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 Forum index » Archive » Archive: Perplex City » PXC: Project Syzygy Pre-Game
[LOCKED] [PUZZLE?] E Numbers
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thebruce
Dances With Wikis


Joined: 16 Aug 2004
Posts: 6899
Location: Kitchener, Ontario
tanner wrote:
as for definitions of numbers they are part and parcel to mathematics and need not apply outside of mathematics

you are of course free to create a new disapline called "thebruces mathematics" where you define numbers differently -- but mathematics and its subset number theory have well established definitions of numbers

Well I'm not changing the definition of numbers. I'm not changing anything. I'm simply saying that the statement A->B so A=B is not precise, absolute math. It's a different kind of math, dealing with irrational numbers. You can never result in a precise, 100% accurate practical finite value when dealing with an infinite equation. I'm not saying it's incorrect or false in the entire mathematics genre (if I said that, then I apologize, it's not what I meant - otherwise I contadicted myself by saying I don't disagree with any of this math Smile). To be more specific, again, you simply can't result in a finite, accurate value when dealing with an equation that requires a remainder infinite calculation (ie not cancelled in other math formats like algebra or calculus, which don't involve removing the absolute precision of any value within). I'm not a pro mathematician, so maybe a lot of what I say simply confuses the issues because I don't use the right terminology... but I truly hope you can grasp the concept I'm putting forth... it's not a concept I've come up with myself, it's been around for a long time, by nature I'm discussing the different areas of math, the different kinds.

In dealing with absolutes, A ->B cannot equate to A=B.
In dealing with derived math, it can. And for all practical purposes, it can be used to achieve a sufficiently accurate value to be used.

Quote:
again sorry if you thought id insulted your intelligence

I wasn't referring to you specifically, but thanks. I love challenges, and I'm a very logical person. If I'm wrong, I like to know how and why. If I can't prove something or disprove something that contradicts with a claim, then I'll gladly settle for a stalemate, hoping eventually the result will be brought forth in some other location or source. But some people are absolutely set in their ways that they can't let a disagreement end knowing that their side hasn't been hammered as the only possible answer. So I'm hoping that all parties involved here can at least understand where I'm coming from, and the points I'm trying to lay down. I can see the debate from the other side, and I can see why it would be fought. And I can see the barrier between the two arguments, and why neither can 'win' over the other. Because it deals with an issue outside the realm of the debate. Whatever the terminology may be.

I'm simply dealing with absolute, precise calculations. And the absolute, precise value can never be known when dealing with an infinite equation.
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PostPosted: Mon Jan 03, 2005 1:45 pm
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thebruce
Dances With Wikis


Joined: 16 Aug 2004
Posts: 6899
Location: Kitchener, Ontario
Chris K wrote:
Sine works like this:
- You input a number
- An infinite amount of little numbers are added up
- The sum of these is outputted

Chris... Chris... listen. You're not providing HOW the accurate value of sin is calculated by formula.

Quote:
If the input is x, then the numbers that are added up are:

If the input is 1, then the numbers that are added up are:
1 - 1/6 + 1/120 - 1/5040 + 1/362880 - 1/39916800 ......

You can see that the numbers get drastically smaller. This means that we can get a (very accurate) estimate by adding up, not an infinite amount of numbers, but 10,000.

You are defending my position. Thank you.
You are not resulting in a precise absolute value. You just agreed with me that in dealing with an infinite series, you either round, or calculate to a sufficiently accurate length. This is not a precise value.

Quote:
We cannot add up an infinite amount of terms, but if we did, we would know for certain that with certain inputs (such as 30 degrees) we would get out an exact, finite number.

IF we did, then yes. Until then, you can only assume, based on extremely positive evidence. But that is still not an absolute.

Quote:
In summary:
- We know that the output of sine will be 0.5 with a certain input
- We know that sine is calculated with the sum of an infinite series (ie. by adding up an infinite amount of numbers)

- Therefore we know that the sum of an infinite series can be a finite number.

Sorry, that's a claim based on missing details.
We know that we should be able to create a function which will calculate a precise value of 0.5 given an input of 30. Your claim is a self-supporting claim. Because sin(30) gives 0.5, we know that we can create a function that will give 0.5 given an input of 30.

