[OT] Re: [NBLUG/talk] OT division by zero OT

[Eric Eisenhart]

On Fri, May 30, 2003 at 09:45:20AM -0700, mrp wrote:
> Algebra with infinite (or more accurately, transfinite) numbers is a
> tricky subject George Gammow's book "One Two Three.. Infinity" has a
> good introduction.  (It's probably out of print now.. the copy I have
> was used 20 years ago when I got it.)

Another good book on the topic is "White Light" by Rudy Rucker.  (dig dig)
Originally 1980, reprinted 2001.  ISBN: 1-56858-198-X.  It's a, uhm, wacky
science fiction book that covers the topic pretty well.  For instance, the
main character goes into the infinite hotel and everybody moves up a room
number to make room for a new guest. He also wrote a non-fiction book on the
same basic topic at about the same time: "Infinity and The Mind".

(Rucker is a science fiction author and a mathematician)

> One of the weirdest aspects of infinity is that there are at least 3 different
> values of infinity, and some are bigger than others.  They're known as 
> Aleph_0, Aleph_1 and Aleph_2.  I'm not good enough at transfinite math
> to know if there are more than that. (Aleph is the first letter of the hebrew
> alphabet, and by that time the Latin and Greek alphabets had been used many
> times over for other "constants", so somebody started in on Hebrew.)

I believe that series goes to Aleph_Aleph.  Just don't ask me to comprehend
aleph_17 or anything, ok?

> Left as a execise for the reader:  Prove that the number of integers is
> the same as the number of even numbers (hint.. think of the hotel example.)

Take the set of all integers, multiply each one by 2.  You now have a set of
the same size, but they're all even numbers.

> (Extra credit: prove that the number of all rational numbers is the same
> as the number of integers.)

A rational number is the result of simple math on 2 integers.  int1/int2. 
Implying that the set is all_ints*all_ints, except that 1/2=2/4=3/6=4/8,
etc.  (and 1/3=2/6=3/9 and 1/4=2/8=3/12) -- follow that series and you have
all_rationals = (all_ints*all_ints)/all_ints = all_ints.

> (A level proof: prove that there are more real numbers than rational
> numbers.)

Okay...  wait for it...:
.
.
.
pi!
-- 
Eric Eisenhart
NBLUG Co-Founder & Vice-President Pro Tempore
The North Bay Linux Users Group
http://nblug.org/
eric at nblug.org, IRC: Freiheit at freenode, AIM: falschfreiheit, ICQ: 48217244