Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#201 2006-12-17 17:06:05

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

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

we can't "see" infinitely many decimals
-the problem I proposed in this thread is that infinite decimals, if existed, would have contravasory among themselves. Not as simple as we cannot see.

You have yet to prove or even show that.  Show me that the length of a side of your desk is not an irrational number.

Infinite sequences are a dilema as well. Infinite sequences mean only you can add in new entries without a stop.

So you also have a problem with infinite sequences as well?  Well then, how exactly do you propose we define the reals?  Cause the last I heard, both popular methods involve an infinite sequence of rational numbers.

The rest of your post I could not seem to understand.


"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..."

Offline

#202 2006-12-17 19:14:33

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

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

Simple: If we define one metre on the desk. Then we can examine how many particles the 1 metre has passed through. Let's denote the amount of particles as N. Then how much length is a given distance on the desk? Just count again, and get the number M. Therefore the length we want to know is M/N metres.

Yes, I do have problems with Real as well, which defines numbers as infinite sets, and the defination of many infinite decimals, the root of 2, e and pi, to name a few.

Before this sentence-
"It is with this in mind that we make the following definition"
you do the proof within a finite framework
To be precise, you proved the Series r is always increasing, which is r[sub]n+1[/sub]>r[sub]n[/sub]

After the sentence and the defination, you use the concept of infinity (or infinite decimals).
Once you use the concept of infinite decimals, you cannot explain how the finitth decimals goes to infiniteth decimals. That is quite simple- between finite quantities and infinite "quantities" is an untrancendable gap.


X'(y-Xβ)=0

Offline

#203 2006-12-18 02:07:59

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

So there we have it! No one can say how many .9’s it takes for a number to be or become a whole Number!!

Never forget what the great Philosopher/Mathematician “MATHIUS” (1803) said!

“For someone to prove the end of an Infinite Number, would be the same as someone saying, I have captured all the Stars in the Universe, and put them in this Bag!”

p.s have a Good One “back in the new year!

A.R.B

Offline

#204 2006-12-18 06:54:03

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

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

Anthony, I'm pretty sure that most people are just ignoring you now.

Simple: If we define one metre on the desk. Then we can examine how many particles the 1 metre has passed through. Let's denote the amount of particles as N. Then how much length is a given distance on the desk? Just count again, and get the number M. Therefore the length we want to know is M/N metres.

Interesting.  Because all particles, even in solids, are moving.  Also, the volume (and thus, length) of a solid depends on the temperature.  While these are minute differences, you still must take them into account.

It is not simply that atoms are balls laying there.  They are very active things, moving around.  Your calculations do not take this into effect.

you do the proof within a finite framework
To be precise, you proved the Series r is always increasing, which is rn+1>rn

After the sentence and the defination, you use the concept of infinity (or infinite decimals).
Once you use the concept of infinite decimals, you cannot explain how the finitth decimals goes to infiniteth decimals. That is quite simple- between finite quantities and infinite "quantities" is an untrancendable gap.

I argue that there is no such thing as finite decimals.  That is, 5.38 is really 5.380000000...

Then, there is no "going" to infinite decimals, since everything is.  There is no gap because there is no difference.


"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..."

Offline

#205 2006-12-18 16:15:15

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

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

Ricky wrote:

Interesting.  Because all particles, even in solids, are moving.  Also, the volume (and thus, length) of a solid depends on the temperature.  While these are minute differences, you still must take them into account.

It is not simply that atoms are balls laying there.  They are very active things, moving around.  Your calculations do not take this into effect.

Yes, I admit that would be a problem. But things in the world may be not static as we used to think-we used to think distance is distance, time is time-but now we no longer think so.

Ricky wrote:

I argue that there is no such thing as finite decimals.  That is, 5.38 is really 5.380000000...

Then, there is no "going" to infinite decimals, since everything is.  There is no gap because there is no difference.

It is really interesting to see how you have distorted a rational

into the "number" you like:

And you may call it evolution.

Anthony, the key is to deny infinitesimals, because of the 10×0.999...=9.99... "proof".

Last edited by George,Y (2006-12-18 16:16:56)


X'(y-Xβ)=0

Offline

#206 2006-12-18 16:26:18

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

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

Yes, I admit that would be a problem. But things in the world may be not static as we used to think-we used to think distance is distance, time is time-but now we no longer think so.

Ok, so back to the original question.  Show me that the length of your desk is not irrational.

Um, why is it that you are raising 10 to the infinity?


"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..."

Offline

#207 2006-12-19 01:49:06

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

Hi! did not think I would get the chance to put this forward anyway!

Quote: Ricky " Anthony, I'm pretty sure that most people are just ignoring you now."

Maybe in yours eyes they are! but as long as there are an Infinite Amount of people! I will always have people who are not!

(ENDLESS <> END)  =  (INFINITE 0.9 <> 1) 

p.s have a Good One “back in the new year!!

A.R.B

Offline

#208 2006-12-19 04:35:42

Ehiness
Member
Registered: 2006-12-05
Posts: 6

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

Anthony.R.Brown wrote:

Hi! did not think I would get the chance to put this forward anyway!

