Exploring the Convergence of 0.999... and the Concept of Infinity

[ram2048]

but i come bearing gifts

http://home.earthlink.net/~ram1024/

please do enjoy!

 

[cookiemonster]

Maybe I'm paranoid, but I feel extremely suspicious of that link...

cookiemonster

 

[recon]

Am I missing something here?

 

[Zurtex]

One of your disproofs seem to assume that [itex]0.\overline{0}1[/itex] and similar numbers are real numbers, they are not. Also sums to infinity have been long established and if you believe they do not exist that you do not believe the number [itex]0.\overline{9}[/itex] exists anyway. Furthermore you say that:

[tex]0.\overline{3} \neq \frac{1}{3}[/tex]

You seem to have really lost the plot here and seem to be implying that [itex]0.\overline{3}[/itex] is an irrational number.

Hmm reading further on I see you do conclude that it is an irrational number. I would therefore be interested to see how you define the number [itex]0.\overline{3}[/itex]. As you state it is neither:

[tex]\frac{1}{3}[/tex]

or

[tex]\sum_{n=1}^{\infty} \frac{3}{10^n}[/tex]

 

[ram2048]

Zurtex: i DID state that they are not standard notation... they are numbers just the same, however.

i THINK i explained 1/3 and [itex]0.\overline{3}[/itex] quite well in one of the pages as to how they can or cannot be rational etc.

aha here we go
http://home.earthlink.net/~ram1024/where.html
proof #4

and actually if I'm doing the sigma thing correctly [tex]\sum_{n=1}^{\infty} \frac{3}{10^n}[/tex] does create [itex]0.\overline{3}[/itex]

 

[Zurtex]

Hmm, I've had an idea. Let's for one moment assume you are correct and say that [itex]0.\overline{0}1[/itex] is a real number. Then:

[tex]0.\overline{0}1 = \frac{1}{\infty}[/tex]

As infinity halved is still infinity then:

[tex]\frac{1}{2}\infty = \infty[/tex]

Taking the reciprocal of both sides:

[tex]2 \left( \frac{1}{\infty} \right) = \frac{1}{\infty}[/tex]

Using out identity:

[tex]0.\overline{0}1 = \frac{1}{\infty}[/tex]

Then:

[tex]2(0.\overline{0}1) = 0.\overline{0}1[/tex]

Taking away [itex]0.\overline{0}1[/itex] from both sides:

[tex]0.\overline{0}1 = 0[/tex]

 

[Zurtex]

and actually if I'm doing the sigma thing correctly [tex]\sum_{n=1}^{\infty} \frac{3}{10^n}[/tex] does create [itex]0.\overline{3}[/itex]

What do you mean create? This isn't physics, in maths either a number is equal or it isn't.

 

[ram2048]

As infinity halved is still infinity then:

whoa whoa... where did you get THAT from. 1/2 infinity = infinity? I think not.

1/2 infinity = 1/2 infinity

1/2 infinity < infinity

it's unresolvable in the first place and [tex]\frac{1}{2}\infty = \infty[/tex] is just not logical

 

[chroot]

ram,

Quit being such a ****head and put your topics where they belong. You're allowed to talk about them all you want, as long as you put them in the right place.

- Warren

 

[Zurtex]

whoa whoa... where did you get THAT from. 1/2 infinity = infinity? I think not.

1/2 infinity = 1/2 infinity

1/2 infinity < infinity

it's unresolvable in the first place and [tex]\frac{1}{2}\infty = \infty[/tex] is just not logical

Well at least I now know you really do have no understanding of mathematics even at the philosophical level.

Infinity isn't a real number, do you think it is going to behave like other numbers?

E.g

Person A has an infinite number of bananas. For every 2 bananas person A has, Person B has 1 banana. Does person B have an infinite number of bananas?

 

[ram2048]

What do you mean create? This isn't physics, in maths either a number is equal or it isn't.

well 1/3 creates a "process" as can be described using my expanded notation.

