...

...

8/9 = 0.888888...

9/9 = 0.999999... Pretty neat huh? Not really a proof though, is it? Does this suggest that 1 is .999... ?

**igloo** **myrtilles** **fourmis**

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**George,Y****Member**- Registered: 2006-03-12
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Ricky wrote:

"George, now you are going into the absurd.

0.111.... has no complex part, and thus, it must be part of the reals."

actually, the ultimate disagreement is that i deny such an expression a number. because i cannot calculate it out, nor can i say it's stable. the only way to say it's stable is by inferal of Reached infinite thing. And i deny this concept because of subjectivity. see #43

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

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**cray****Member**- Registered: 2006-10-05
- Posts: 29

Ok heres two things may surprise you

1) I dont get why the people who say .999r is not = 1 cant accept that 1=.999r

I dont get why you think that 1/3 is not = .3333

after all

1/3 =.333

2/3 = .666

3/3 = .999

2) go to google and type this exactly into google : .999999999999 + .999999999999

it will give you an answer of 2

google is also a scientific calc and if you type enough 9's it know you mean its a reccuring number.

then type .999999999999 - 1

the answer is 1

I'm not a person that says its right - google say so

the claim is that 1=1 and 1=.999r

but none of you seem able to provide proof for the opposite claim

namely that 1=1 and 1 != .999r

("!" means, "not")

so I fail to see why people cant accept that 1 can = both

its obvious that it can

there is no difference between 1/3 and .333r

so why you think theres a difference between 3/3 and .999r ?

Personally I think some of you believe that the missing number is lost somewhere back in the 9's

an infinite distance back - my own theory is that 0 is not a number and the rel sequence of numbers is therefore

3 2 1 -1 -2 -3

think of those on a east to west axis

but the missing .1 is somewhere in an infinite number on a north south axis between 1 and -1

*Last edited by cray (2006-10-05 13:17:42)*

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

my own theory is that 0 is not a number

Then the real numbers (or any numbers) are not a group with respect to addition, and entire fields of mathimatics are thrown into chaos, such as abstract algebra.

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**polylog****Member**- Registered: 2006-09-28
- Posts: 162

George,Y wrote:

The poor of induction

Russel once said this story

a hen saw her master giving her food one day.

she saw the same thing the 2nd day, the 3rd day...

after many many days, she concluded an ultimate truth based on induction

- he will always give me food.

the next day, she waskilledby her master.

That's hilarious!

Good thing the natural numbers don't have such killer abilities!

Seriously though, this thread is fascinating.

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