Quote: Ricky " Anthony, I'm pretty sure that most people are just ignoring you now."

Maybe in yours eyes they are! but as long as there are an Infinite Amount of people! I will always have people who are not!

(ENDLESS <> END)  =  (INFINITE 0.9 <> 1) 

p.s have a Good One “back in the new year!!

A.R.B

Ok I am 99.999...% sure that you have no idea what infinite means.

Offline

#209 2006-12-19 18:14:27

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

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

Ricky wrote:

Ok, so back to the original question.  Show me that the length of your desk is not irrational.

Simple, because the length  is basically an integar, the length of any line made of particles is rational, or put another way, discrete. You may compose a right triangle by actually two line segments of 1,000,000,000 particles, then you wanna find the longest side length. You may use the ruler made of the same material, then find the side of 1,414,213,562 or 1,414,213,563 particles, to be the most precise.
This would, for sure, cause some problem of imperfection in regard of the length. But it would be a lesser problem than regarding 1,000,000,000(√2) particles composing the side.


Ricky wrote:

Um, why is it that you are raising 10 to the infinity?

Again you don't understand what you mean. Well, first I would like to explain that writing 0.000... is not just a game. Nor does it simply mean zero. It infact represents 0(1)+0(1/10)+0(1/100)+...+0(1/10^∞). The simble of infinity may scare you, but it is true when you say infinite digits, and if you like, we can replace it by the super quantity q, as what I used in a previous post. Then how can I add them together? By unifying the denominator. So the addition simply turns to 0/10^∞, here I could add 538/100 in, making the notation of the one you don't like.

Or simply put, the infiniteth digit means a time of 1/10^infinity. Astounded? You creat it.

Initially people had integers. Then they had rationals, litterally one integer devided by another. Later decimals was used. Decimals are indeed helpful to guage a complex rational, but imediately people found lots of times a rational cannot be turned into finite decimals. Then some smart guy had the idea of recurring to trace the pattern for roungding conveniency. But the further idea of infinite decimals is misleading. Infinite, litterally means endless and irreachable came into use.

It is very funny to accept an no-end as an end. How many digits does an infinite decimal contain? Larger than any integer.
What is larger than any integer? Cannot figure, but is no integer , no limitation.
But how does an infinite decimal stand static rather than forever growing (1 decimal, 2 decimals, 3 decimals and so on)?
By simply an amount of digits larger than any amount.


X'(y-Xβ)=0

Offline

#210 2006-12-20 04:59:59

Devantè
Real Member
Registered: 2006-07-14
Posts: 6,400

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

Anthony, 0.333..., as you said (I think), multiplied by 3, is 0.999... . Now, [sup]1[/sup]/[sub]3[/sub] multiplied by 3 is [sup]3[/sup]/[sub]3[/sub], right? If [sup]1[/sup]/[sub]3[/sub] = 0.333..., tell me what you answer is when you multiply that my 3. And stop straying from the answer.

Last edited by Devanté (2006-12-20 05:00:24)

Offline

#211 2006-12-20 11:53:45

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

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

Simple, because the length  is basically an integar, the length of any line made of particles is rational, or put another way, discrete.

You already agreed that the length of a particle is not a constant value (uranium is "longer" than hydrogen).  Also, the particles are constantly moving and vibrating.  Again, you attempt to ignore these very basic facts.

It is very funny to accept an no-end as an end.

Please show me where I have stated that something with infinite digits has an end.

George, I want to move past 0.999... = 1 and onto more basic stuff.  Do you believe that irrationals should not exist in mathematics?  That rationals are all we need?

If so, we have nothing further to discuss.  You simply disagree on philosophic principle, not mathematics.  Our definitions would thus be entirely different, and so we would constantly be talking past one another.  I see no use for that.

If not, please explain to me how it is that you believe the real numbers, an ordered field with completeness, should be constructed.


"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..."

Offline

#212 2006-12-21 14:45:12

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

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

"You already agreed that the length of a particle is not a constant value (uranium is "longer" than hydrogen).  Also, the particles are constantly moving and vibrating.  Again, you attempt to ignore these very basic facts."

--Initially the defination of One Metre is the length of a bronze ruler. The ruler was kept by the Franchmen in Paris. Every metre defined in other countries was the duplication of this ruler. Do you mean just because microly the particles of the ruler is constantly viberating, the defination of the metre is wrong?
Indeed, microly changing distants do not cause too much macro problems-the particles of the words now you read are moving. However, temperature would be a problem-how did they deal with the expansion subject to temperature? They further defined the length under 20°C is the real length of one metre. Wait, how did they define 20°C? They used mercury. They defined the height of the mercury in a thermometre dipped in a mixture of water and ice to represent 0°C, and the height when it is in boiling water to represent 100°C. Then the differiential of the height is devided by 100 equal intervals-wait, how can we divide such intervals? By compasses, perhaps. Again the intervals are constantly changing.

