Is 0 divided by 0 equal to any number?

[Zurtex]

What is the result of 9.9999.../0.9999... ?

Well as 0.9999... = 1 and 9.9999... = 10 * 0.9999... = 10 * 1 = 10

Then you are asking what the result of 10 / 1 is.

Edit:

I suppose another way of looking at the problem is saying that it is the same as:

\lim _{x \rightarrow \infty} \frac{10 - 10^{-x}}{1-10^{-x}}

Where x is a natural number.

 

[Organic]

9.9999.../0.9999... = 10 , which is the ratio between 9.9999... and 0.9999...
9.9999 .../1.0000... = 9.9999... , which is the ratio between 9.9999... and 1.0000...

Now let us check this arithmetic:

x = 0.99999999...
10x = 9.999999...
10x - x = 9.9999... - 0.999999...
9x = 9
x = 1

A question: How we can be sure that the result of 0.99999... - 0.99999...
is exactly 0 where there is no right side to begin the subtraction operation?

 

[matt grime]

Because addition and subtraction can be put into an algorithm for infinitely long decimals if you need to do so. However as you're asking what the result of x-x is it doesn't depend on x having a nice form, it is zero by definition. Why do people confuse numbers and their decimal representations?

 

[matt grime]

Or if you don't like that answer then how about doing it in terms of Cauchy sequences, which is after all the most useful construction of the Real numbers as a mathematical object.

 

[Organic]

Please prove that there is no connection between a number and its stuctural represention upon infinitely many scales.

 

[matt grime]

I didn't say there wasn't a connection (though what you mean by that is unclear) but that the addition of two real numbers is independent of the choice of decimal expansion, should it have two. The simple proof of this fact follows from the definition of the real numbes as the completion of the rationals. Go and get a basic analysis book. Just because you do not know it, Organic, does not mean it is not true or known by other better informed people. Proof: let x_n and y_n be two equivalent Cauchy sequences. This means x_n-y_n converges to zero. Let w_n and v_n ba any other pair of equivanlent cauchy sequences.

then the element of R that [x_n-w_n] coresponds to is the same as the class (real number) [y_n-v_n]

proof: we are to show x_n-w_n-y_n+v_n converges to zero, but that is trivially true since |x_n-w_n_y_n+v_n| < |x_n-y_n| +|w_n-v_n| and both those terms can, be made arbitrarily small by hypothesis,a nd we have proved subtraction of two real numbers is indpendent of the Cauchy sequences we pick to represent them. OK?

 

[Organic]

What you show is a rough jump that forces infinitely long sequence to become finitely long, and than you use subtraction after you created an artificial right side (which cannot exist in infinitely long sequence) in a non-logical way for your own purpose.

 

[matt grime]

No, this is the rigorous proof that addition is well defined on the Real numbers. Perhaps you ought to go and learn some mathematics?

 

[Zurtex]

Have you ever just thought about converting 0.3333333... to base 3 to get 0.1 which when multiplied by 3 you get 1.

 

[Janitor]

Zurtex,

I would anticipate that Organic's objection to your suggestion is that "there is no right side to begin the base-conversion operation."

 

[Organic]

Hi Zurtex,

This is exactly my point of view on this case, a number is not just a quantity but has also an internal structute that cannot be ignored, for example look at this paper:

http://www.geocities.com/complementarytheory/Complex.pdf

Perhaps you ought to go and learn some mathematics?

I cannot agree with mathematics which is based on forcing methods.

 

[Integral]

If it is an integer it is rational. It doesn't need to be an integer. Try it with .000012121212... with the repeating12 pattern. you get .001200000... which is rational. It just produces from a recurrent decimal y, a terminating decimal, r, ie rational, satisfying y(10^n-1)= r hence y is a rational divided by an integer, thus a rational.

very good! Thank you.
This gets us the next gem also, observe that:

y = \frac r {10^n -1}

The denominator will consist of n 9's

 

[matt grime]

Organic, the question was posed in the field of real numbers with their mathematical definition. Your opinions as to whether that is not the correct object are irrelevant to the question, and its answer. Or would you like to point out where the proof that 0.99999 =1 is wrong when working in the usual definition of the real numbers as equivalences classes of cauchy sequences? I do not need to use all the things that are true about some object to prove things about it. For instance, every digit in 0.99999... is a prefect square, I didn't use that fact. I didn't use the fact that considered as curves embedded in the plane the digits involved all have fundamental groups that aren't trivial. I didn't use the fact that 9 is 6+3 where 3 is the smallest odd prime and 6 is the smallest order of a non-abelian group. If you aren't prepared to learn what mathematics involves then how can you possibyl answer questions about it? I mean, there is a theory where 0.9999... is not equalt to 1. Perhaps you want to learn about Abraham Robinson's non-standard analysis? Whereof you do not know do not speak?




