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BoBzOmBiE
Boot

Joined: 30 Nov 2004
Posts: 18
Location: Somewhere beneath the underpass

Take that same line, split it in to thirds, write them in decimal, 0.33333recurring add them together again, and you have the same line with a length of... hang on a moment... 0.999999recurring . now who said decimals are better than fractions.
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Posted: Sat Jan 01, 2005 6:11 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

 neon snake wrote: I'm still keen before moving much further to have a definitive list of the points you disagree with/interpret differently to us. Reason being, the points we seem to disagree on are changing.

Again, I don't think it's a matter of any points we're disagreeing on, just a difference between mathmetical certainty and assumed certainty... every point we discuss all comes down to a matter of assuming, for all intents and purposes, that any infinite number that approaches another, is that number. 0.9~ -> 1, therefore 0.9~ = 1. That's not a mathematical (as in calculatable) certainty, but an assumed certainty, because we cannot know by calculation for certain, because it deals with calculating a value we cannot calculate. I'm not saying it's incorrect to use that assumption in math in order to come to a workable result, I'm simply saying that by making that assumption, you don't have a 100% absolutely precise value. All of the topics we've discussed up until now come down to that simple principle.

 Quote: For example the statement - 'in an infinitely long string of random digits, the probability of any possible finite string being found is 100%.' - seems to have divided us - I say its 100%, you say its both 100% and 0%.

Well, really it's not 100% AND 0%, that's half what I said... I said we can't know for certain what the absolute chance is, because the chance -> 100%... so no matter how long you try, how many attempts you try, there will always be a chance that it might not happen, so what's the point in calculating? You could say that with 100 attempts the chance could be a specific amount. Or with 1,000,000 attempts. But you can't say what the chance would be after infinite attempts. Because you can never reach infinite attempts. You can say after any certain amount of attempts what the chance would be. That's the difference. As soon as you remove infinity, you are no longer dealing with an infinite value, obviously.

 Quote: We then moved on to various discussions whether there is such a thing as an infinite series, or an infinite string, and how you can only prove a string or series is infinite by calculating all the possible values (?)

In a sense. But we consider numbers infinite by definition. We can state that 0.3 repeating is an infinite number because we imply it by the number's definition. 1/3 we consider to be infinite because as far as we calculate it, it's an infinite number. Consider it a matter of checking the remainders when long dividing. The remainders never change, so we consider it an infinite number. Because we never end up with 0 remainder, we can never know the true value of 1/3, except by defining it as 0.3 repeating. When we calculate values using 0.3 repeating, we represent it as 1/3 so that we can work with the number in order to remove the infinite, repeating aspect; in order to result in a number which we can calculate. 1/3*2 doesn't do that; 1/3*4 doesn't do that; but we know that 1/3*n does where n is a whole integer divisible by 3.

 Quote: Various people have voiced opinions, proofs and facts, and I'm starting to lose track of which bits you agree with, and which you don't. And indeed, what we're discussing, as we keep moving the goalposts.

The only thing I've ever disagreed with in order to return a precise value, is the claim that if A -> B, then A = B. Any proof that assumes that will return an imprecise value by calculation.

 Quote: Some of the argument (and it was as difficult to track as this thread) seemed to revolve around the following- 1 divided by 3 is one third. In decimal notation we say 0.3recurring, because there exists no other way to write it. However, 0.3recurring multiplied by 3 IS NOT 0.9recurring - its 1 (exactly 1). The only way you get to 0.9recurring is to multipy each of the 3s in the string by 3. The problem lies here. If you do that, then you have ignored the recurring part of the number, and have just multiplied 0.3 by 3, and then added the recurring part back on afterwards. Try it on a calculator (NOT IN EXCEL). You'll get 1 divided by 3 =0.333333333. Now multiply by 3. You'll get 0.999999999. This is because the calculator works out 0.3recurring to a set number of decimal places, but can't handle true recurrence. Now try in excel, which is better at maths than a calculator, and can handle true recurrence, understands that it's a mere representation, and gives an exact value of 1! Magic.

