Quote (taekvideo @ May 12 2012 09:24pm)

There are the same number of real numbers as complex numbers.
The complex numbers are basically a plane, so |C| = |RxR| = |R| (finite cartesian products don't change cardinality)

The wording of "more numbers" gets very confusing when talking about infinite sets lol.

I still remember back when I first learned how to prove that .9999999(repeating) = 1... that's when infinities first started fascinating me.



I think it was alcohol not pot.

Quote (taekvideo @ May 12 2012 09:24pm)

There are the same number of real numbers as complex numbers.
The complex numbers are basically a plane, so |C| = |RxR| = |R| (finite cartesian products don't change cardinality)

The wording of "more numbers" gets very confusing when talking about infinite sets lol.

I still remember back when I first learned how to prove that .9999999(repeating) = 1... that's when infinities first started fascinating me.

Quote (carteblanche @ May 12 2012 10:30pm)

i thought the 1/3 + 1/3 + 1/3 proof was most convincing since it's simple. but the other algebra/calculus proofs work too.

have you heard of the infinity hotel? i found that interesting

Quote (carteblanche @ May 12 2012 09:30pm)

i thought the 1/3 + 1/3 + 1/3 proof was most convincing since it's simple. but the other algebra/calculus proofs work too.

have you heard of the infinity hotel? i found that interesting

Quote (Voyaging @ May 12 2012 09:25pm)

Well I would say to say there are an equal number is pretty much meaningless?

Like the old riddle:

Take two sets of numbers, one of every whole number:

1, 2, 3, 4, 5, 6...

one of every even number

2, 4, 6, 8, 10, 12...

Both continue to infinity, but there must be two of the first set for every one of the second set.

Obviously there are an infinite number in both sets. To say there is an equal number in two infinite sets is really misleading and really meaningless.

Quote (taekvideo @ May 12 2012 03:12pm)

There are a lot more numbers than just real and imaginary numbers.

Quote (taekvideo @ May 13 2012 10:49am)

The 1/3 + 1/3 + 1/3 proof isn't really a "proof"... since first you'd need to show that .333333333r = 1/3 which isn't any easier than showing .9999999999r = 1
This only points out the illogic of people who take that .333333333r = 1/3 for granted but object when you say .99999999r = 1

The most simple/elegant proof (imo) is just to multiply by 10 and subtract.  A = .99999999r, 10A = 9.999999999r, then 10A - A = 9 (everything after decimal cancels), thus 9A = 9 so A = 1.

Quote (taekvideo @ May 13 2012 10:49am)

If there are an equal number in 2 infinite sets then there are also twice as many in each as the other, or 17 times as many, or pi times as many, whatever you want.
But at the same time, if you actually compute the limit of a sequence which composes those infinite sets that you described, you'd end up with the 1/2, so there is a difference.
But.. not all infinite sets have an equal number of elements.  Some infinite sets have zero times as many elements as other infinite sets (ie the infinity is so much smaller it's negligible in comparison).
Consider the rationals (fractions of integers)... they're not only infinite they're dense (ie for any 2 numbers on the real line, there is a rational number between them).  Yet there are so few rationals compared to real numbers that with regard to integration you can simply ignore what the function does at the rational points and only integrate on the irrationals and you always get the same answer.
For example, the (Lebesgue) integral of the Dirichlet function (1 at rationals, 0 at irrationals)... without knowing that the rationals have the same number of terms as the natural numbers (ie they're countable) we'd have a much harder time computing it.  But since we do know that the rationals are countable, we can just ignore those points and integrate on the irrationals... and the integral is 0 on any set.
So.. it's not really meaningless to compare infinities, just a bit confusing at first.

Quote (Derkaderk @ May 13 2012 01:45pm)

are there? what are they?

Quote (Voyaging @ May 13 2012 02:10pm)

I don't understand. It's impossible to say there are more or less anything in a INFINITE set.

Quote (taekvideo @ May 13 2012 04:57pm)

The field of modern/abstract algebra deals with most of that.  This gives a decent history:
http://en.wikipedia.org/wiki/Hypercomplex_number
I don't particularly care for it though... possibly because of the pitifully boring professor who taught all those classes at my school
I specialize in real analysis (measure theory in particular).



I know it's hard :(  Took me awhile to come to terms with it myself lol... but it's true.  Cardinality of infinities is an important part of higher mathematics.  Just have to stop thinking of infinity as a number... especially since all infinities aren't created equally heh
The most basic distinction is between a countable infinity and an uncountable infinity (since countable infinity is the "smallest" infinity.. and anything bigger is uncountable).  If you can grasp that the rest becomes easier.
For instance using a diagonalization argument we can show that the interval [0,1] is uncountable.
First note that every real number in [0,1] can be uniquely defined by its infinite decimal representation (so 0 is .00000r, 1 is .99999r, and if for instance you have .8322, you would write it as .8321999999r, so every decimal is infinite).
Then suppose you have an infinite sequence which attempts to "count" the numbers in [0,1].  Then you can define a new number x as follows:
Let the 10th's place of x be any digit different than the 10th's place of the first number in the sequence... let the 100th's place be different than the 100th's place of the 2nd number in the sequence, and so on...
Now it's easy to see that x is missed by the sequence since for the Nth number in the sequence, the Nth digit is different, so no number in the sequence is the same as x no matter how far you go.
(In fact there are an uncountably infinite number of points missed, though that's a bit more complicated to show)
So no matter how you try to "count" the numbers between 0 and 1, you'll fail to get them all (hence the term uncountable).

On the other hand, the rationals are easy to count.  A picture is the best way to see this:
http://www.learner.org/courses/mathilluminated/images/units/3/table2.png
Note that this creates duplicates, but you can always skip those if you want.  Just following that path you'll eventually hit every rational number, and given any rational number you can easily acquire the index in the sequence for when it's counted, so the rationals are a countably infinite set.

Quote (Voyaging @ May 13 2012 05:36pm)

Countable infinites... does not compute... not possible.

Quote (bentherdonethat @ May 13 2012 07:11pm)

It actually is. If I remember right from my intro to mathematical proofs course, a set is countably infinite if there exists a bijection that can map that set onto the natural number line. It's called countably infinite because you can provide a methodology where you can progress forward and never, ever miss one. The natural numbers are countably infinite because if you start at 1 (the first natural number) and just add one repeatedly, you can eventually "count" them all.