Once again, the second claim is not known, because it cannot be proven. It cannot be calculated. Otherwise, I'm wrong, so tell me how the accurate, precise value of sin can be calculated.

I believe sin can give an accurate, precise value. Because sin(30) gives 0.5. So, tell me how the function gives that result! it shouldn't be so hard to do. How can I rephrase that request so it's understood?

Please answer me these two questions:
1. How does a calculator calculate the sine of a number?
2. And is the result a 100% precisely accurate value, or is it a sufficiently accurate result for the practical purposes of a consumer calculator?
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PostPosted: Mon Jan 03, 2005 1:55 pm
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tanner
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Quote:
It's a different kind of math, dealing with irrational numbers. You can never result in a precise, 100% accurate practical finite value


i think we may be at the crux of the matter --- pure mathematics does not claim to be *practical* ---- it is engineers and physicists who insist on applying mathematics to the *real* world , not mathematicians who are quite happy with the definition of a number that they have and therefore do not even consider it neccessary to calculate it

pure mathematics is true unto itself and logic -- that is all
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PostPosted: Mon Jan 03, 2005 1:56 pm
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yanka
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Joined: 06 Oct 2003
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 		I thought I was out of this thread as well, but since, I guess (hope?), this will be locked rather soon, I would like to say this.

thebruce wrote:
I'm not claiming to know your intelligence, or lack of it, so if you claim to know my intelligence or lack of it, you very quickly lose respect in my eyes, and likely many in this community.

thebruce, I am not sure who you were addressing, but I would rather think that that person was making an assessment of your level of math knowledge - not your intelligence. It is rather obvious that you are not familiar with what constitutes a "proof", or a "value" in math; and so the argument you want to have is one that would address your understanding of these concepts - not the concepts as they are universally understood. [edit] I guess tanner already said that. [/edit] A person who points it out to you, and suggests that you lack the proper knowledge base to engage in this discussion is not very likely to lose respect of many in this community, as talking (and with such persistence, I might add) about something that you lack the proper knowledge or understanding of is considered bad form anywhere.
thebruce wrote:
This is, and has been, to the best of my knowledge, an attempted civil debate. And apart from a few derogatory comments, it's been quite enjoyable, even though an absolute answer has not been achieved.

To whom has it been enjoyable? To me it has been tedious and frustrating. Through the last few pages, there has been nothing intellectually challenging, thought-provoking, engaging, enlightening, or otherwise positive about this discussion. Watching a number of people (including myself) argue with you over - really - definitions is not enjoyable. You have made a number of nonsensical statements and demanded that people address them (how could they, if they have no idea what you're saying??) and a number of blatantly wrong statements that people repeatedly refuted - yet you insist that they keep refuting them. Why? So that you could ask them to refute them again? Frankly, I am as puzzled about what aspect of this you find "enjoyable" as I am about the fact that people still are engaged in this utterly pointless debate.
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PostPosted: Mon Jan 03, 2005 2:09 pm
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Chris K
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Joined: 15 Dec 2004
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 		An infinite series is the only way of calculating a value for sin.
A calculator will simply get close to 0.5

I have provided proof that the sum of infinite series can be an exact finite number.

Quote:
IF we did, then yes. Until then, you can only assume, based on extremely positive evidence. But that is still not an absolute.


YOU DO NOT NEED TO CALCULATE THE INFINITE SERIES TO KNOW THAT IT WILL COME TO 0.5!!!!!
Look at the picture!! Are you honestly disagreeing with this:
- sin is calculated using an infinite series
- sin can give out exactly 0.5

This is a cornerstone of mathematics. You simply have to accept, even though you don't seem to understand it. Maybe when you have been taught more mathematics, you will understand it.

Quote:
If you, or anyone can show me how you arrive at 5, by adding up an infinite series, I will gladly give up my stance.


Seeing as you obviously don't know it, here is the equation for the sum to infinity of a geometric series:
S = a / 1 – r

If you really want me to I will show you how to get this equation.

Will you now 'give up your stance'?