Precisely what I'm saying.

The way I picture it is that in decimal, expressing in any way some form of infinity, means the decimal is not workable.

What people were doing was taking 0.3 repeating, and working with that decimal value, which cannot be done, because you can't work, in decimal, with 'repeating'.

so, 1/3 = 0.3~
*2
2/3 = 0.6~
*3
3/3 = 1

but you can't go
0.3~ = 1/3
*2
0.6~ = 2/3
*3
0.9~ = 3/3

I visualize it like the letter 'E'. The left side is the fraction, the right is the decimal. you can't connect the decimals by calculation if they include infinity so you end up with dead ends, but you can connect them indirectly by working with their fraction counterparts which make the decimals more complex in order to work with whole numbers which don't include infinity.
but without infinity, you can have
1/5 = 0.2
*3
3/5 = 0.6
*5
5/5 = 1

also work as
0.2 = 1/5
*3
0.6 = 3/5
*5
1 = 5/5

it's simply the concept of infinity in decimal that is a dead end, because decimal is already assumed to be a practical, calculated number, otherwise it's technically a dead end.

 Chris K wrote: Rest assured, that if you keep adding up these terms: 1 + 0.8 + 0.64 + 0.512 + 0.4096 ... where every term is 0.8 times the one before, to infinity, you will come to exactly 5. If you can't accept that then you can't accept trigonometry or calculus.

I can accept the theory that it does. But until you can calculate it yourself to know for certain, you will never know the exact value of the sum, because it's not an exact value you can calculate.
you have - 1, 1.8, 1.94, 2.452, 2.8616, etc... the number will never pass 5, but it will never reach 5, and that I can guarantee. The proof includes the assumption that because A->B, therefore A=B.

If you state the sum, and that final statement, then I accept the proof, because it assumes to jump the gap between the infinite and the finite.

 Quote: - Sin of a number is calculated by adding up an infinite number of numbers

A machine cannot calculate a sum of an infinite series by adding up each value.

Tell me the process for how a machine calculates the result of a sin value.

 Quote: If you want to find out the length of the side marked x in this picture: Then add up all of these terms: 1 - 1/6 + 1/120 - 1/5040 + 1/362880 - 1/39916800 ......

As I said, that cannot be how a calculator or any kind of computer arrives at an answer, without first calculating the limit of the equation, and assuming that as A->B, then A=B

 Quote: Real quick. Not getting in to an argument, but I love this example. take a line (its easier to illustrate) and assign it a langth of 1 membit (fake unit)

I like this example too .

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024 + ... = 1 membit
assumes you can cut a physical (or metaphysical) object into an infinite amount of pieces. This you cannot practically do. Theoretically, the theory can work, but the theory includes the assumption of infinity. "Assuming you cut up the membit into an infinite number of halves, then we know that the sum will always equal 1"

That's the not the same as the statement "A 1 membit line cut up into 1000 (or insert any finite value) halves, will always add up to the original 1"

Which itself is not true. Because the finite equations you cited include the finite end of adding a second of the final fraction. This you cannot do with an infinite sum, otherwise it would not be infinite.
Therefore, just because
1/2+1/2=1
1/2+1/4+1/4=1
1/2+1/4+1/8+1/8=1
does not deduce that
1/2+1/4+1/8+1/16+... =1

also, in the former quote, you can never transverse from a finite equation (1/2+1/4+...+1/8192+1/8192=1) to an infinite equation (1/2+1/4+...=1) because your equation already assumes an infinite series, and the equation will never be complete. The equation can never be proven because it assumes that as the fractions -> 0, the sum -> 1, therefore the sum = 1. The finite equations you cited can be proven, because by nature they are not infinite, and each value can be calculated.

and, your sigma notation (the sum of 1/n as n = 1 to infinity is not = 1, but it is the limit of 1. Again, you're assuming that as A -> B, therefore A = B.

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

I guess, as I started out, the only thing I disagree with in this thread, is simply the claim that as A -> B, then A = B. It seems that's all this math comes down to.