Integral, yes, but, there's no need for the r in there to be an integer, which is what you wanted originally I seem to recall. And ignores the x. Example .011111 = 1/90, arguably a recurrent decimal and not consisting eintrely of 9s in the denominator.

 

[matt grime]

Have you ever just thought about converting 0.3333333... to base 3 to get 0.1 which when multiplied by 3 you get 1.

In base 3 one has other problems such as .22222222... =1

Forget decimals, or any other system of representation like that. Just operate with the definitions of the real numbers. That's how mathematics works, practically.

 

[Integral]

Matt,
Perhaps now you are beginning understand why my proof does not perform operations on non finite digits. People such as Organic, who should be restricted to posting in Theory Development, simply will not accept any proof you can provide that non finite operations are allowable. Beyond that I was taught in my analysis courses that such operations should not be included in fundamental proofs.

Organic is special in that he has his own number system which he cannot separate from the Reals that the rest of us use.

 

[Integral]

Integral, yes, but, there's no need for the r in there to be an integer, which is what you wanted originally I seem to recall. And ignores the x. Example .011111 = 1/90, arguably a recurrent decimal and not consisting entirely of 9s in the denominator.

So simply factor out the non repeating part.

.0111... = .111... x 10^{-1} = \frac 1 9 x 10^{-1}

So we have a multiple of 10 and a rational with 9's in the denominator. This is validation of the methods mentioned up thread using 9's in the denominator. We have shown that every repeating decimal can be represented as a factor of 10 and the repeating portion over 9s.

 

[Integral]

In base 3 one has other problems such as .22222222... =1

Forget decimals, or any other system of representation like that. Just operate with the definitions of the real numbers. That's how mathematics works, practically.

Matt,
The Reals are not base dependent, a base 2 or 3 or 16 system is the same as a base 10 representation as far as the real system is concerned. Yes, different bases have different rationals as repeating decimals. For example

.1 (base 10) = .0001100110011...(base 2) (I think I got the right number of leading zeros). This is of significance because it means your computer must round off .1 .

EDIT: Opps, Matt I just reread your post, I got different meaning the 2nd time. I am trying to say the same thing you are. The Reals are Base independent.

 

[matt grime]

Where did I say the reals are base dependent? If everyone remembered what the real numbers acutally are then none of these recurring (pun intended) nightmares would happen. It's amazing how often this question comes up, isn't it?

 

[ShawnD]

Have you ever just thought about converting 0.3333333... to base 3 to get 0.1 which when multiplied by 3 you get 1.

That's sort of how the fraction representation works. 1/3 x 3 = 1.

 

[Organic]

Organic is special in that he has his own number system which he cannot separate from the Reals that the rest of us use.

Let us take the circle's equation: (x-h)^2 + (y-k) = r^2
http://www.xavierhs.org/departments/Mathematics/PreCal/Conics/conics.htm

solid is a "one piece" state

r=radius

h=x center

k=y center

But the interesting variables are x and y, where x is the entire x-axis and y is the entire y-axis.

x-axis or y-axis are "actual form of infinity" as we can see in this model:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

To construct the circle we have to break the solid states of both x-axis and y-axis and define a sequence of unique pairs of R members, which are used as x,y coordinates of the circle.

The point here is that we have no R members before we break the solid state of x-axis and y-axis, and only after we break them we get R members.

The same state is an information form of, for example, 0.9999999...

It cannot be in both states of finite and infinite sequence of non-zero values upon infinitely many scales.

Therefore there is a XOR condition between 1.0 and 0.9999... exactly as there is a XOR condition between a solid state and a broken state.

 

[Integral]

Organic,
Please restrict your posts to the topic at hand.

