From Moonlight Sent Fri, Jun 11th 1999, 06:26
From Moonlight Sent Fri, Jun 11th 1999, 06:26
>Or, think of it this way. You can count integers 1-2-3-4... But how do >you "count" fractions? There's always an infinite number of fractions >between any two given fractions. Wrong! (sorry, a bit of math elitism), you can count rational numbers like this: (written to show the pattern) 0/1, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 1/4, -1/4, 2/3, -2/3, 3/2, -3/2, 4/1, -4/1, 1/5, -1/5,... Then, just pair them up with the natural numbers (1,2,3...) by counting them. Also, the set of algebraic numbers (all rationals and all of their rational powers , like -(457/3)^(241024/243) ) is _still_ countable, (just do the same thing with the rationals, start with zero, then +/-1 (almost like IDM content!), then go to all configurations (+/-)(a/b)^(c/d) where a+b+c+d=5, and make sure not to repeat, you wouldn't want to show that the cardinality of natural numbers is greater than that of all algebraic numbers. >The second, infinity of harmony and infinity of >tone--a double infinity, if you will. Well, you're only taking the cartesian product of countable sets, which is what you're doing when you go from integers to rationals or algebraic numbers, you end up with another "larger," yet still countable set. >Sam. High school math nerd. Adam. College math nerd. For more info, go to a math history book and look up Georg Cantor, he was all over this kind of stuff about 100 years ago. _________________________________ Adam Roesch / xxxxxx@xxxxxxxx.xxx University of Idaho / Moscow / ID / USA Visit my Fila Brazillia/Pork Recordings fan site: http://dogbert.augsburg.edu/~roesch/pork/ "Because success needs killing" TRICKY