Is 0.999... Truly Equal to 1 in the Realm 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 0.\overline{0}1 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 0.\overline{9} exists anyway. Furthermore you say that:

0.\overline{3} \neq \frac{1}{3}

You seem to have really lost the plot here and seem to be implying that 0.\overline{3} 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 0.\overline{3}. As you state it is neither:

\frac{1}{3}

or

\sum_{n=1}^{\infty} \frac{3}{10^n}

 

[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 0.\overline{3} 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 \sum_{n=1}^{\infty} \frac{3}{10^n} does create 0.\overline{3}

 

[Zurtex]

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

0.\overline{0}1 = \frac{1}{\infty}

As infinity halved is still infinity then:

\frac{1}{2}\infty = \infty

Taking the reciprocal of both sides:

2 \left( \frac{1}{\infty} \right) = \frac{1}{\infty}

Using out identity:

0.\overline{0}1 = \frac{1}{\infty}

Then:

2(0.\overline{0}1) = 0.\overline{0}1

Taking away 0.\overline{0}1 from both sides:

0.\overline{0}1 = 0

 

[Zurtex]

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

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 \frac{1}{2}\infty = \infty 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 \frac{1}{2}\infty = \infty 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.

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

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

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 0.\overline{3} (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 \infty+1 \neq \infty, 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 \infty + 1 \neq \infty Which means you must have some default value for infinity, for example: \infty_d such that:

0.\overline{0}1 = \frac{1}{\infty_d}

Am I correct so far? If so how do you define this \infty_d 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 \infty + 1 \neq \infty Which means you must have some default value for infinity, for example: \infty_d such that:

0.\overline{0}1 = \frac{1}{\infty_d}

Am I correct so far? If so how do you define this \infty_d 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 0.\overline{0}1 = \frac{1}{\infty_d} 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 \infty_e. likewise were the transforms to result in negative "gain" then the resulting infinity would be less than the original infinity. let's use \infty_c

we can accurately say \infty_c &lt; \infty_d &lt; \infty_e.

<crosses fingers for tex coding>

 

[Zurtex]

You have not defined the original infinity so how can you say that? Nor have you defined a relationship, for example would:

\infty_d - \infty_c = a

Where a is some finite number and according to your system it would be a specific finite value, what would it be? Even if it is not a finite value according to your system it would be a specific infinite number, what would it be? I am of course assuming a > 0 under the system you have stated.

So I wonder would:

\frac{1}{{\infty_d}^{\infty_c}} = 0.\overline{0}n ?

Where n is some finite digit? Or would this actually be 0?

Furthermore your notation is flawed, you are only going to have a finite amount of infinites where as I can think of an infinite number of ways to increase the value of a number. That's all the questions I can write down at the moment as I have to go, have at least another few dozen questions to see how your system works.

 

[matt grime]

You ought to at least understand that hyper-real, or infinitesimal numbers, are not real numbers and so attempting tp cite them in defence of your ramblings of ignorance about the real numbers won't win you any arguments.

 

[Hurkyl]

CHAPTER 3

Proof #1

In this proof they call upon a previously existing summation theorem that assumes to be correct and then proceed to apply the variables to come out with the conclusion. NO ONE can accurately sum any series to infinity to begin with. This entire proof stems from that fallacy.

Would you care to explain what you mean? Do recall that the DEFINITION of the sum of an infinite series is:

<br /> \sum_{i=0}^{\infty} a_i = \lim_{m \rightarrow \infty} \sum_{i=0}^{m} a_i<br />

As you correctly point out, one cannot directly add an infinite number of terms... which is why we're not silly enough to do it that way.


Proof #2

on the third line. x or .9999... is subtracted from both sides and the assumption is made that you will get an even number of 9 as the result.

Actually, it is fairly easy to do this in a perfectly rigorous way.. In the way mathematicians define things, the position of each digit in a decimal number is labelled by an integer.

To give a hint of the flavor of how things are done done, I'll ask you this question:

In what position does 0.999~ have a '9' that doesn't appear in 9.999~?

(again, let me remind you; by the mathematical definition, that position has to be an integer... and no integers are infinite)

Proof #3

something INFINITELY SMALL is never NOTHING.

Incorrect; zero is the only infinitessimal real number. FYI, the definition of "infinitessimal" is:

x is an infinitessimal if and only if |x| is smaller than any positive real number.

Since the decimal numbers are constructed so that they are a model of the real numbers, it follows that zero is the only infinitessimal decimal number.


Proof #4

Go ahead and do the division for yourself. One divided by three. .33333333~ forever. Where is it wrong?

It's not.

well three does not divide evenly into one no matter how many digits you take it to, so that remainder of 1 will always exist just as the digits will never terminate.

