Addition, multiplication and exponentiation are
simply higher orders of the same function:
Each arrow starting from exponentiation
forms the higher operators
Numerical examples (the operator is right-associative):
We can grow larger numbers by simply
adding more arrows onto the expression
The idea is not to generate the largest number,
per se, but rather the largest growing function...
Many different styles: Conway's chained arrow,
hyper-geometric, and of course, Knuths Up-Arrow
We have primitive brains. For small numbers we can only think
spatially, 4 cows, 3 hens etc... Abstract numerical systems
allow us understand larger quantities.
If you build it... large numbers systems
were invented because of their necessity.
Grahams number is so big that even Knuths up arrow notation
is insufficient to contain it. It is the best known
upper-bound to the problem:
This is an upper bound to the problem. It has been proven that the lower bound solution is at least 11. The authors (modestly) state that there is some room for improvement.
You, at the conclusion of this talk
(originally invented by Euler)
Let be a collection of simple closed curves drawn in the plane. The collection is said to be an independent family if the region formed by the intersection of is nonempty, where each is either the interior or exterior of .
If, in addition, each such region is connected and there are only finitely many points of intersection between curves, then is a Venn diagram, or an -Venn diagram if we wish to emphasize the number of curves in the diagram.
In other words every subset built from a collection of n objects has to be represented only once.
Not Venn as is not represented.
Still known as an Euler diagram.
Constructing graphs allows different Venn diagrams of the same order to be compared. Diagrams are isomorphic if their graphs are isomorphic.
Must display n-fold symmetry.
Can be shown that these only exist when n is prime.