[tex]\sum_{n=1}^{\infty} \frac{3}{10^n}[/tex] creates .3 + .03 + .003 etc etc which does NOT include the process.

just trying to see if it's safe to say that [tex]\sum_{n=1}^{\infty} \frac{3}{10^n}[/tex] = [itex]0.\overline{3}[/itex]

and i think it is

 

[ram2048]

Person A has an infinite number of bananas. For every 2 bananas person A has, Person B has 1 banana. Does person B have an infinite number of bananas?

yes. but he has LESS than person A

not EQUAL

Infinity + 1 > Infinity

 

[Zurtex]

yes.

ty, my point is proven.

 

[Integral]

So you have concluded that .333... is a irrational number.

Does that mean that .1 (base 3) is also irrational? Just what is an irratioal number to you?

 

[ram2048]

Does that mean that .1 (base 3) is also irrational? Just what is an irratioal number to you?

that's a different notational system.

just as 1/3 is a different system than decimal base 10.

1/3 converts perfectly to .1(base 3) but NOT perfectly as [itex]0.\overline{3}[/itex] (base 10)

 

[Zurtex]

not EQUAL

Infinity + 1 > Infinity

So you think there is some default value for infinity?

 

[ram2048]

Chroot: don't hate, it's not healthy man. everything is fine.

 

[chroot]

No, ram, everything is not fine. If you continue to disobey our rules here, things will rapidly become less fine.

Our rules are really not that restrictive. Please follow them, and make both our lives easier.

- Warren

 

[ram2048]

i think in order to use infinity you have to define a default value and extrapolate logical movements from that position.

this being an alternative to illogically assuming that any values transformed on it have no effect.

hence you could have > and < expressions detailing conversions in the value of infinity

 

[ram2048]

Chroot: i had no earthly idea where to find this mythical "theory development" page in the first place. so just be calm

 

[Zurtex]

Assumptions, speculation and purely going off what you feel is intuitive. I suggest taking a course on mathematical proof and how it works.

 

[ram2048]

you forgot disproving current mathematical theorems and analysis and building a true logical system that works :D

hell ya

 

[Zurtex]

you forgot disproving current mathematical theorems and analysis and building a true logical system that works :D

hell ya

If you understood what true proof was you would know that in maths you can't disprove something that has been rigorously proved.

 

[ram2048]

If you understood what true proof was you would know that in maths you can't disprove something that has been rigorously proved.

sure you can. that makes no sense whatsoever :D

that's like saying if a criminal is tried and convicted you can't appeal his case if new evidence is found that would vindicate his position

 

[Zurtex]

sure you can. that makes no sense whatsoever :D

that's like saying if a criminal is tried and convicted you can't appeal his case if new evidence is found that would vindicate his position

Thanks, you've just shown you don't know what maths proof is, won't be replying to you ever again if you carry on thinking like that.

 

[ram2048]

your loss...

bye then

 

[Integral]

They both represent the same point in the Real Number system. This is what you fail to understand. A decimal or binary number (or any other base) is simply a representation of a point in the Real numbers. .1(base3) represents the same point as .333... base 10. It is not about the representation but the point being described. Like wise 1 represents the same point as .990... (base 10) or an infinitely repeating representation of the largest digit in any base.

Sorry that you do not like the way the real numbers are constructed. But the fact is that is the way it is. The basic properties of the Real number system is INDEPENDENT of ANY representation. Representations of a point on the real number line is more like the image on a movie screen. It gives you something to look at, but is not the same as the screen itself. The image can change but the screen remains the same. You can only discuss the image because you have no knowledge of the screen.

I am sure that you have spent hours thinking about the properties you wish to give infinity. What you have is not useful or even very interesting. Sorry. My opinion, but then you are simply expressing your opinion.

 

[Hurkyl]

Ram, can your interpretation of the decimals (exactly) solve the equation 3x=1? If not, why should anyone use your interpretation instead of one that can exactly solve this equation?