But, the way mathematicians define the decimals, they don't have remainders. So 1/3 is 0.333~.

so 1/3 would equal 0.33~ R1/3

So this begs the question, in your system, what is 1/3? We can keep substituting:

1/3 = 0.33~ R1/3 = 0.33~ R0.33~ R 1/3 = 0.33~ R0.33~ R 0.33~ R 1/3

but we'll never come to a satisfactory representation.

But consider a much more pressing issue; how can a remainder possibly make sense if we want a decimal representatino of &radic;2, or pi?



Also, allow me to suggest something for "fun". Let's assume for a moment that you actually have a well-defined and consistent number system.


Allow me to invent a new number system by decreeing that any two numbers that have the same decimal part are equal (even if they have different remainders), and furthermore I decree true any statement that can be proved from this decree. (So, for instance, 0.999~ = 1 because you've been able to prove 0.999~ R3/3 = 1 R0) I wonder what properties this number system has?

 

[Hurkyl]

Hrm.

4/3 = 4 * 0.333~ R(1/3) = 1.333~ R(4/3)
4/3 = 1 + 1/3 = 1 + 0.333~ R(1/3) = 1.333~ R(1/3)

Problem...

 

[ram2048]

\frac{1}{{\infty_d}^{\infty_c}} = 0.\overline{0}n ?

i don't know that looks complicated. is it actually a real question or are you coming up with new inventive ways to waste my time? :D

CHAPTER 3

Proof #1

Would you care to explain what you mean? Do recall that the DEFINITION of the sum of an infinite series is:

<br /> \sum_{i=0}^{\infty} a_i = \lim_{m \rightarrow \infty} \sum_{i=0}^{m} a_i<br />

As you correctly point out, one cannot directly add an infinite number of terms... which is why we're not silly enough to do it that way.


Proof #2



Actually, it is fairly easy to do this in a perfectly rigorous way.. In the way mathematicians define things, the position of each digit in a decimal number is labelled by an integer.

To give a hint of the flavor of how things are done done, I'll ask you this question:

In what position does 0.999~ have a '9' that doesn't appear in 9.999~?

shifting the decimal for such a number is tricky business because it causes an inequality in the infinities in both expressions.

(again, let me remind you; by the mathematical definition, that position has to be an integer... and no integers are infinite)

Proof #3

i'm kinda going through this with someone in another thread at the moment as well and there is no satisfactory answer yet thus far. but expression 1 would have infinite 9's and expression 2 would have infinite+1 9's.

Incorrect; zero is the only infinitessimal real number. FYI, the definition of "infinitessimal" is:

x is an infinitessimal if and only if |x| is smaller than any positive real number.

Since the decimal numbers are constructed so that they are a model of the real numbers, it follows that zero is the only infinitessimal decimal number.


But, the way mathematicians define the decimals, they don't have remainders. So 1/3 is 0.333~.

So this begs the question, in your system, what is 1/3? We can keep substituting:

1/3 = 0.33~ R1/3 = 0.33~ R0.33~ R 1/3 = 0.33~ R0.33~ R 0.33~ R 1/3

but we'll never come to a satisfactory representation.

But consider a much more pressing issue; how can a remainder possibly make sense if we want a decimal representatino of √2, or pi?

the remainder is only used to create rational decimals for rational fraction values that would otherwise create non-terminating irrationals

the r(1/3) is not at the end of the series, rather it is part of the last digit. so you wouldn't be able to string it like that.

you could consider the last digit to be 10/3 rather than 3 <basically>

Also, allow me to suggest something for "fun". Let's assume for a moment that you actually have a well-defined and consistent number system.


Allow me to invent a new number system by decreeing that any two numbers that have the same decimal part are equal (even if they have different remainders), and furthermore I decree true any statement that can be proved from this decree. (So, for instance, 0.999~ = 1 because you've been able to prove 0.999~ R3/3 = 1 R0) I wonder what properties this number system has?

sounds like you're describing the current number system...

Hrm.

4/3 = 4 * 0.333~ R(1/3) = 1.333~ R(4/3)
4/3 = 1 + 1/3 = 1 + 0.333~ R(1/3) = 1.333~ R(1/3)

Problem...

you messed it up :D

4/3 = 4 * 0.333~ R(1/3) = 1.333~ R(1/3)
3x.333~ made 999~ the 3/3 completed the 1 which was part of the last digit 9. creating 10/10 that completes all the way to the top to make 1.

the process remainder is part of the last digit not after it.

 

[Integral]

3x.333~ made 999~ the 3/3 completed the 1 which was part of the last digit 9.

You now need to share with us what you mean by a digit. How is something "part of a digit"

 

[ram2048]

sorry if it wasn't clear enough.

when you divide out 1/3 each digit is created by a process of 10/3. each digit = 9/3 however. that 1/3 left over is the remainder I'm talking about.

 

[Integral]

You are correct, it is not clear. What do you mean when you use the word "digit"?