 

[Organic]

Hi Integral,

My previous post fits exactly to subject of this thread.

Please read all of it and see for yourself.

 

[Integral]

Organic,
We are all talking about the Real numbers, you are talking about the organic numbers, you are off topic. Please take your garbage back to theory development where it belongs.

 

[Zurtex]

That's sort of how the fraction representation works. 1/3 x 3 = 1.

I know but organic seems to be struggling with the concept of converting decimals into fractions when they recur.

In base 3 one has other problems such as .22222222... =1

Forget decimals, or any other system of representation like that. Just operate with the definitions of the real numbers. That's how mathematics works, practically.

Well convert 0.111111... to decimal to get 0.5, multiply by 2 to get 1

 

[Zurtex]

O.K, merging what I put earlier with my work on base numbers.

Lets say we are working in base b, and let's us say that x is a number in base b and that x is a natural number. n is a digit in the decimal number x such that n = b -1.

So if:
x = 0.n = 1 - 10^{-1} = \frac{b-1}{b}

Or if:

x = 0.nnnnn = 1 - 10^{-5} = \frac{b^5-1}{b^5}

If n occurs p number of times:

x = 0.nnnnnnnn... = 1 - 10^{-p} = \frac{b^p-1}{b^p}

If n occurs an infinite number of times:

x = 0.nnnnnnn... = \lim _{p \rightarrow \infty} 1 - 10^{-p} = 1
Or x = \lim _{p \rightarrow \infty} \frac{b^p-1}{b^p} = \lim _{p \rightarrow \infty} (b^{-p})(b^{p}) - 1(b^{-p}) = \lim _{p \rightarrow \infty} 1 - b^{-p} = 1

Now your equation x - x: x - x = x(1-1) = x*0 = 0 As long as x has a numerical value regardless of how it is expressed.

 

[matt grime]

Don't think you're going to get any sympathy there, Organic, especially given the God-awful choice of font, colour, and spelling. You are posting off topic, something we're probably all guilty of at some point admittedly, but you are also posting utter tripe that has no place in a mathematics thread. I'm amazed that, as a moderator, Integral didn't just delete your incoherent rubbish. You aren't some innocent posting a silly question and being dismissed out of hand. You are a recidivistic poster whose answers are tantamount to vandalization and are entirely unmathematical and plain wrong, yet you keep making them.

 

[HallsofIvy]

Well, if it were me instead of integral, I certainly wouldn't have used the word "garbage"!

(The word I would have used would have gotten all "knocked off the air"!)

 

[Organic]

Matt and HallsofIvy,

I invite you to show your skills and prove that my ideas have nothing to do with Math language development.

here it is again:

Let us take the circle's equation: (x-h)^2 + (y-k) = r^2
http://www.xavierhs.org/departments/Mathematics/PreCal/Conics/conics.htm

solid is a "one piece" state

r=radius

h=x center

k=y center

But the interesting variables are x and y, where x is the entire x-axis and y is the entire y-axis.

x-axis or y-axis are "actual form of infinity" as we can see in this model:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

To construct the circle we have to break the solid states of both x-axis and y-axis and define a sequence of unique pairs of R members, which are used as x,y coordinates of the circle.

The point here is that we have no R members before we break the solid state of x-axis and y-axis, and only after we break them we get R members.

The same state is an information form of, for example, 0.9999999...

It cannot be in both states of finite and infinite sequence of non-zero values upon infinitely many scales.

Therefore there is a XOR condition between 1.0 and 0.9999... exactly as there is a XOR condition between a solid ("one piece") state and a broken state.

 

[matt grime]

We only need to show that what you wrote doesn't have anything to do with the question as asked.

Item 1. Your bloody post which has nothing to do with the question "why does 0.9999..=1?"

Item 2. Your bloody post which has nothing to do with the question "why does 0.9999..=1?"

Strictly speaking they are the same, but I thought it important enough to mention twice. This is a question about the real numbers. Do you know what they are? Evidently not judging by your bilge of a repost.

Edit: deleted silly thing about bandwidth. Still think you're a moron though.

 

[matt grime]

Can some moderator please lock this thread? I think it has been adequately answered as to why there is nothing reomtely controversial about 0.9999.. being the same as 1, and all mathematical issues arising therefrom seem to have been sorted.