nooooo! Spambot! Curse you for starting this! CURSE YOUUU!
So just to clarify on the 0.99... thing, in the real numbers, 0.99...=1. No ifs/ands/buts. But 0.99... isn't really... it's a decimal representation. It's weird. Decimals are weird, they're our representations of the reals, in the form of a sum.
If d(n) is a function that gives the nth digit, and 10^(-n) is... well, ten-to-the-power-negative-n, then the real number we're talking about is actually the infinite sum d(n)*10^(-n), summed over all n. But wait, you can't just sum over all n, because infinity is weird, so here's how math deals with it:
You can take the partial sums, so you say A(n)=[sum of d(i)*10^(-i) from i=1 to i=n]. Then, A(n) forms a sequence, and you can take the limit of this sequence as n goes to infinity, "rigorously" stated by using, "there exists" and "for all". (A(n)->L as n->infinity means that "for every e>0, there exists an integer N, such that n>N implies |A(n)-L|<e".
THAT'S the rigorous definition of a limit used, so there's a LOT going on behind the scenes. If you don't like it come up with a better definition. (maybe using
http://www.youtube.com/watch?v=wsOXvQn3JuE as inspiration?)
But even if you did come up with a 'better' definition of a limit, there is no real number infinitely close to 1 but not equal to 1. The vihart video I linked goes through some of the reasoning if you assume there is (0.999...999...999...123... heh), and maybe those properties could be made valid, but at the very least they probably break all the proofs of convergence and what-not, so no using e^x or sine and cosine until you've worked out all the details.
Then there's something called "nonstandard analysis"
http://en.wikipedia.org/wiki/Hyperreal_number which I know nothing about, so the idea does have some credence, they're just
not reals.
Quote: "But acording to that, the positive numbers are always twice as many as the whole numbers. Even though both are infinite. Whenever you add one more positive number (...1), you will add 2 more whole numbers (-...1). Dosn't that mean that infinity can infact be added to. infinity+1 != infinity even though both of them are infinity"
AAAAAAH.
So yeah, which infinity are we talking about, cardinal numbers, extended reals, or ordinal numbers, or extended complexes? Or hyperreals. xD
The easiest infinity is the extended reals. That's just R (the reals), plus two elements, positive infinity and negative infinity, and mathematicians cheat and just define them to have the property inf+1=inf, and a<inf for all real a, and -inf<a, etc,
leaving "inf-inf", "inf/inf" and "0*inf" undefined. This gives R a nice property called 'compactness'. Every infinite sequence in the extended reals has some subsequence that converges. And a bunch of other stuff but that's the easiest to explain.
Then the complex extended ones... I think it also makes the complex numbers compact. The cool thing here is that you just take C (complex) with ONE extra element, inf. No difference between -inf and +inf anymore. With THIS definition, you can define 1/0!! You just
define a/0=inf, when a is not 0, and it's handy and consistent and has nice properties and makes sense so mathematicians do that.
Then the ordinals... I know little about these, but they're basically infinite integers.
Then the cardinals which are the coolest to explain w/ sizes of sets. What you were saying about sizes of sets. You can define an operator "~" and "<~" and "~>". Where "~" is similar to equals, "<~" is similar to less-than-or-equal, and "~>" is similar to greater-than-or-equal, and it has all the same properties of >= = and <=. (definitions below). You find that these operators work perfectly well for finite sets. {a,b,c}~{1,2,3} , {1,2,3,4,5}~>{1,2,3} etc. They also work perfectly well on infinite sets! You find that N~Z (where N is the naturals, Z is the integers), Z~ZxZ (where ZxZ is the set of all ordered pairs of integers (a,b)), AND awesomely, N~Q (where Q is the set of all rationals), AND even more awesomely, N<~R. (you can prove that "N~R" is false, so that R is strictly greater than N)
That's because if you take {1,2,3,4,5,...} and {0,1,-1,2,-2,...}, well, the notion of 'size' is blurry, and the most sensible notion is to map one set on the other. 1 maps to 0, 2 maps to 1, 3 maps to -1, etc, and you cover all the integers, using just the naturals! You CAN'T do this with the reals though. If you try to make 1 map to 0.10010101000... and 2 maps to 0.001001... etc, you can show that there will always be real numbers not covered by any natural number. (for some real x, there exists no natural n, such that f(n)=x, where f is ANY function at all)
So the rationals, integers, and naturals are the same size, but are bigger than any finite set, and smaller than the reals.
You can prove that if A is a set, then the set of all subsets of A is strictly larger than A. 'the set of all subsets' is denote P(A) (for 'power set'), so using this you can determine that P(N)~R, and R<P(R), so that you can get infinities-upon-infinities.
If you have a problem w/ it, note that the process of a function mapping is very real. It's like you take one set that's a bucket of water, and dump it into another bucket, and you can never fill the other bucket. Clearly the other bucket is bigger in some sense!
(the definition involves "one-to-one" and "onto" functions. A~B if there exists a one-to-one onto function between them. A<~B if there exists a function from B
onto A, and A~>B if there exists a one-to-one function from B to A.
This has all the properties you'd expect. If A<~B and B<~C then A<~C, if A~B and B~C then A~C, etc.)
And this is all straightforward (ehhh, except, for example, the ordinals, you get to the point where taking 'the set of all ordinal numbers' leads to a contradiction, so you get into weird stuff with the continuum hypothesis and russel's paradox and the incompleteness theorem Q_Q. But the, "screw it, I don't want to go through a book on foundational set theory" approach is very straightforward.) All straightforward. If you really want to learn about it I suggest how to prove it by Velleman. Cool stuff. Makes you think better.
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