PostPosted: Mon Jan 03, 2005 2:32 pm
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thebruce
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Joined: 16 Aug 2004
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tanner wrote:
i think we may be at the crux of the matter --- pure mathematics does not claim to be *practical* ---- it is engineers and physicists who insist on applying mathematics to the *real* world , not mathematicians who are quite happy with the definition of a number that they have and therefore do not even consider it neccessary to calculate it

pure mathematics is true unto itself and logic -- that is all


That I can agree with. Throwing everything else out the window to this point, my only disagreement is that the limit of a number is precisely that number. In dealing with absolutes, that does not equate.

Quote:
thebruce, I am not sure who you were addressing, but I would rather think that that person was making an assessment of your level of math knowledge - not your intelligence.

Ok, I was referring specifically to Chris as he seemed to be the one to turn to a personal attack.
Quote:
you clearly have no idea what you are talking about
...
Therefore, you don't know what you are talking about
...
You clearly have little knowledge of fairly simple mathmatics or even mathmatical language and notation.
...
Secondly, it is painfully obviously that you know very little about this subject that you are so vehemently arguing
...
That is absolutely meaningless
...
Just another piece of gibberish, I'm afraid

etc, including quite emotional outbursts, but I won't keep quoting...

neon, tanner, and others I'm happy to discuss with, because they sincerely wanted to understand my position, which I did my best to explain.

Quote:
A person who points it out to you, and suggests that you lack the proper knowledge base to engage in this discussion is not very likely to lose respect of many in this community, as talking (and with such persistence, I might add) about something that you lack the proper knowledge or understanding of is considered bad form anywhere

Yes I admit I'm not educated when it comes to terminologies, to specific memories of mathematical proofs and evidences, but once again, I understand whent they are presented. Even so, I'm not disagreeing or arguing against any processes or mathematical evidences or proofs, I'm simply distinguishing between calculatable, absolute values, and derived results from other forms of math. That's not a hard concept to grasp, even though I don't have the knowledge required to quote formula after formula of tried and true historic math.

My sole point through this whole debate is simply, once again, this:
'A -> B therefore A=B' is a math that doesn't deal with precise values. You cannot state that the limit of a finite number absolutely is equal to the number. You can say it in other forms of maths, but, as tanner put it, in practical purposes, just one area of maths, it cannot be a true statement. And I take that into account.
For all intents and purposes, if a chance approaches 100% at infinity, then sure, it's 100%. But accurately, the true chance can never be known.

I don't see how that requires years of mathematical education to grasp. There is a clear difference between both statements, and depending on which kind of math you value, either one may be true to you.

Quote:
To whom has it been enjoyable? To me it has been tedious and frustrating.

Enjoyable emotionally, sure it's been redious and frustrating for me too, because somehow some people can't even fathom the points I'm trying to make, yet to me they seem so simple. And the same for those people trying to get me to agree to their point, when I can't fathom how they can define it as an absolute. So emotionally on both sides, yes, it's frustrating. Intellectually for me it's been enjoyable, because I'm learning a lot about math I don't know, and confirming a lot I already know.

The main thing I'm taking away from this thread - there are people who will vehemently stand by and defend what they know to be true, without necessarily being able to defend from different perspectives. Simply because the perspectives conflict by nature.

Which is why I'm simply stating - let's agree to disagree. The genre of math has different areas, some that deal with absolutes, and some that do not. Those that do not simply cannot be applied or defended in an absolute sense. Thus arguing either side is pointless because with both maths being true and accurate in their own areas, will never be able to defend or disprove either side when pitted against each other.

If you view my last couple of paragraphs as 'gibberish', then so be it. But as I said, this is not a viewpoint I am alone with. Simply for the fact that this debate can be found throughout the internet, all over the world. So, it is pointless debating here, assuming any of us have the absolute answer.

thebruce is envisioning a room full a greek philosophers arguing over the existence of 'absolute truth', and the concept of absolutes, waving scrolls and parchments in the air at each other as they yell in disagreement.
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PostPosted: Mon Jan 03, 2005 2:45 pm
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tanner
Entrenched


Joined: 21 May 2003
Posts: 875
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 		ok i think it may be time to lock this thread
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PostPosted: Mon Jan 03, 2005 2:51 pm
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thebruce
Dances With Wikis


Joined: 16 Aug 2004
Posts: 6899
Location: Kitchener, Ontario
Chris K wrote:
A calculator will simply get close to 0.5

So sin rounds the value to 0.5?

