[NBLUG/talk] OT division by zero OT

[mrp]

On Fri, May 30, 2003 at 09:06:59AM -0700, Cal Herrmann wrote:
> Apologies for another comment, but would this help the understanding?
> Consider the limit of 1/x as x increases toward infinity.  Would the
> value of 1/x approach zero?
> This is the inverse of the posed problem, but should have a consistent
> result. If 1/"infinity" is zero, then 1/0 should be infinity, from common
> algebra.

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.)

Consider these examples from the infinite hotel:

You're the night manager at the infinite hotel, with rooms number 1,2,3..
all the way ot infinity. All the rooms are full.  A new guest arrives.
How do you make room for him?

(Answer:  You have everybody move to the next room up. That frees up 
room 1, so you can put him there.)

A little while later, an inifinite bus pulls up, and disgourges an infinite
number of passengers.  How do you make room for all of them?

(Answer: You have everybody move to a room with number twice that of their
current room.  This frees up all the odd rooms for the new guests.)

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.)

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.)

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

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

  -- Mitch