Your favorite bits of Math

[WillyBrandt]

My favorite is Goedel's proof that every sufficiently rich formal system contains true statements that are not provable within that system--the part I like is how he uses ad hoc number theory to prove his point.

Also like the myriad proofs reducing computational systems to Turing machines.

And Russell's paradox: Let A be the set containing all sets that are not members of themselves; is A a member of itself. Or, if the barber of Seville shaves every man of Seville who does not shave himself--does the barber shave himself?

On edit: I remember thinking Dedekind cuts were interesting, but I kind of forgot how all that worked.

[phaseolus]

1. Ah, here it is...

"...he defined the Dedekind cut (German: Schnitt), a new idea to represent the real numbers as a divisions of the rational numbers. An irrational number is a cut separating all rational numbers into two classes, an upper and lower class (set) For example, the square root of 2 is a cut putting the negative numbers and the numbers with square smaller than 2 into the lower, and the positive numbers with square greater than 2 into the higher class. This is now one of the standard definitions for the real numbers."
http://en.wikipedia.org/wiki/Dedekind

A further link, a little more formal language --
http://en.wikipedia.org/wiki/Dedekind_cut

Whoa -- what's this?? I never heard of surreal numbers before...
"The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger or smaller than any real number. They also include "infinitesimal" numbers that are closer to zero than any real number, and each real number is surrounded by surreals that are closer to it than any real number. In this, the surreals are similar to the hyperreal numbers, but their construction is very different and the class of surreals is larger and contains the hyperreals as a subset. Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.

"Surreal numbers were first proposed by John Conway and later detailed by Donald Knuth in his 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is actually a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea has been first presented in a work of fiction."
http://en.wikipedia.org/wiki/Surreal_number

You know, I can't really do much math beyond what they taught me in college, (though I try to teach myself abstract algebra once in a while to keep the brain cells happy,) but I can follow links on wikipedia and read *about* math for hours...

[WillyBrandt]

2. Read about math! Yes!

That's what I like to do. Actually doing the crap is hard as hell, and takes too much devotion. But watching a Nova episode or something about math--I could do that all day! :)

[phaseolus]

3. really basic group theory seems pretty easy and fun

compared to learning plain ol' algebra in high school. I'm not sure what practical use it is, but you might have fun checking it out...