PS BobZombie, welcome to the thread; may want to catch up by reading the past pages 0.3~ * 3 = 0.9~ = 1 has already been discussed, so you can draw your own conclusions
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Posted: Sun Jan 02, 2005 9:29 pm
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

one quick final... in simple terms
from the equations above.

"the sum of 1/n where n = 1 to X + (1/X) = 1" is true
"the sum of 1/n where n = 1 to infinity = 1" is not true
"the sum of 1/n where n = 1 to infinity -> 1" is true
- because you can't add the final 1/infinity value that the 1st equation requires to = 1

therefore, you're arguing that given X=infinity,
"the sum of 1/n where n = 1 to infinity + (1/infinity) = 1" would be true. But 1/infinity is an impossible number. Or, it approaches -> 0.

the only true statement there is
"the sum of 1/n where n = 1 to infinity + (X where X -> 0) -> 1"
that's the only true, precise equation. Which cannot be calculated. The limit is 1, but the value only approaches 1 for infinity.

A -> B... not A = B
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Posted: Sun Jan 02, 2005 10:13 pm
Chris K
Boot

Joined: 15 Dec 2004
Posts: 37

Oh my god.
Pleeease stop writing out huge posts which don't go anywhere.

Firstly, you have said a lot of stuff that is absolutely completely wrong (such as sin being a 'lookup' function and sin being 'an infinite series that is always between -1 and 1') - however - I will not comment on these.

Secondly, it is painfully obviously that you know very little about this subject that you are so vehemently arguing. It blows my mind that you would go on so long about this when you have been opposed by literately dozens of people who evidently know more maths than you.

Thirdly, I have provided numerous mathematical proofs for what I have said. You seem to wave these away 'yeah but that relies on...' or 'but you've assumed...'. I don't care what I've assumed!!. They are proofs. They are absolutely 100% totally watertight - you simply cannot argue with a mathematical proof. You however, have babbled on for hundreds of lines about hazy, constantly changing ideas that you have not explained at all. For example this:
 Quote: Theoretically, the theory can work, but the theory includes the assumption of infinity
means NOTHING. That is absolutely meaningless.

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

Anyway, whatever, I don’t really care but I will attempt to answer your questions and make you understand.

 Quote: the number will never pass 5, but it will never reach 5, and that I can guarantee.

OK. I will try and say this as simply as is possible:
- The sum, to infinity, of that series, is exactly, 5.

Please do not dispute that any more, it is an absolute certainty.

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

To get the exact value for sin you need to perform an infinite sum. This is impossible. A computer will therefore add up the first, 1000 or maybe 10000 terms. This will give it very accurately.

However, because of the nature of sine, (it was invented for triangles) we can see that some results – such as the 0.5 one I showed – will exactly equal a ‘non-infinite number’ as you would put it. I gave this example as a proof that an infinite sum can exactly equal a finite number, even if we cannot calculate it. I bewilders me that you still dispute it.

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

Finally, you keep disputing the ‘as A => B then A = B’ thing. Well, at infinity, then yes – it does equal it.

EDIT ============

OK. This thing:
 Quote: every point we discuss all comes down to a matter of assuming, for all intents and purposes, that any infinite number that approaches another, is that number. 0.9~ -> 1, therefore 0.9~ = 1. That's not a mathematical (as in calculatable) certainty

is really annoying me.
0.9 recurring is exactly 1.

Nothing is 'assumed' in maths.

 Quote: That's not a mathematical (as in calculatable) certainty

This is the bit which really gets me.
It absolutely definitely is a mathematical certainty.

Here's a link to start you off:

http://mathworld.wolfram.com/RepeatingDecimal.html

Posted: Mon Jan 03, 2005 6:03 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

 Chris K wrote: Firstly, you have said a lot of stuff that is absolutely completely wrong (such as sin being a 'lookup' function and sin being 'an infinite series that is always between -1 and 1') - however - I will not comment on these.