 

[ram2048]

Ram, can your interpretation of the decimals (exactly) solve the equation 3x=1? If not, why should anyone use your interpretation instead of one that can exactly solve this equation?

3x=1
x=1/3
x=.333r(1/3) exactly and rational

furthermore going backwards

x=.333r(1/3)
3x=.333r(1/3) x 3
3x=.999r(3/3)
3x=1

 

[uart]

3x=1
x=1/3
x=.333r(1/3) exactly and rational

Hehe 1/3 = .333r(1/3), it just gets funnier. You mean 1/3 of 1/infinity there don't you RAM.

You know I knew that you'd believe that [tex]\infty+1 \neq \infty[/tex], such a belief is actually an inevitable consequence of believing the 0.999 is not equal to 1.

For example, what would RAM say 1-(1+0.999)/2 equals. It's obvious that he reply that it was 0.0...05, where the "0...0" denotes (infinity + 1) zero's (to distinghish it from the infinity zeros that 1 - 0.999 = 0.0...01 has).

I'll give you one thing RAM, you're a great entertainer.

 

[ram2048]

Hehe 1/3 = .333r(1/3), it just gets funnier. You mean 1/3 of 1/infinity there don't you RAM.

no. i mean Remainder 1 divided by 3

this expanded notation is part of the last calculated digit in a non-terminating series created by a rational fraction.

it is simply a way to create the true rational decimal value

it's detailed in the link... kinda (still prettifying the page)

 

[russ_watters]

To be more general than Hurkyl: So what? So you've got a different way of defining infinity. Why should we care?

 

[ram2048]

you should care because the traditional way of viewing it leads to inaccuracies. such that something cut infinitely results in nothing.

it's fine by me if you want to embrace something that produces the wrong results due to fallacious logic.

go ahead

 

[Zurtex]

ram2048, I've been thinking about this and you've almost convinced me.

However there is just one little flaw that I see that you may be able to patch up. You are of the belief that [itex]\infty + 1 \neq \infty[/itex] Which means you must have some default value for infinity, for example: [itex]\infty_d[/itex] such that:

[tex]0.\overline{0}1 = \frac{1}{\infty_d}[/tex]

Am I correct so far? If so how do you define this [itex]\infty_d[/itex] and furthermore for this to hold true you must have a way of defining all other infinities right? Otherwise your system would just fall apart.

If you can show me this way of mathematically creating relationships of all infinites with one natural infinity and describe their mathematical relationship to real numbers then I will believe you

 

[ram2048]

However there is just one little flaw that I see that you may be able to patch up. You are of the belief that [itex]\infty + 1 \neq \infty[/itex] Which means you must have some default value for infinity, for example: [itex]\infty_d[/itex] such that:

[tex]0.\overline{0}1 = \frac{1}{\infty_d}[/tex]

Am I correct so far? If so how do you define this [itex]\infty_d[/itex] and furthermore for this to hold true you must have a way of defining all other infinities right? Otherwise your system would just fall apart.

i'm not certain that [tex]0.\overline{0}1 = \frac{1}{\infty_d}[/tex] they are infinitessimals (someone used that words somewhere I'm just going to assume it's a real word) but the way they are defined and created are quite different, thus i don't think you can create and equality between them. let me consider.

If you can show me this way of mathematically creating relationships of all infinites with one natural infinity and describe their mathematical relationship to real numbers then I will believe you

cute but i doubt your sincerity :D

in any case, let's assume a default infinity to begin with. if all rational transforms upon that infinity result in a net positive gain then the resulting infinity is greater than the original and we will use a different notation for the new infinity [tex]\infty_e[/tex]. likewise were the transforms to result in negative "gain" then the resulting infinity would be less than the original infinity. let's use [tex]\infty_c[/tex]

we can accurately say [tex]\infty_c < \infty_d < \infty_e [/tex].

<crosses fingers for tex coding>