Ricky wrote:

George, as promised, I will be answering your posts either tomorrow or Saturday, so don't worry, they won't go ignored.  I ask you (as a friend in a debate, not as a moderator) not to post until I have, otherwise I fear there will just be utter confusion.

Till now, you have disagreed on decimal expansion on the real numbers, and I always figured the problem was higher up, for example definition of the convergence of an infinite series, which would intuitively come after a consensus of real numbers and decimal expansions have been reached.  Let me address this now.

I believe we all agree on the following things:

1. A decimal expansion of any real number r takes the form of:

2. For any real number r, given an epsilon (e), we can find a finite number of decimal digits d such that:

But we have problems with this.  Specifically we can't write all real numbers or even all rational numbers with a finite number of digits. So what do we do about this?

One thing is to do nothing.  Accept this failure of decimals.  Personally, I feel unsatisfied with this.  Is there no way we can save decimal expansion of infinite decimals?  Certainly, we can't "see" infinitely many decimals.

But wait!  We have this really cool method of dealing with infinite things invented/discovered by Cauchy et. al.  Limits.  They allow us to handle infinite sequences of numbers.  And in the end, isn't this all we're dealing with?

So let's just explore this possibility for the time being.  How can we use limits?  Certainly it may not be possible, but lets just give it a try.

For any real number r, let us define a sequence of real numbers (using the same notation as before):

It should be clear that:

For any integer n.  So this sequence is monotonely increasing, as well as bounded.  By the monotone-bounded convergence theorem, this sequence must converge.  As we (hopefully) agreed in #2 (see above), it does in fact converge to r.

So where are we at?  Given any real number, we can use the decimal expansion to approach it, and get arbitrarily close, a limit per say.

It is with this in mind that we make the following definition:

Note that the notation above simply means an infinite amount of decimal points.

The results of this definition are the following:

1. No contradiction with any math that I'm aware of
2. Every real number has at least one decimal expansion equal to it.
3. Some decimal expansions are not unique.

3 can be considered a problem.  I certainly do.  But I argue (with only opinion, not pure logic) that having multiple decimal expansions are a lesser problem than not being able to represent some real numbers with a decimal expansion.

An immediate consequence of this definition is that 0.999... = 1.

Most people accept this definition by intuition alone.  You're the first person I've ever seen which did not.

Edit: I don't think my signature has every applied to any one of my posts greater than this one.

Specifically we can't write all real numbers or even all rational numbers with a finite number of digits.
-glad you admit this

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.

it does in fact converge to r.
- do you interpret "converge" or abitarily proximate as "reach"? The same old topic again.

an infinite amount of decimal points.
-again contravasory and no sense.

The most important thing I insist is that an amount that you can tell as a determined number, such 2, 3.5, has no way to transform into either infinitesimal or infinity. Infinite sequences are a dilema as well. Infinite sequences mean only you can add in new entries without a stop.

If you mean an infinite series itself can be defined as a decimal expansion, then infinite digit numbers are at least non-static.

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_n+1>r_n

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.

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.

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?

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.

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

Last edited by Devanté (2006-12-21 04:00:24)

"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.

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.

-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.

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.

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.

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.

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.