 

[ram2048]

suppose i wanted to get the true decimal value of 1/3 (one divided by three)

i would start doing division (the tedious way)

[pre]
0.333~ r 1
3|1.0
9
10
9
10
[/pre]

each "digit" of 3 in the decimal notation resulting was created by dividing 10 by 3.

to use hurkyl's notation that he brought up
f(n)=0 where n ≥ 1
f(n)=3 where n < 1

those digits.

except in this case

f(n)=(10/3) where n = ∞

hope that formatting works

 

[Integral]

Ok, how is something "part" of a digit? Is not a digit a single bit of information how am I to interpret something as part of a single piece?

You have shown me how you generate digits but you still have not told me what a digit is.

 

[ram2048]

okay... a digit(in base ten) is one integer of values 0 through 9 which exist in sequence to describe real numbers such that these digits multiplied by various powers of 10 can be summed to create the number they describe.

i'm going to assume that your follow-up question will be "if you just said the digit is integers 0 through 9 how can you have a digit that is 10/3"

the answer is, this is new notation

 

[Hurkyl]

shifting the decimal for such a number is tricky business because it causes an inequality in the infinities in both expressions.

Can you state yourself in a precise way?

Here are two interesting questions for your system:

What is 9 + 0.99~?
What is 0.99~ * 10?

Can you even divine a difference in your system?

In the standard mathematical definition of decimals, the two are the same string of digits (and thus equal as decimals), because for every integer n, the digit in the n-th place in the first is equal to the digit in the n-th place in the second.

i'm kinda going through this with someone in another thread at the moment as well and there is no satisfactory answer yet thus far. but expression 1 would have infinite 9's and expression 2 would have infinite+1 9's.

Mathematicians do study these kinds of things; they're called order types. There really is a way to rigorously say that a sequence has length "infinity + 1" (where "infinity" is the order type of the positive integers). However, there is no such thing as an "infinity - 1"; the problem is that in order to take things off of the end of an ordering, the ordering has to have an end.

Because the digits to the right of the decimal place are indexed by the positive integers (or negative, if you prefer), this ordering doesn't have an end from which things can be removed. This is why it is possible to left shift all of the digits and have "one more" than you started with. (Mathematically, it still has the same order type and cardinality)

You, like most others who have similar ideas, seem to be imagining an order type that does have an end. Now, if you were to define the decimals as having two groups of digits to the right of the decimal place, an "a group" and a "b group" so that the positions went like this:

| 1a, 2a, 3a, ... | ... 3b, 2b, 1b |

(where I've used pipes (|) to denote the boundary of a group)

then when you multiplied this new type of decimal by 10, you would indeed "lose" a digit off the right end; the 1b position would be filled with a zero.

sounds like you're describing the current number system...

Exactly right. I was trying to demonstrate that even if you did have a consistent number system with infinitessimals, you can recover a number system that behaves exactly like the standard mathematical system. The point is to show relative consistency; if you believe the standard system to be flawed, then your system must be flawed as well because your system can be used to create the standard system.

 

[Hurkyl]

Here's a puzzler. If you are only introducing remainders in the case of dividing things, then what is e^(-ln 3)?

 

[Integral]

okay... a digit(in base ten) is one integer of values 0 through 9 which exist in sequence to describe real numbers such that these digits multiplied by various powers of 10 can be summed to create the number they describe.

i'm going to assume that your follow-up question will be "if you just said the digit is integers 0 through 9 how can you have a digit that is 10/3"

the answer is, this is new notation

So once again what do you mean by part of a digit? You are going to need a better definition then simply saying it is notation. The ellipsis is simply notation, I can define it in a few words; it stands for a infinitely repeating pattern. Now what do you mean by "part of a digit" ?

You seem to have agreed that infinity must have some default value, which to me says that you have confused infinity with a large finite number, all of your work is consistent with that conclusion. If you replaced infinity with any large finite number all of your work would make sense and you would have little disagreement from anyone on this board.

Your belief that 1 - .999... >0 means that the interval (.999...,1) has a finite non zero length, thus there must be a hole in the real line at this point, are you saying that .999... is the hole? are there other holes? If there is a hole at .333... why is it that .1(base 3) exists is it not right in the middle of the hole that you have placed at .333...?

I think it is your system that is full of holes, it simply does not work. Your time would be better spent actually attempting to learn how the real number system works rather then trying to reinvent a square wheel. It's your time.

 

[Hurkyl]

Consider this.

Choose any of the 9's in 0.999~.
Consider the 9 in the next position to the right.
This 9 gets shifted underneath the 9 you chose when you multiply by 10.
Thus, 9.999~ has a 9 in the same position as the 9 you chose in 0.999~.

If this argument holds for any 9 you choose, then we have proved that 0.999~ cannot have a 9 in a position that 9.999~ does not.


Again, this works because decimal numbers don't have right ends; each position has a next position to the right.