No, sin is not a function which literally 'looks up' a value. My point was that sin is not a value in itself, it retrieves a value between a known limit of -1 and 1. However that is calculated, it's a value within a limited range. It's not finding the value within an infinite series, depending on what you consider an infinite series. In this case, the infinite series being a series with a beginning (ie 1) and no ending (ie infinity), not a series which contains and infinite amount of real numbers. sin can retrieve any real number between -1 and 1, but again, you need show the process for those numbers being calculated, using the formula which calculates as 'sin'.

 Quote: Secondly, it is painfully obviously that you know very little about this subject that you are so vehemently arguing. It blows my mind that you would go on so long about this when you have been opposed by literately dozens of people who evidently know more maths than you.

If you read my posts correctly, I am not debating the validity of any proofs, except the claim the A->B means A=B. I am not arguing that math is wrong. Please stop putting words in my mouth.

 Quote: Thirdly, I have provided numerous mathematical proofs for what I have said. You seem to wave these away 'yeah but that relies on...' or 'but you've assumed...'. I don't care what I've assumed!!. They are proofs.

You are therefore saying 'they are' because you say they are. If you want to offer proof, you need to support that proof. Show the process so the proof can be understood. Simply stating a proof and not offering your understanding of why the proof is correct is totally blind faith.

Still you have not shown how the equation can be calculated to result in a final value given an input number, such as 0.5 from 30.

 Quote: They are absolutely 100% totally watertight

Given a finite equation assuming the statement A->B so A=B, then yes they are. But they are not precise. That is my argument. They are 'essentially' correct values. Rather, those equations, depending on who you speak to, may be either written as the sum of an infinite series has the limit of X, or the sum of the infinite series equals. Depending on whether or not they accept the statement that A-.B so A=B

Quote:
you simply cannot argue with a mathematical proof. You however, have babbled on for hundreds of lines about hazy, constantly changing ideas that you have not explained at all. For example this:
 Quote: Theoretically, the theory can work, but the theory includes the assumption of infinity
means NOTHING. That is absolutely meaningless.

Howso? Back up your claim. Again I'm not disagreeing with any proofs where the answer provides for all known truths, and does not jump a line with still in dispute, ie A->B so A=B.

Quote:
 Quote: the number will never pass 5, but it will never reach 5, and that I can guarantee.

OK. I will try and say this as simply as is possible:
- The sum, to infinity, of that series, is exactly, 5.

Please do not dispute that any more, it is an absolute certainty.

I'm sorry, just because you say so is not evidence for me. 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.

 Quote: About sine. To get the exact value for sin you need to perform an infinite sum. This is impossible. A computer will therefore add up the first, 1000 or maybe 10000 terms. This will give it very accurately.

THANK YOU.
You know how accurate 1000/infinity is? If you were using sin in an equation to fly to the jupiter, you could fly past your mark by thousands of miles... I think anyone here will agree that rounding a variable does NOT result in a precise value. Machines calculating sin may round to a 'sufficiently accurate' decimal in order to return a finite result - this is not a precise value, and is no longer the sum of an infinite series. By your claim, adding the first 1000 or 10000 terms of 1/2+1/4+etc will not equal 5.

 Quote: However, because of the nature of sine, (it was invented for triangles) we can see that some results – such as the 0.5 one I showed – will exactly equal a ‘non-infinite number’ as you would put it. I gave this example as a proof that an infinite sum can exactly equal a finite number, even if we cannot calculate it. I bewilders me that you still dispute it.

sine uses an equation, which may or may not result in an infinite number. Sine is not an infinite equation. A machine cannot calculate an infinite series. Which why I would like you to find out how programmers program the sin function in calculators and machines, in order to find out precisely what equation they use in order to retrieve a specific value. Find out if they assume A->B so A=B, or if they round to a sufficiently accurate decimal. Or if they actually use an equation which will allow for the cancellation of infinity in the equation so it is then a calculatable result.

I guarantee you that if you find that equation, we will then be able to discuss why sin can result in a finite or an infinite equation.

 Quote: Finally, you keep disputing the ‘as A => B then A = B’ thing. Well, at infinity, then yes – it does equal it. 0.9 recurring is exactly 1.