Then technology improved. We have got laser beams.Now one metre is defined according to the length a specil laser beam could travel within a given time. The time is defined by radioactive decay. The metre, however, is still not accurate. Because 1) the light does not always travel in the same speed, according to recent surveys 2) the light does not travel in absolute line due to the gravity, according to Einstein.

Perhaps the most ratinal thinking about measure is that it cannot be absolute exact. The exactness only exists in some mathematicians' imagination.

It is not that I don't like irrationals or reals, but that I believe their existence is inferior to logic consistency.


X'(y-Xβ)=0

Offline

#213 2006-12-21 14:48:22

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

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

Correction:
specil->special
recent surveys->recent studies


X'(y-Xβ)=0

Offline

#214 2006-12-21 16:34:00

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

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

Perhaps the most ratinal thinking about measure is that it cannot be absolute exact. The exactness only exists in some mathematicians' imagination.

Exactly!  The universe is exact, our measurements are approximations.  So when we do mathematics, should we be exact, or only close enough that we can base engineering on it?

My answer is be exact.  It's the closest you can get.


"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..."

Offline

#215 2006-12-21 17:22:50

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

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

Exactly!  The universe is exact, our measurements are approximations.  So when we do mathematics, should we be exact, or only close enough that we can base engineering on it?

-Well, I find you are really a Platoist- The real world is imperfect, but the concept of it could be; Every chair is just an approximation of the concept of chair, which is exact.

It is philosophic stands that underly our disagreement.


X'(y-Xβ)=0

Offline

#216 2006-12-21 17:25:48

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

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

-Well, I find you are really a Platoist- The real world is imperfect, but the concept of it could be; Every chair is just an approximation of the concept of chair, which is exact.

Nope.  Reality is perfect.  Our measurements of it are not.


"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..."

Offline

#217 2006-12-21 17:38:17

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

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

Do you really believe the universe continous? Or time continuous? distance?

If none of them is, exactness of the continuum has no sense. Think of the electric charge.

Zeno's paradox is clear to demonstrate the impossibility of continuous quantity(distance).


X'(y-Xβ)=0

Offline

#218 2006-12-21 18:29:47

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

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

There do exist unsolved (or unavoided) paradoxes with the assumption of the continuous universe.  However, we have yet to observe that the universe is discrete.  Thus, the statement, "The universe is continuous" has yet to be falsified.  Since this statement is part of science, I accept it in the standard Popperian way.


"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..."

Offline

#219 2006-12-21 18:48:44

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

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

Whereas I choose the Ockham's Razor.


X'(y-Xβ)=0

Offline

#220 2006-12-22 07:54:26

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

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

How does Occam's Razor apply here?  There is absolutely no evidence for a discrete universe.  However, each and every time our measurement ability increases by a factor of 10, we still find that everything at least *seems* continuous.

Occam's Razor only applies to two competing theories of equal evidence and explanatory power.


"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..."

Offline

#221 2006-12-22 15:24:24

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

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

Ricky wrote:

However, each and every time our measurement ability increases by a factor of 10, we still find that everything at least *seems* continuous.

The same as Aristotle thought of gold?

A study:
Atoms of Space and Time, pp55~65, Scientific American, January 2004

Ricky wrote:

Occam's Razor only applies to two competing theories of equal evidence and explanatory power.

Because the error is so minor that we always ignore. (What's the difference between one glass of water and one glass of water plus one water particle?)Thus we lack the technique advanced enough to examine which is correct.


X'(y-Xβ)=0

Offline

#222 2006-12-22 16:45:12

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

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

The same as Aristotle thought of gold?

A study:
Atoms of Space and Time, pp55~65, Scientific American, January 2004

I'm sorry, I don't seem to understand what you are trying to say.  Could you quote or paraphrase the study?

Because the error is so minor that we always ignore. (What's the difference between one glass of water and one glass of water plus one water particle?)Thus we lack the technique advanced enough to examine which is correct.

You should become an engineer.


"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..."

Offline

#223 2006-12-22 17:38:28

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

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

The study mainly proposed a model of discrete space.

What I want to emphasize is that when you calculate the volume of a barrel of gasoline by a formula involving Pi,  you really want to know how many burnable particles in it. And the preciseness of the formula involving Pi under common occasions doesn't prove the gasoline is continuous nor that the continuum model of gasoline is superior than the discrete model. Because you haven't see microly enough.


X'(y-Xβ)=0

Offline

#224 2006-12-22 18:40:31

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

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

When you count things such as the number of particles, those are discrete quantities and thus, decimals don't apply.

When you count discrete things, yes, it should be obvious only integers matter.  However, volume, as defined by the distance from the top to bottom of a certain container, is not discrete.  At least, that's what we are arguing over.


"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..."

Offline

#225 2006-12-24 17:51:11

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

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

Just read this on a site:

Equations and the Real World

    * Engineers think that equations approximate the real world.
    * Scientists think that the real world approximates equations.
    * Mathematicians are unable to make the connection...

Thought it would be appropriate.


"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..."

Offline

Board footer

Powered by FluxBB