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[BoBzOmBiE]

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.

[thebruce]

[thebruce]

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

[Chris K]

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:

[thebruce]

[SilentAvenger]

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*

[thebruce]

[Chris K]

[neon snake]

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:

[thebruce]

[thebruce]

[tanner]

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

[thebruce]

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.

[tanner]

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

[Chris K]

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.