*sigh* that is EXACTLY my point. You are taking one side in an argument that's been raging for ages - if A->B then A=B.

 Quote: This is the bit which really gets me. It absolutely definitely is a mathematical certainty.

Sorry, the link you provied right off the bat states simply that 0.5~ = 0.50~ = 0.49~. That's not proof. That's a foundation from which they derive their math proofs. They don't offer proof that 0.49~->0.5 so 0.49~=5

The only support for that claim is a philosphical one, by definition, through which 'math breaks down', not by my own words. Basically, A can only be B if there are no real numbers between A and B. If anyone can offer a real number between 0.9~ and 1, then we'll know that 0.9~ does not equal 1. Otherwise, they are the same number.

So you see, the claim that A->B so A=B is entirely a philosophical one - ie, arguing by our definition of numbers, not by offering mathematical proof. Because mathematically, calculation breaks down when dealing with infinite decimal values.

And stop saying I know absolutely nothing of what I'm talking about. I'm trying to keep this discussion civil, and you're turning it into a flaming contest, meaning this thread could very well get locked.
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Posted: Mon Jan 03, 2005 8:10 am
SilentAvenger
Boot

Joined: 23 Oct 2004
Posts: 44

thebruce, I'm sorry to tell you, but 0.999~ = 1.

I will display the proof again.

x = 0.999~
10x = 9.999~
10x - x = 9.999~ - 0.999~
9x = 9
x = 1
1 = 0.999~

This proves 0.999~ = 1. You can try to argue it, its your choice.

Also, 0.999~ is exactly:

9*10^-1 + 9*10^-2 .... + 9*10^-n as n -> infinity, or,
sigma: n=1, n -> infinty, 9 * 10^-n

*waits as thebruce argues about that, too*

Posted: Mon Jan 03, 2005 8:32 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

 SilentAvenger wrote: thebruce, I'm sorry to tell you, but 0.999~ = 1. I will display the proof again.

*sigh* I'm sorry to tell you, but we've already gone over that, and I already addressed why that cannot be possible when working with decimals. And the links I provided also discussed why that's a math that breaks down.
And I'm not the only one who has argued against that being a valid proof.

 Quote: Also, 0.999~ is exactly: 9*10^-1 + 9*10^-2 .... + 9*10^-n as n -> infinity, or, sigma: n=1, n -> infinty, 9 * 10^-n

In other words, the sum of 9*10^(-n) as n = 1 to infinity approaches 1. You are adding to the statement, based on the AB claim, that it therefore = 1, where in reality, by number, not by definition, it only approaches 1.

 Quote: *waits as thebruce argues about that, too*

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Posted: Mon Jan 03, 2005 9:06 am
Chris K
Boot

Joined: 15 Dec 2004
Posts: 37

 Quote: They don't offer proof that 0.49~->0.5 so 0.49~=5

Just another piece of gibberish, I'm afraid. You seem to have some trouble in understanding what an infinite series is and what a repeating decimal is.
You see, you have basically said a number tends to a number. That's impossible. The sum of a series can tend to a number, but a number can't tend to or approach anything. That's like saying 1 => 2.

 Quote: sine uses an equation

No!!! It uses an infinite series, which I have already provided.
That is the series which computers use to calculate sin.

We can never use sin to get exact answers, however, we already know what the outcome of sin(30) will be before we calculate it.

This is how we know that the infinite sum that is performed by sine, will outcome in the finite number of 0.5, it won't be rounded to it, it won't tend to it, it won't approach it - it will be exactly it.
--------
What is wrong with this proof (provided many times already)?
Point out one hole in the logic:

x = 0.9~
10x = 9.9~
Subtract, all the 9s cancel
9x = 9
x = 1
---------
 Quote: I'm sorry, just because you say so is not evidence for me. 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.

Posted: Mon Jan 03, 2005 9:49 am
neon snake
Veteran

Joined: 18 Mar 2004
Posts: 70
Location: Chelmsford, UK

I think the difficulty we're having here lies in a difference in level of education.

