#3562. Infinite infinities as factors of integers

Mathematics, level: Senior
Posted Mon Sep 12 20:35:16 PDT 2005 by Randall Hudson (Mathtutor234@aol.com).
Springdale, USA
Activity Time: 60 min
Concepts Taught: Math structure

This is a lesson plan to strech our conceptions of numbers. Everyone 'knows' that finite numbers do not have infinity as a factor. But the following geometric progression shows otherwise.

All integers except for -1, 0, and 1 have an infinite number of infinities as factors of themselves.

n=(n+1)(Sn) where Sn=1/n to the zero power -1/n to the first power+1/n to the second power-1/n to the third power...+1/n to the infinity power. So here is one factor of infinity in integers.

Now we can multiply the first side of the above equation by (n+1) and multiply the second side by the equivalent expression of (n+2)(Sn+1) and obtain:

n=(n+2)(Sn)(Sn+1)
We can continue and by this method find the generalized equation of:

n=(n+z)(Sn)(Sn+1)(Sn+2)...(Sz-1)

and we have an infinite number of infinite factors.