Quote:
YOU DO NOT NEED TO CALCULATE THE INFINITE SERIES TO KNOW THAT IT WILL COME TO 0.5!!!!!
Look at the picture!! Are you honestly disagreeing with this:
- sin is calculated using an infinite series
- sin can give out exactly 0.5

This is a cornerstone of mathematics. You simply have to accept, even though you don't seem to understand it. Maybe when you have been taught more mathematics, you will understand it.

So, your saying I have to trust by blind faith that the sum of an infinite series is a calculatable value? Sorry, can't do that. If you can, good for you. Thus the argument is ended, unless you can show me how you know for absolute precise calculation, that the sum = 0.5.
But, if you don't use absolute math and I do, then there's no point arguing, and we should let the matter drop.

Quote:
Seeing as you obviously don't know it, here is the equation for the sum to infinity of a geometric series:
S = a / 1 – r

If you really want me to I will show you how to get this equation.

Please do, I like to know how. (like I'm in court, 'yes, I'm going somewhere with this')
I'll tell you now that the proof for that equation would end in something similar to (a/1-r)->S therefore (a/1-r)=S
Which crosses from absolute precision to derived results.
Both equally valid, both argued in this thread, neither with a leg up in winning the debate.
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PostPosted: Mon Jan 03, 2005 2:52 pm
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thebruce
Dances With Wikis


Joined: 16 Aug 2004
Posts: 6899
Location: Kitchener, Ontario
tanner wrote:
ok i think it may be time to lock this thread


agreed.
Although I'm sure I'll continue to hear the repercussions of this in other threads (I've already read mocking comments in other threads regarding e and infinity, which doesn't help me respect that person(s) any more than now)
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PostPosted: Mon Jan 03, 2005 2:54 pm
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Chris K
Boot

Joined: 15 Dec 2004
Posts: 37

 		The thing is, I think most people, no wait everyone, would agree that I was right and that I showed considerable patience in trying to explain something to you, coming equations and proofs left, right and centre while you simply ignored everything I said and spewed pages and pages of crap, which I would then trawl through in an attempt to see what you meant.

What did you mean?

No idea.

-------------

EDIT

- Get an equilatrial triangle
- Cut it in half
- See for yourself that sin(30) is exactly 0.5

PostPosted: Mon Jan 03, 2005 2:55 pm
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thebruce
Dances With Wikis


Joined: 16 Aug 2004
Posts: 6899
Location: Kitchener, Ontario
Chris K wrote:
The thing is, I think most people, no wait everyone, would agree that I was right and that I showed considerable patience in trying to explain something to you, coming equations and proofs left, right and centre while you simply ignored everything I said and spewed pages and pages of crap, which I would then trawl through in an attempt to see what you meant.

What did you mean?

No idea.


If you couldn't see my clear point which I stated numerous times, then there truly is no point in keeping this thread active.

One last time
For practical, precise values, the statement 'A -> B therefore A = B' cannot be true.
If you can't see why that's true, then please, someone lock this thread.

Thank you, others, for a civil discussion, frustrating though it has been.
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PostPosted: Mon Jan 03, 2005 2:58 pm
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SilentAvenger
Boot

Joined: 23 Oct 2004
Posts: 44

 		Ok, I have done some research on the material and have come the following conclusions: (these facts are not my own, I asked around some math teachers/people, and looked online)

1. This has been very well verified, .999~ does, indeed, equal 1. There is no point discussing this.
2. thebruce, you are correct, a->b is not the same as a = b. It means, a = b +/- (epsilon), where epsilon is ->0, and is not exactly 0.
3. This doesnt change the fact that infinite regressing series can, indeed, have a finite sum. This is a subject well taught in high-school mathematics, (as a matter of fact, their going to teach it in my school in about a week or two).

