Post your favorite theorem and discuss it a little.
I don't have a favorite theorem necessarily, but here's one of my favorites: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
This is a completely counterintuitive notion of which I have virtually no understanding, but it has that wonderful mysterious and paradoxical quality that can only be achieved by mathematics. Uli, why don't you list one of your favorites?
Does anyone have any idea how the extra credit questions work? Can we turn them in at any time or are certain ones due by a specific date?
My personal favorite are Zeno's paradoxs.http://en.wikipedia.org/wiki/Zeno_paradox. Every one of them seems to be valid mathematically, but all of them defy common sense. The reason for this is that Zeno bases the the assumption that you cannot infinitely divide something. Eventually you get to the smallest unit of time or distance, and then you can't break it down anymore.
I dont have any theorems in mind but i always wanted to derive the general formula of a fibonnaci sequence based of basic math like not using the ideas of progression or any other general forms.If we could do it any way that is similar to Gauss.
I was thinking about Zeno's paradox and I think that someone(either my math teacher or Archimedes :)) said that it is possible to break down Zeno's problem into infinitely many decreasing terms and then summing the resulting geometric progression to get a finite number and therefore, solve the problem. This seems mathematically valid but, nevertheless, rather counter-intuitive.
TQU: My favorite theorem, well it's not a theorem, it's a principle: The Pigeonhole Principle we just learned in class. It has a funny name and I love pigeons. It states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. One interesting thing about this principle is that it explains what injection means to me in another math class I am taking. The pigeonhole principle can be extended to infinite sets by phrasing it in terms of cardinal numbers: if the cardinality of set A is greater than the cardinality of set B, then there is no injection from A to B.
It falls between a theory and a proof, but the Interesting Number paradox is my favourite. The theory says that "All [natural] numbers are interesting", that is have some property that sets them apart from all other numbers. The proof is through contradiction: If you had a set, S which contained all natural numbers in order and it had a subset U which contained all "uninteresting numbers" also in order. A paradox arises because the first number in I would be the smallest uninteresting number, which gives it some property that sets it apart from the rest, therefore it cannot be in U.