Before doing my A-Levels, maybe even before University, I'd have had the same difficulty understanding the principles of infinity, and the ability to sum infinite series to a finite sum.

I must admit, when I saw this:
 Quote: 1 + 0.8 + 0.64 + 0.512 + 0.4096 ... where every term is 0.8 times the one before

...I had to have a quick refresher to gen up on summing infinite series.

Most of the proofs we have presented, whilst we understand their validity, required us to have had a couple of years prior education in order that we follow them.

I think for someone without the same grounding, the idea that 0.9recurring is the same number as 1 probably is quite bewildering.
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Posted: Mon Jan 03, 2005 10:23 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

Chris K wrote:
 Quote: They don't offer proof that 0.49~->0.5 so 0.49~=5

Just another piece of gibberish, I'm afraid. You seem to have some trouble in understanding what an infinite series is and what a repeating decimal is.
You see, you have basically said a number tends to a number. That's impossible. The sum of a series can tend to a number, but a number can't tend to or approach anything. That's like saying 1 => 2.

Ok I really do not know what you're saying here.
You quoted that url as proof that A->B so A=B. They don't offer proof, the math they're discussing simply states off the bat that 'because 0.49~=0.5' They don't offer proof. I was not saying that myself, I was quoting what they were saying. So I'm guessing from what you just said above, you're agreeing with me that 0.49~ does not = 0.5. ?? You're jumping around now...

Quote:
 Quote: sine uses an equation

No!!! It uses an infinite series, which I have already provided.
That is the series which computers use to calculate sin.

You just said sine is not a lookup in a series. A computer cannot 'find' a value in an infinite series. It can calculate a specific value or round to a value. In order to return a value, sin must be calculated in some way. A calculator cannot store an infinite series in order to find the best value. You're contradicting yourself. A machine must have some formula or equation in order to calculate a value based on an input number. It cannot store an infinite series in memory and find the best value. It must be calculated. Tell me how a machine would calculate the sin of any specific number, and tell me it does this with 100% accuracy, inclusive of an infinite equation.

 Quote: We can never use sin to get exact answers, however, we already know what the outcome of sin(30) will be before we calculate it.

What?! That is ridiculous and is not any kind of statement that can be used as even a semblance of proof in any statement. It's true because we know it's true?? please... if you are a mathematician, you should be able to offer mathematical evidence, by number, by calculation, that a proof is stable.

 Quote: This is how we know that the infinite sum that is performed by sine, will outcome in the finite number of 0.5, it won't be rounded to it, it won't tend to it, it won't approach it - it will be exactly it.

How do you know? That is what I'm asking.

 Quote: What is wrong with this proof (provided many times already)? Point out one hole in the logic: x = 0.9~ 10x = 9.9~ Subtract, all the 9s cancel 9x = 9 x = 1

I've referred to that, quoted it, even said the same thing in my own words before getting to that link. I'm not alone in knowing that there's a flaw in that proof. But it comes entirely down to the belief in whether or not A->B so A=B. If you don't believe it, it's not true. If you do, it is. This is not a matter of math by absolute numbers.
So, I can't prove it, because my comments state that you cannot work in decimal notation with an infinite number; Because you cannot calculate using a precise infinite decimal - the calculation would never end.

Quote:
 Quote: I'm sorry, just because you say so is not evidence for me. 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.

I'm asking you to prove your claim. The burden of evidence is on you. You state that the sum of the infinite equation 1/2+1/4+1/8+... = 5. So prove it. Without equating A->B so A=B.
I state that the limit of that equation is 5. Mathematicians will agree. But the sum of the equation does not equal 5. Some mathematicians will take the step further in saying that because the limit is 5, the answer = 5. Some will not.

Why is this such a hard thing to agree on?

Why don't we just agree to disagree on that one single point of contention, the A to B claim. It's not math, it's philosophy, for lack of a better term. Math ends at the result A->B. It can go no further.
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Posted: Mon Jan 03, 2005 10:24 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

 neon snake wrote: I think for someone without the same grounding, the idea that 0.9recurring is the same number as 1 probably is quite bewildering.