None of these have anything to do with the facts that:
1. In a infinite fully (unflawed, unlimited) random stream of numbers, you will always find every single permutaion of finite length of numbers, an infinite number of times.
2. In a infinite, either non-random, or not-fully random stream of numbers, each permutation of finite length will either exist (100%), or not exist (0%). This is commonly reffered to as the 0 1 law.
3. We (in this thread), have come to the conclusion that e is not fully random, and therefore, each finite string has a chance of either 0 or 1 to appear in e.

4. It is Q1-2005 and PPC hasent started yet. Thats not ok. It should have started at lim t -> 0, from 00:00 31/12/04 GMT -12, but nobody asks me.
5. We are fairly capable of having civic discussions, and should do so more often, but need to know where to stop Smile

Thanks for the fun discussion, hope we can do so again.

PostPosted: Mon Jan 03, 2005 3:13 pm
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thebruce
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SilentAvenger wrote:
1. This has been very well verified, .999~ does, indeed, equal 1. There is no point discussing this.

I've searched for proofs as well, and I haven't found one proof that hasn't been debated, or isn't currently under debate, or has come to an accepted conclusion. If you know of one, please quote.

Quote:
2. thebruce, you are correct, a->b is not the same as a = b. It means, a = b +/- (epsilon), where epsilon is ->0, and is not exactly 0.

Agreed

Quote:
3. This doesnt change the fact that infinite regressing series can, indeed, have a finite sum. This is a subject well taught in high-school mathematics, (as a matter of fact, their going to teach it in my school in about a week or two).

Well I hope they teach that it's a limit that you're calculating, not a finite sum. As long as a value is continuously added to the previous sum, a finite sum will never be reached. But a limit can always be found. Whether you consider the limit to the be the finite sum is the point in question. Depending of course, on the nature of the series. The series 1+0+-1+0+etc can be shown to have a sum 0 for every 4th and 4th-1 item in the repeating series. But a sum of 1+1/2+1/3+1/4+etc can never reach a finite result, because a value is constantly added to each previous sum.

Quote:
None of these have anything to do with the facts that:
1. In a infinite fully (unflawed, unlimited) random stream of numbers, you will always find every single permutaion of finite length of numbers, an infinite number of times.

If the definition of the random number says that each number must be used at least once, then yes essentially the chance is 100%.
If the definition simply says that the only numbers that are usable are between 0 and 9, then there's a chance that the random number will be entirely made of 5's, so the sequence 5551 will never be found. In other words, a truly random number will always have a chance, granted -> 0%, that that a sequence may never be found.

Quote:
2. In a infinite, either non-random, or not-fully random stream of numbers, each permutation of finite length will either exist (100%), or not exist (0%). This is commonly reffered to as the 0 1 law.

Agreed.

Quote:
3. We (in this thread), have come to the conclusion that e is not fully random, and therefore, each finite string has a chance of either 0 or 1 to appear in e.

Agreed.

Quote:
4. It is Q1-2005 and PPC hasent started yet. Thats not ok. It should have started at lim t -> 0, from 00:00 31/12/04 GMT -12, but nobody asks me.
5. We are fairly capable of having civic discussions, and should do so more often, but need to know where to stop Smile

Agreed and agreed Smile

Quote:
Thanks for the fun discussion, hope we can do so again.

ditto... hopefully on not such a controversial topic Smile hehe
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PostPosted: Mon Jan 03, 2005 3:38 pm
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Olorin
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Joined: 04 Nov 2004
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 		So, have you guys managed to prove that
sin(30) = 0.9999~
?
Twisted Evil

F.O.R.

PostPosted: Mon Jan 03, 2005 3:51 pm
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ariock
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 		Some Clarifications
First off, I haven't seen this clarified anywhere, so I am going to do it:

The Infinite series that Chris posted earlier depends on using angles measured in RADIANS, not degrees. If you put 30 into it, you will get garbage(well, for the first 30 or so n. After that, it converges to near -0.9880, but that is 30 radians, not degrees). If you put a number like pi/2 into it, the series will converge to 1 at infinity.

