PUZZLaTOMictM

The Number Family
 

The number family above falls from top to bottom or rises from bottom to top.
It falls as follows:
Octonions can produce
Quaternions can produce
Complex Numbers can produce
Real Numbers can produce
Rational numbers can produce
Integers can produce
Natural numbers with zero then
Natural numbers with no zero

Hovering above and meandering around and through all these simple quantity numbers are the infinities or Alephs of Cantor (uses the Hebrew symbol for N, N0 being the set of natural numbers number fields while N1 is thought to be the set of real numbers.  These infinite numbers go on forever and have a whole hierarchy of infinite numbers many infinities upon infinites larger. There are also new concepts (relatively speaking!!) such as algebraic numbers, and other things that can be thought of simply as kinds of numbers such as limits, series, Zeta function of Riemann, Prime types, the list is much larger and always growing. It is a big field but still a perfectly manageable thing and a beautiful machine of great power. Without this growing mathematical machine future and present technology would probably eventually come to a screeching halt!
Perhaps someday we will have a computer that we can pose mathematical questions to, and by asking just the right questions, very creative and inspired questions, the computer will go to work and out will come wonderful new mathematics from time to time!!

Copyright 2013 by Douglas A. Engel, Littleton, CO, U.S.A.