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[Shish]

The difference between 1 and I contains I-1 blocks, and the difference between 0 and 1 is I blocks? Something like that seems to make sense, I'm unsure of the terminology though :/

[Olorin]

[SilentAvenger]

neon snake, the number of REAL numbers between 0 and infinity and the number of REAL numbers between 0 and 1 is the same, but the number of INTEGERS from 1 to infinity is smaller than the number of REAL numbers between 0 and 1.

And you know whats evil? That function by the guy who's name I can only pronounce but not spell, starts with a D:

A function which is 1 for every irrational number, and 0 for every rational number. See, THAT is evil.

[thebruce]

edit: sorry, this is long... I just rambled on and on... I promise I won't keep going on about this, I don't want this to turn into a heated debate, which tends to happen when discussionary threads like this start getting long posts like this...

[SilentAvenger]

thebruce,
1. Mathematics today has many tools to deal with infinity. As a matter of fact the whole concept of derivatives and integrals rely on the fact that you can sum up and infinite number of parts, each with the width of zero. You need to understand that when we are dealing with an infinite random number, we are dealing with the abstract concept of one, and do not need to go calculate an actual RNG to prove the theory. Nobody went and checked that for every value, 1 = sin(X)^2 + cos(X)^2, but it has been proven to be true. Furthermore, nobody went and checked that there are an infinite number of primes, but it has been proven. This is the magic of math.

2. When you have a limit, x -> n, you may very well treat x as if it were n, if it was mathematically feasible. For example:
sin(x)/x lim x->0. You cannot directly enter 0 into this equation, but you can calculate the projected that this function will have at x = 0. Because of that, if lim x->1, if you are taking x out to infinity, its value is truly one.

[tanner]

thank you avenger -- you have said exactly what i wished to say if i had the patience and eloquence that you obviously have

the various proofs and the mathematicians who devised them are easy to find if a little difficult to understand without some knowledge of number theory

cantor and hilbert come immediately to mind.

[Sygerrik]

This is very enlightening, but does it help us find the cube? If there were an inifinite number of messages to read, this would be urgent and important. As it is, it's sort of deviating from the topic at hand. What are we going to do to find the missing messages? What use are we going to make of the messages we already have? These are the questions that face us.

[Chris K]

[thebruce]

[SilentAvenger]

I'm not going to ramble on and on for a long time here, as its not going to help anyone, and PPC starts in a few days anyways

I suggest we go and chat sometime thebruce, I'm hochberggSPLAThotmail.com for MSN and SAvenger7 for AIM, if you ever feel like doing so.

I'm just going to say that you seem to have misunderstood the meaning of "proof". When you prove something mathematically, you are saying that because of the basic set of axioms at the base of mathematics, the thing you say will ALWAYS happen.

For example, for 1/3, I can prove, via induction, that no matter what digit you calculate, the next one will always be 3. This means that yes, I know for a FACT that it never ends, and keeps repeating. I dont only know to the last calculated digit, but to any digit.

Furthermore, infinity is a very well defined concept in math, and there is no use arguing against that.

One before last, I think it has been proven that pi and e are irrational. This means that, yes, you know that no matter how many digits you take it out, it'll never repeat. This is a fact that is derived out of the axioms of math, and therefore is, and will be, (as long as the axioms exist) completely and undeniably true, and no matter how many digits you calculate, it'll never repeat. This is the meaning of proof.

Last, we (in this discussion), dont know enough about e and its properties to say if there truly is every finite-length number in e. We would have to ask people who deal with number theory all the time, and know their stuff to really get a good answer, and it might change between mathematicians

[tanner]

well said avenger -- trouble is some folks mistake evidence for proof and vice verci

the only absolutes are mathematical -- the hypothetical (if axioms are true ) and the proof resulting in a conclusion is all that exists with maths

mathematics is NOT an evidence based science --- its structure is proof

mathematical theorems are utturly different from scientific theories

IMO

[Olorin]

[OT]

[Chris K]

Oh dear.
That was the longest, wrongest post I've ever read.
How can you talk about the beauty of mathmatics and get it so wrong?

So we can never know that some numbers never end?
Rubbish.
I'll prove it now.

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

Do you agree that if a decimal number ends, it can be written as a fraction?
You would just move the decimal point to the end and divide by 100000000 or whatever.

For example: 4.5125 = 45125 / 10000

OK.
So now, if a numbers can't be written as a fraction then it never ends.

I'll prove it with SQRT( 2 ).

SQRT( 2 ) = a / b ............... let's say that it can be written as a fraction.
2 = a ^ 2 / b ^ 2
2b ^ 2 = a ^ 2 ............. This means that a^2 is even, so a must be even, so we can introduce a simpler number which is half of a.
2b ^ 2 = ( 2c ) ^ 2
2b ^ 2 = 4c ^ 2
b ^ 2 = 2c ^ 2 ............ Now we'll do the same with b.
4d ^ 2 = 2c ^ 2
2d ^ 2 = c^2
2 = ( c ^ 2 ) / ( d ^ 2 )
SQRT( 2 ) = c / d

We are now left with a simpler fraction than we started with. You can see that we could just do this again. and again and again.

Of course, you can't keep simplifying a fraction.

Therefore you can't write SQRT( 2 ) as a fraction, because if you could, you would be able to simplify it infinately.

[tanner]

[Nightmare Tony]

One time as a kid for giggles, I tried for making pi into a fraction. I forget the method exactly but it was that if the fraction result was lower or higher, you would double the upper lower term and adjust the upper term to be closer to pi. This kept going to the calculator's limit, so I ended up with a fraction to the 7's digit place on each term. If I remember, you start with 22/7 and go from there....