I don't think it takes years of understanding to see that 0.9~ -> 1. It's a limit. The next step is a statement which isn't absolute math, in other words it's bridges a gap which mathematics can't fathom. ie the definition of what a number is. Many agree and many don't. In math, calculations are absolute, it's a 'universal language' per se. When dealing with the infinite, it's not an absolute. Math can represent the concept, but it can't calculate it. The limit of 0.9~ is 1. Stating that therefore 0.9~ = 1 is a statement that claims to know what infinity is in order to calculate it. We then bridge from inarguable math to arguable philosophy. Which is precisely why this argument has gone on for ages, and no result has come of it. The absolute result is the statement that a number approaches another, that B is the limit of A.

To attempt to finalize the topic, the chance of any sequence of numbers appearing in e approaches 100% at infinity. Essentially, it is. Absolutely, we will never know. That is as far as absolute math can possibly take that issue.

Let's just simply agree to disagree on this one point, because it will never be resolved.
If you or I or anyone else in this thread claims to know the absolute answer to this point of contention, we claim to know more than all those previous who came before us who have continually argued both sides in the name of precise, absolute mathematics. Let this debate end.
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Posted: Mon Jan 03, 2005 10:30 am
tanner
Entrenched

Joined: 21 May 2003
Posts: 875
Location: (x,y,z,t,i, ...)+

thebruce
edit --- replying to your post before the last one --- still applies tho

that post was rubbish -- pure mathematics has nothing to do with cutting a piece of wire or any other measuring stuff -- no measurement is ever totally accurate but is only accurate within tolerances

a pure mathematical theorem is true if the proof and the axioms are true -- this has nothing to do with engineering, physics or anything else outside of mathematics

in fact its a wonder and a marvel that tools discovered by pure mathematicians have been so useful to everyone else

mathematics is not an art of measurement or calculcation (arithmetic) but an art of constructing a logical proof --- i dont know where you learnt your maths but i would seriously try to find another school if i were you

i have now broken my new years resolution --

but ok i'm outa here

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Posted: Mon Jan 03, 2005 10:42 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6775
Location: Kitchener, Ontario

again, I'm not disagreeing with anything in any of the mathematics shown here. What we're doing now is arguing definitions, not arguing calculatable math. We're arguing the definition of a number. We're arguing at whether the limit of an infinite number IS the number. That's what it all comes down to. Whether it's a matter of math vs philosophy, or whether mathematics is the art of constructing a logical proof or an art of measurement or calculcation (arithmetic). What it comes down is a matter of absolute results. The math we're discussing here, the agreements, the debatability, all comes down to whether A=B if A->B.

Regardless of definitions, regardless of theorems, calculations, logic, proof, what have you... A=B is absolute. A->B is not. The two cannot absolutely equate. That's simply where our argument ends. In which case, as I said, neither of us can prove satisfactorily to the other side that the chance of any sequence appearing at some point in e is either 100% or approaching 100%. Because we're arguing a belief, or lack of belief, that the limit of an infinite number IS the number.

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. 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.
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Posted: Mon Jan 03, 2005 11:09 am
tanner
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i didnt mention your intelligence, just your school -- im sorry if you read it otherwise --- 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

again sorry if you thought id insulted your intelligence
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Posted: Mon Jan 03, 2005 11:26 am
Chris K
Boot

Joined: 15 Dec 2004
Posts: 37

OK. I will walk you through it.
-------------------------
Forget infinite series, sine, recursion, everything.

Imagine this triangle:

We know that these value are exactly true. That is an absolute, we do not need anything to calculate those numbers, it is simply half an equilatrial triangle.

OK, now someone really clever invented a way of finding out lengths of triangles, he called his function sine.

Sine works like this:
- You input a number
- An infinite amount of little numbers are added up
- The sum of these is outputted

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.

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.
------------
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.

Posted: Mon Jan 03, 2005 11:38 am
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