Second. thebruce, you have posted a couple of links to sites that supposedly refute the Identity of 0.99~ = 1. This is not actually the case. Neither site uses a valid mathematical proof to disprove the assertion. They merely use some philosophical hand-waving that, unfortunately for them, is trumped by mathematical proofs.

Seriously, the second link you posted actually uses the line, ""=" in this case does NOT mean the same as." This is, in all honesty, a joke. Now = doesn't mean equal?

Another interesting point from your first link: Harold says, "So, 0.3 repeating times 3 should equal 1, correct? But, it doesn't. It gives you 0.9 repeating. But we all know three thirds is a whole, so what happened?" What did happen? Unfortunately, Harold doesn't tell us.

1/3=0.33~
3*0.33~=0.99~
3*1/3=1

Therefore 0.99~=1

Note that the symbol ~ indicates that the digit preceding it is repeated infinitely. This is NOT an approximation. It is not like saying 0.99(with another 50 billion 9's following it) when to say that invites the criticisms you have made. 0.99~ does not say that. It says explicitly and by definition that there MUST be an INFINITE number of trailing 9s. Since there is no such thing as "Infinity plus 1," our proof is logical and valid.

I will agree that 0.99~ isn't as clean a representation as 1, but they are one and the same. If a mathematical proof is incorrect, you can find a flaw: did they divide by 0? are they completely removing x from the equation? In a little bit, I will get to what I imagine they believe must be the flaw, and an explanation.

Another quote from Harold: "Thirds exist, but not for the number 10."

This is a patently ridiculous assertion. And it is one with no support. Obviously, to EXACTLY divide 10 cookies among 3 friends would be nigh impossible. I concede that the error of measurement would make it highly unlikely that it would be done exactly. But to do the same thing with 1 cookie between 2 friends is eqally impossible. And yet you have no problem in saying that 2*0.5=1. If you use one as proof of the problem of thirds, then there must also be a problem with halves.

He continues, "Because of that the number 0.3 with the number 3 repeating for infinity is not really one third. It is an approximation." Here is another problem. If you base an argument on a patently ridiculous assertion, everything that follows is equally ridiculous.

So lets try it...Long Division style. 3 into 1. 3 goes into 1.0, 0.3 times, with a remainder of 0.1. 3 goes into 0.10, 0.03 times with a remainder of 0.01. At this point, we have 0.33 with a remainder of 0.01. I know that if I do this again, I will have 0.333 with a remainder of 0.001. And so on. As we continue, the line of 3s gets longer and longer, and the remainder gets smaller and smaller.

Now HERE is where the dichotomy occurs. I say that if we assume that this continues infinitely that we will have an infinite string of 3s and that there will therefore be no remainder. Do you believe in an infinitely small remainder? What would that mean? -infinity? or 0? or an infinitesimal?

Let's assume an infinitesimal. We will call it (tesm). So:
1/3=0.33~+(tesm)
3*{0.33~+(tesm)}=0.99~+3*(tesm)
so 0.99~+3*(tesm)=1??

You need 3 infinitesmals to get from 0.99~ to 1? What does that even mean?

How about 1/7?
1/7=0.142857142857~6characters+(tesm).
7*{0.142857142857~6characters+(tesm)}=0.99~+7*(tesm)
this time, 0.99~+7*(tesm)=1???

Is this the same infinitesmal? Are there degrees of infinitesmals? We need 7 of these infinitesmals to get back to one from 0.99~. And this is the same identical 0.99~ in both cases. Maybe (tesm) is some kind of transcendental number that will always be the same, no matter what number you multiply by it.

Do you want to know the only number that remains the same no matter what number is multiplied by it?

0

zero.

And here is the thing. It really, honestly is zero. At infinity, there is no remainder. Because there can't be. A remainder implies stopping. Since there is no stopping, there can never be a remainder.

By the way, Calculus is based on the concept of infinite series of very small numbers adding up to discrete values. And even though Mathematics has symbols for "less than," "greater than," and "approximately equal to," you will not find them used in the calculus.

PostPosted: Mon Jan 03, 2005 5:00 pm
Last edited by ariock on Mon Jan 03, 2005 5:46 pm; edited 2 times in total
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